ABCF->ab-angle a

Percentage Accurate: 18.6% → 60.7%

Time: 55.9s

Alternatives: 13

Speedup: 6.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Initial Program: 18.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Alternative 1: 60.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot t\_0\\ t_2 := \mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)\\ t_3 := -t\_0\\ t_4 := \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_4\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_4 - {B\_m}^{2}}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;-{\left(\sqrt{\sqrt{2 \cdot \left(F \cdot \frac{C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{t\_2}\right)}}\right)}^{2}\\ \mathbf{elif}\;t\_5 \leq -5 \cdot 10^{-217}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_1} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{t\_3}\\ \mathbf{elif}\;t\_5 \leq 10^{+134}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(4 \cdot C - \frac{{B\_m}^{2}}{A}\right)}}{t\_3}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{F \cdot \frac{t\_2}{C}}}{A}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* F t_0))
        (t_2 (fma -4.0 (* A C) (pow B_m 2.0)))
        (t_3 (- t_0))
        (t_4 (* (* 4.0 A) C))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_4) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_4 (pow B_m 2.0)))))
   (if (<= t_5 (- INFINITY))
     (-
      (pow
       (sqrt (sqrt (* 2.0 (* F (/ (+ C (+ A (hypot B_m (- A C)))) t_2)))))
       2.0))
     (if (<= t_5 -5e-217)
       (/ (* (sqrt (* 2.0 t_1)) (sqrt (+ A (+ C (hypot (- A C) B_m))))) t_3)
       (if (<= t_5 1e+134)
         (/ (sqrt (* t_1 (- (* 4.0 C) (/ (pow B_m 2.0) A)))) t_3)
         (if (<= t_5 INFINITY)
           (* 0.5 (/ (sqrt (* F (/ t_2 C))) A))
           (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = F * t_0;
	double t_2 = fma(-4.0, (A * C), pow(B_m, 2.0));
	double t_3 = -t_0;
	double t_4 = (4.0 * A) * C;
	double t_5 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_4) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_4 - pow(B_m, 2.0));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = -pow(sqrt(sqrt((2.0 * (F * ((C + (A + hypot(B_m, (A - C)))) / t_2))))), 2.0);
	} else if (t_5 <= -5e-217) {
		tmp = (sqrt((2.0 * t_1)) * sqrt((A + (C + hypot((A - C), B_m))))) / t_3;
	} else if (t_5 <= 1e+134) {
		tmp = sqrt((t_1 * ((4.0 * C) - (pow(B_m, 2.0) / A)))) / t_3;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = 0.5 * (sqrt((F * (t_2 / C))) / A);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(F * t_0)
	t_2 = fma(-4.0, Float64(A * C), (B_m ^ 2.0))
	t_3 = Float64(-t_0)
	t_4 = Float64(Float64(4.0 * A) * C)
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_4) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_4 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(-(sqrt(sqrt(Float64(2.0 * Float64(F * Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) / t_2))))) ^ 2.0));
	elseif (t_5 <= -5e-217)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_1)) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / t_3);
	elseif (t_5 <= 1e+134)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(4.0 * C) - Float64((B_m ^ 2.0) / A)))) / t_3);
	elseif (t_5 <= Inf)
		tmp = Float64(0.5 * Float64(sqrt(Float64(F * Float64(t_2 / C))) / A));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$0)}, Block[{t$95$4 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$4), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$4 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], (-N[Power[N[Sqrt[N[Sqrt[N[(2.0 * N[(F * N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), If[LessEqual[t$95$5, -5e-217], N[(N[(N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$5, 1e+134], N[(N[Sqrt[N[(t$95$1 * N[(N[(4.0 * C), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(0.5 * N[(N[Sqrt[N[(F * N[(t$95$2 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0 25.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

   5. Simplified 61.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{F \cdot \frac{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 61.3%

         \[\leadsto -\color{blue}{\sqrt{\sqrt{2} \cdot \sqrt{F \cdot \frac{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}}} \cdot \sqrt{\sqrt{2} \cdot \sqrt{F \cdot \frac{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}}}} \]

      2. pow2 61.3%

         \[\leadsto -\color{blue}{{\left(\sqrt{\sqrt{2} \cdot \sqrt{F \cdot \frac{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}}}\right)}^{2}} \]

      3. sqrt-unprod 61.5%

         \[\leadsto -{\left(\sqrt{\color{blue}{\sqrt{2 \cdot \left(F \cdot \frac{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}\right)}}}\right)}^{2} \]

      4. associate-+l+ 63.5%

         \[\leadsto -{\left(\sqrt{\sqrt{2 \cdot \left(F \cdot \frac{\color{blue}{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}\right)}}\right)}^{2} \]

   7. Applied egg-rr 63.5%

      \[\leadsto -\color{blue}{{\left(\sqrt{\sqrt{2 \cdot \left(F \cdot \frac{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}\right)}}\right)}^{2}} \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000002e-217

   1. Initial program 95.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 95.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 95.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. associate-+r+ 95.3%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. hypot-undefine 95.3%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \color{blue}{\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. unpow2 95.3%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2}} + \left(A - C\right) \cdot \left(A - C\right)}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 95.3%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + \color{blue}{{\left(A - C\right)}^{2}}}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. +-commutative 95.3%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{\left(A - C\right)}^{2} + {B}^{2}}}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. sqrt-prod 97.3%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      8. *-commutative 97.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot 2} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      9. associate-+l+ 97.4%

         \[\leadsto \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2} \cdot \sqrt{\color{blue}{A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Applied egg-rr 97.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if -5.0000000000000002e-217 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 9.99999999999999921e133

   1. Initial program 20.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 43.5%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 43.5%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(\color{blue}{\frac{-1 \cdot {B}^{2}}{A}} + 4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Applied egg-rr 43.5%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(\color{blue}{\frac{-1 \cdot {B}^{2}}{A}} + 4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 43.5%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(\frac{\color{blue}{-{B}^{2}}}{A} + 4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. distribute-neg-frac 43.5%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(\color{blue}{\left(-\frac{{B}^{2}}{A}\right)} + 4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. distribute-neg-frac2 43.5%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{-A}} + 4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   9. Simplified 43.5%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{-A}} + 4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 9.99999999999999921e133 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 37.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 15.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 15.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 15.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Taylor expanded in B around 0 15.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 15.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   10. Simplified 15.3%

       \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   11. Taylor expanded in F around 0 32.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{A} \cdot \sqrt{\frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}{C}}\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 32.8%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}{C}}}{A}} \]

       2. *-lft-identity 32.8%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}{C}}}}{A} \]

       3. associate-/l* 44.0%

          \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{F \cdot \frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{C}}}}{A} \]

       4. fma-define 44.0%

          \[\leadsto 0.5 \cdot \frac{\sqrt{F \cdot \frac{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}{C}}}{A} \]

   13. Simplified 44.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{F \cdot \frac{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}{C}}}{A}} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 18.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 18.2%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 18.2%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 18.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sqrt-div 22.4%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]

   7. Applied egg-rr 22.4%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 22.3%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F}}{\sqrt{B}}} \]

      2. pow1/2 22.3%

         \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F}}{\sqrt{B}} \]

      3. pow1/2 22.3%

         \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{F}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 22.4%

         \[\leadsto -\frac{\color{blue}{{\left(2 \cdot F\right)}^{0.5}}}{\sqrt{B}} \]

   9. Applied egg-rr 22.4%

      \[\leadsto -\color{blue}{\frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 22.4%

          \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}} \]

       2. *-commutative 22.4%

          \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}} \]

   11. Simplified 22.4%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-{\left(\sqrt{\sqrt{2 \cdot \left(F \cdot \frac{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}}\right)}^{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-217}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+134}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C - \frac{{B}^{2}}{A}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{F \cdot \frac{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}{C}}}{A}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 1.15 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{t\_0 \cdot \left(F \cdot \left(4 \cdot C\right)\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;B\_m \leq 4.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 1.15e-168)
     (* (sqrt (* t_0 (* F (* 4.0 C)))) (/ -1.0 t_0))
     (if (<= B_m 4.7e+109)
       (/
        (* (sqrt (* 2.0 (* F t_0))) (sqrt (+ A (+ C (hypot (- A C) B_m)))))
        (- t_0))
       (- (* (sqrt (* 2.0 F)) (pow B_m -0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 1.15e-168) {
		tmp = sqrt((t_0 * (F * (4.0 * C)))) * (-1.0 / t_0);
	} else if (B_m <= 4.7e+109) {
		tmp = (sqrt((2.0 * (F * t_0))) * sqrt((A + (C + hypot((A - C), B_m))))) / -t_0;
	} else {
		tmp = -(sqrt((2.0 * F)) * pow(B_m, -0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 1.15e-168)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * Float64(4.0 * C)))) * Float64(-1.0 / t_0));
	elseif (B_m <= 4.7e+109)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / Float64(-t_0));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) * (B_m ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.15e-168], N[(N[Sqrt[N[(t$95$0 * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.7e+109], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.14999999999999993e-168

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 21.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   8. Step-by-step derivation

      1. div-inv 20.7%

         \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

      2. associate-*l* 20.8%

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   9. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   if 1.14999999999999993e-168 < B < 4.69999999999999998e109

   1. Initial program 23.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 31.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 31.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. associate-+r+ 30.6%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. hypot-undefine 23.2%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \color{blue}{\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. unpow2 23.2%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2}} + \left(A - C\right) \cdot \left(A - C\right)}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 23.2%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + \color{blue}{{\left(A - C\right)}^{2}}}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. +-commutative 23.2%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{\left(A - C\right)}^{2} + {B}^{2}}}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. sqrt-prod 31.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      8. *-commutative 31.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot 2} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      9. associate-+l+ 32.0%

         \[\leadsto \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2} \cdot \sqrt{\color{blue}{A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Applied egg-rr 44.5%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 4.69999999999999998e109 < B

   1. Initial program 7.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 53.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 53.9%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 53.9%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 53.9%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sqrt-div 71.6%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]

   7. Applied egg-rr 71.6%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 71.5%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F}}{\sqrt{B}}} \]

      2. pow1/2 71.5%

         \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F}}{\sqrt{B}} \]

      3. pow1/2 71.5%

         \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{F}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 71.8%

         \[\leadsto -\frac{\color{blue}{{\left(2 \cdot F\right)}^{0.5}}}{\sqrt{B}} \]

   9. Applied egg-rr 71.8%

      \[\leadsto -\color{blue}{\frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 71.8%

          \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}} \]

       2. *-commutative 71.8%

          \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}} \]

   11. Simplified 71.8%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   12. Step-by-step derivation

       1. div-inv 71.7%

          \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot \frac{1}{\sqrt{B}}} \]

       2. pow1/2 71.7%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \frac{1}{\color{blue}{{B}^{0.5}}} \]

       3. pow-flip 71.7%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \color{blue}{{B}^{\left(-0.5\right)}} \]

       4. metadata-eval 71.7%

          \[\leadsto -\sqrt{F \cdot 2} \cdot {B}^{\color{blue}{-0.5}} \]

   13. Applied egg-rr 71.7%

       \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {B}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.15 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 1.55 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{t\_0 \cdot \left(F \cdot \left(4 \cdot C\right)\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 1.55e-168)
     (* (sqrt (* t_0 (* F (* 4.0 C)))) (/ -1.0 t_0))
     (if (<= B_m 1.85e+109)
       (*
        (sqrt (* 2.0 (* F t_0)))
        (/ (sqrt (+ (+ A C) (hypot (- A C) B_m))) (- t_0)))
       (- (* (sqrt (* 2.0 F)) (pow B_m -0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 1.55e-168) {
		tmp = sqrt((t_0 * (F * (4.0 * C)))) * (-1.0 / t_0);
	} else if (B_m <= 1.85e+109) {
		tmp = sqrt((2.0 * (F * t_0))) * (sqrt(((A + C) + hypot((A - C), B_m))) / -t_0);
	} else {
		tmp = -(sqrt((2.0 * F)) * pow(B_m, -0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 1.55e-168)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * Float64(4.0 * C)))) * Float64(-1.0 / t_0));
	elseif (B_m <= 1.85e+109)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) / Float64(-t_0)));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) * (B_m ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.55e-168], N[(N[Sqrt[N[(t$95$0 * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.85e+109], N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.55e-168

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 21.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   8. Step-by-step derivation

      1. div-inv 20.7%

         \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

      2. associate-*l* 20.8%

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   9. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   if 1.55e-168 < B < 1.8500000000000001e109

   1. Initial program 23.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 31.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 31.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. associate-+r+ 30.6%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. hypot-undefine 23.2%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \color{blue}{\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. unpow2 23.2%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2}} + \left(A - C\right) \cdot \left(A - C\right)}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 23.2%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + \color{blue}{{\left(A - C\right)}^{2}}}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. +-commutative 23.2%

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{\left(A - C\right)}^{2} + {B}^{2}}}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. sqrt-prod 31.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 2} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      8. *-commutative 31.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot 2} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      9. associate-+l+ 32.0%

         \[\leadsto \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2} \cdot \sqrt{\color{blue}{A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Applied egg-rr 44.5%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 44.6%

         \[\leadsto \color{blue}{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2} \cdot \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

      2. *-commutative 44.6%

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)} \cdot 2} \cdot \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. *-commutative 44.6%

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)}} \cdot \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 44.6%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}} \cdot \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. associate-+r+ 43.5%

         \[\leadsto \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \frac{\sqrt{\color{blue}{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Applied egg-rr 43.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   if 1.8500000000000001e109 < B

   1. Initial program 7.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 53.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 53.9%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 53.9%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 53.9%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sqrt-div 71.6%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]

   7. Applied egg-rr 71.6%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 71.5%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F}}{\sqrt{B}}} \]

      2. pow1/2 71.5%

         \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F}}{\sqrt{B}} \]

      3. pow1/2 71.5%

         \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{F}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 71.8%

         \[\leadsto -\frac{\color{blue}{{\left(2 \cdot F\right)}^{0.5}}}{\sqrt{B}} \]

   9. Applied egg-rr 71.8%

      \[\leadsto -\color{blue}{\frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 71.8%

          \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}} \]

       2. *-commutative 71.8%

          \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}} \]

   11. Simplified 71.8%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   12. Step-by-step derivation

       1. div-inv 71.7%

          \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot \frac{1}{\sqrt{B}}} \]

       2. pow1/2 71.7%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \frac{1}{\color{blue}{{B}^{0.5}}} \]

       3. pow-flip 71.7%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \color{blue}{{B}^{\left(-0.5\right)}} \]

       4. metadata-eval 71.7%

          \[\leadsto -\sqrt{F \cdot 2} \cdot {B}^{\color{blue}{-0.5}} \]

   13. Applied egg-rr 71.7%

       \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {B}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.55 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{F \cdot \frac{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}{C}}}{A}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+143}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+57)
   (* 0.5 (/ (sqrt (* F (/ (fma -4.0 (* A C) (pow B_m 2.0)) C))) A))
   (if (<= (pow B_m 2.0) 1e+143)
     (- (sqrt (/ F (- A))))
     (- (* (sqrt (* 2.0 F)) (pow B_m -0.5))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e+57) {
		tmp = 0.5 * (sqrt((F * (fma(-4.0, (A * C), pow(B_m, 2.0)) / C))) / A);
	} else if (pow(B_m, 2.0) <= 1e+143) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -(sqrt((2.0 * F)) * pow(B_m, -0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+57)
		tmp = Float64(0.5 * Float64(sqrt(Float64(F * Float64(fma(-4.0, Float64(A * C), (B_m ^ 2.0)) / C))) / A));
	elseif ((B_m ^ 2.0) <= 1e+143)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) * (B_m ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+57], N[(0.5 * N[(N[Sqrt[N[(F * N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+143], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999972e57

   1. Initial program 19.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 27.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 29.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 29.0%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 29.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Taylor expanded in B around 0 26.2%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 26.2%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   10. Simplified 26.2%

       \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   11. Taylor expanded in F around 0 22.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{A} \cdot \sqrt{\frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}{C}}\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 22.0%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}{C}}}{A}} \]

       2. *-lft-identity 22.0%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}{C}}}}{A} \]

       3. associate-/l* 27.3%

          \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{F \cdot \frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{C}}}}{A} \]

       4. fma-define 27.3%

          \[\leadsto 0.5 \cdot \frac{\sqrt{F \cdot \frac{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}{C}}}{A} \]

   13. Simplified 27.3%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{F \cdot \frac{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}{C}}}{A}} \]

   if 4.99999999999999972e57 < (pow.f64 B #s(literal 2 binary64)) < 1e143

   1. Initial program 32.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 33.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 7.1%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around 0 13.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

   7. Taylor expanded in B around 0 13.6%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 13.6%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   9. Simplified 13.6%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 1e143 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 29.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 29.5%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 29.5%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 29.5%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sqrt-div 35.6%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]

   7. Applied egg-rr 35.6%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 35.6%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F}}{\sqrt{B}}} \]

      2. pow1/2 35.6%

         \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F}}{\sqrt{B}} \]

      3. pow1/2 35.6%

         \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{F}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 35.7%

         \[\leadsto -\frac{\color{blue}{{\left(2 \cdot F\right)}^{0.5}}}{\sqrt{B}} \]

   9. Applied egg-rr 35.7%

      \[\leadsto -\color{blue}{\frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 35.7%

          \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}} \]

       2. *-commutative 35.7%

          \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}} \]

   11. Simplified 35.7%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   12. Step-by-step derivation

       1. div-inv 35.7%

          \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot \frac{1}{\sqrt{B}}} \]

       2. pow1/2 35.7%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \frac{1}{\color{blue}{{B}^{0.5}}} \]

       3. pow-flip 35.7%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \color{blue}{{B}^{\left(-0.5\right)}} \]

       4. metadata-eval 35.7%

          \[\leadsto -\sqrt{F \cdot 2} \cdot {B}^{\color{blue}{-0.5}} \]

   13. Applied egg-rr 35.7%

       \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {B}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{F \cdot \frac{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}{C}}}{A}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+143}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_0 \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 1.85 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 1.55e-142)
     (/ -1.0 (/ t_0 (sqrt (* t_0 (* F (* 4.0 C))))))
     (if (<= B_m 1.85e+72)
       (- (sqrt (/ F (- A))))
       (- (* (sqrt (* 2.0 F)) (pow B_m -0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 1.55e-142) {
		tmp = -1.0 / (t_0 / sqrt((t_0 * (F * (4.0 * C)))));
	} else if (B_m <= 1.85e+72) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -(sqrt((2.0 * F)) * pow(B_m, -0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 1.55e-142)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(t_0 * Float64(F * Float64(4.0 * C))))));
	elseif (B_m <= 1.85e+72)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) * (B_m ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.55e-142], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(t$95$0 * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.85e+72], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.55e-142

   1. Initial program 17.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 21.4%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 21.4%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 21.4%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   8. Step-by-step derivation

      1. clear-num 21.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}}} \]

      2. inv-pow 21.3%

         \[\leadsto \color{blue}{{\left(\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}\right)}^{-1}} \]

      3. associate-*l* 21.4%

         \[\leadsto {\left(\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}}\right)}^{-1} \]

   9. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{{\left(\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 21.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}}} \]

   11. Simplified 21.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}}} \]

   if 1.55e-142 < B < 1.8500000000000001e72

   1. Initial program 23.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 17.6%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around 0 14.4%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

   7. Taylor expanded in B around 0 14.2%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 14.2%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   9. Simplified 14.2%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 1.8500000000000001e72 < B

   1. Initial program 10.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 54.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sqrt-div 69.2%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]

   7. Applied egg-rr 69.2%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 69.1%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F}}{\sqrt{B}}} \]

      2. pow1/2 69.1%

         \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F}}{\sqrt{B}} \]

      3. pow1/2 69.1%

         \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{F}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 69.4%

         \[\leadsto -\frac{\color{blue}{{\left(2 \cdot F\right)}^{0.5}}}{\sqrt{B}} \]

   9. Applied egg-rr 69.4%

      \[\leadsto -\color{blue}{\frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 69.4%

          \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}} \]

       2. *-commutative 69.4%

          \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}} \]

   11. Simplified 69.4%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   12. Step-by-step derivation

       1. div-inv 69.3%

          \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot \frac{1}{\sqrt{B}}} \]

       2. pow1/2 69.3%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \frac{1}{\color{blue}{{B}^{0.5}}} \]

       3. pow-flip 69.3%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \color{blue}{{B}^{\left(-0.5\right)}} \]

       4. metadata-eval 69.3%

          \[\leadsto -\sqrt{F \cdot 2} \cdot {B}^{\color{blue}{-0.5}} \]

   13. Applied egg-rr 69.3%

       \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {B}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 7.2 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2e-168)
   (/
    (sqrt (* (* F (fma B_m B_m (* A (* C -4.0)))) (* 4.0 C)))
    (* (* 4.0 A) C))
   (if (<= B_m 7.2e+72)
     (- (sqrt (/ F (- A))))
     (- (* (sqrt (* 2.0 F)) (pow B_m -0.5))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2e-168) {
		tmp = sqrt(((F * fma(B_m, B_m, (A * (C * -4.0)))) * (4.0 * C))) / ((4.0 * A) * C);
	} else if (B_m <= 7.2e+72) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -(sqrt((2.0 * F)) * pow(B_m, -0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2e-168)
		tmp = Float64(sqrt(Float64(Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * Float64(4.0 * C))) / Float64(Float64(4.0 * A) * C));
	elseif (B_m <= 7.2e+72)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) * (B_m ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2e-168], N[(N[Sqrt[N[(N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.2e+72], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if B < 2.0000000000000001e-168

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 21.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Taylor expanded in B around 0 19.8%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 19.8%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   10. Simplified 19.8%

       \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   if 2.0000000000000001e-168 < B < 7.20000000000000069e72

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 18.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around 0 15.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

   7. Taylor expanded in B around 0 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 15.1%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   9. Simplified 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 7.20000000000000069e72 < B

   1. Initial program 10.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 54.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sqrt-div 69.2%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]

   7. Applied egg-rr 69.2%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 69.1%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F}}{\sqrt{B}}} \]

      2. pow1/2 69.1%

         \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F}}{\sqrt{B}} \]

      3. pow1/2 69.1%

         \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{F}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 69.4%

         \[\leadsto -\frac{\color{blue}{{\left(2 \cdot F\right)}^{0.5}}}{\sqrt{B}} \]

   9. Applied egg-rr 69.4%

      \[\leadsto -\color{blue}{\frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 69.4%

          \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}} \]

       2. *-commutative 69.4%

          \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}} \]

   11. Simplified 69.4%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   12. Step-by-step derivation

       1. div-inv 69.3%

          \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot \frac{1}{\sqrt{B}}} \]

       2. pow1/2 69.3%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \frac{1}{\color{blue}{{B}^{0.5}}} \]

       3. pow-flip 69.3%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \color{blue}{{B}^{\left(-0.5\right)}} \]

       4. metadata-eval 69.3%

          \[\leadsto -\sqrt{F \cdot 2} \cdot {B}^{\color{blue}{-0.5}} \]

   13. Applied egg-rr 69.3%

       \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {B}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 4.3 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.2e-168)
   (/ (sqrt (* (* 4.0 C) (* (* A -4.0) (* C F)))) (* (* 4.0 A) C))
   (if (<= B_m 4.3e+72)
     (- (sqrt (/ F (- A))))
     (- (* (sqrt (* 2.0 F)) (pow B_m -0.5))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.2e-168) {
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 4.3e+72) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -(sqrt((2.0 * F)) * pow(B_m, -0.5));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 3.2d-168) then
        tmp = sqrt(((4.0d0 * c) * ((a * (-4.0d0)) * (c * f)))) / ((4.0d0 * a) * c)
    else if (b_m <= 4.3d+72) then
        tmp = -sqrt((f / -a))
    else
        tmp = -(sqrt((2.0d0 * f)) * (b_m ** (-0.5d0)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.2e-168) {
		tmp = Math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 4.3e+72) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = -(Math.sqrt((2.0 * F)) * Math.pow(B_m, -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.2e-168:
		tmp = math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C)
	elif B_m <= 4.3e+72:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = -(math.sqrt((2.0 * F)) * math.pow(B_m, -0.5))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.2e-168)
		tmp = Float64(sqrt(Float64(Float64(4.0 * C) * Float64(Float64(A * -4.0) * Float64(C * F)))) / Float64(Float64(4.0 * A) * C));
	elseif (B_m <= 4.3e+72)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) * (B_m ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.2e-168)
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	elseif (B_m <= 4.3e+72)
		tmp = -sqrt((F / -A));
	else
		tmp = -(sqrt((2.0 * F)) * (B_m ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.2e-168], N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(N[(A * -4.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.3e+72], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if B < 3.20000000000000006e-168

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 21.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Taylor expanded in B around 0 19.8%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 19.8%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   10. Simplified 19.8%

       \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   11. Taylor expanded in B around 0 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]
   12. Step-by-step derivation

       1. associate-*r* 19.6%

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   13. Simplified 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   if 3.20000000000000006e-168 < B < 4.3000000000000001e72

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 18.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around 0 15.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

   7. Taylor expanded in B around 0 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 15.1%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   9. Simplified 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 4.3000000000000001e72 < B

   1. Initial program 10.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 54.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sqrt-div 69.2%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]

   7. Applied egg-rr 69.2%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 69.1%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F}}{\sqrt{B}}} \]

      2. pow1/2 69.1%

         \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F}}{\sqrt{B}} \]

      3. pow1/2 69.1%

         \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{F}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 69.4%

         \[\leadsto -\frac{\color{blue}{{\left(2 \cdot F\right)}^{0.5}}}{\sqrt{B}} \]

   9. Applied egg-rr 69.4%

      \[\leadsto -\color{blue}{\frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 69.4%

          \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}} \]

       2. *-commutative 69.4%

          \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}} \]

   11. Simplified 69.4%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   12. Step-by-step derivation

       1. div-inv 69.3%

          \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot \frac{1}{\sqrt{B}}} \]

       2. pow1/2 69.3%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \frac{1}{\color{blue}{{B}^{0.5}}} \]

       3. pow-flip 69.3%

          \[\leadsto -\sqrt{F \cdot 2} \cdot \color{blue}{{B}^{\left(-0.5\right)}} \]

       4. metadata-eval 69.3%

          \[\leadsto -\sqrt{F \cdot 2} \cdot {B}^{\color{blue}{-0.5}} \]

   13. Applied egg-rr 69.3%

       \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {B}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.65 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 4.4 \cdot 10^{+71}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.65e-168)
   (/ (sqrt (* (* 4.0 C) (* (* A -4.0) (* C F)))) (* (* 4.0 A) C))
   (if (<= B_m 4.4e+71)
     (- (sqrt (/ F (- A))))
     (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.65e-168) {
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 4.4e+71) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.65d-168) then
        tmp = sqrt(((4.0d0 * c) * ((a * (-4.0d0)) * (c * f)))) / ((4.0d0 * a) * c)
    else if (b_m <= 4.4d+71) then
        tmp = -sqrt((f / -a))
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.65e-168) {
		tmp = Math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 4.4e+71) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.65e-168:
		tmp = math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C)
	elif B_m <= 4.4e+71:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.65e-168)
		tmp = Float64(sqrt(Float64(Float64(4.0 * C) * Float64(Float64(A * -4.0) * Float64(C * F)))) / Float64(Float64(4.0 * A) * C));
	elseif (B_m <= 4.4e+71)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.65e-168)
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	elseif (B_m <= 4.4e+71)
		tmp = -sqrt((F / -A));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.65e-168], N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(N[(A * -4.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.4e+71], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.6500000000000001e-168

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 21.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Taylor expanded in B around 0 19.8%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 19.8%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   10. Simplified 19.8%

       \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   11. Taylor expanded in B around 0 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]
   12. Step-by-step derivation

       1. associate-*r* 19.6%

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   13. Simplified 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   if 1.6500000000000001e-168 < B < 4.39999999999999989e71

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 18.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around 0 15.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

   7. Taylor expanded in B around 0 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 15.1%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   9. Simplified 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 4.39999999999999989e71 < B

   1. Initial program 10.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 54.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sqrt-div 69.2%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]

   7. Applied egg-rr 69.2%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 69.1%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F}}{\sqrt{B}}} \]

      2. pow1/2 69.1%

         \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F}}{\sqrt{B}} \]

      3. pow1/2 69.1%

         \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{F}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 69.4%

         \[\leadsto -\frac{\color{blue}{{\left(2 \cdot F\right)}^{0.5}}}{\sqrt{B}} \]

   9. Applied egg-rr 69.4%

      \[\leadsto -\color{blue}{\frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 69.4%

          \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}} \]

       2. *-commutative 69.4%

          \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}} \]

   11. Simplified 69.4%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.65 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{+71}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.45 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 4.3 \cdot 10^{+71}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.45e-168)
   (/ (sqrt (* (* 4.0 C) (* (* A -4.0) (* C F)))) (* (* 4.0 A) C))
   (if (<= B_m 4.3e+71)
     (- (sqrt (/ F (- A))))
     (* (sqrt (/ 2.0 B_m)) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.45e-168) {
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 4.3e+71) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 3.45d-168) then
        tmp = sqrt(((4.0d0 * c) * ((a * (-4.0d0)) * (c * f)))) / ((4.0d0 * a) * c)
    else if (b_m <= 4.3d+71) then
        tmp = -sqrt((f / -a))
    else
        tmp = sqrt((2.0d0 / b_m)) * -sqrt(f)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.45e-168) {
		tmp = Math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 4.3e+71) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = Math.sqrt((2.0 / B_m)) * -Math.sqrt(F);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.45e-168:
		tmp = math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C)
	elif B_m <= 4.3e+71:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = math.sqrt((2.0 / B_m)) * -math.sqrt(F)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.45e-168)
		tmp = Float64(sqrt(Float64(Float64(4.0 * C) * Float64(Float64(A * -4.0) * Float64(C * F)))) / Float64(Float64(4.0 * A) * C));
	elseif (B_m <= 4.3e+71)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(F)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.45e-168)
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	elseif (B_m <= 4.3e+71)
		tmp = -sqrt((F / -A));
	else
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.45e-168], N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(N[(A * -4.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.3e+71], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.45e-168

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 21.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Taylor expanded in B around 0 19.8%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 19.8%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   10. Simplified 19.8%

       \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   11. Taylor expanded in B around 0 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]
   12. Step-by-step derivation

       1. associate-*r* 19.6%

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   13. Simplified 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   if 3.45e-168 < B < 4.29999999999999984e71

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 18.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around 0 15.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

   7. Taylor expanded in B around 0 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 15.1%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   9. Simplified 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 4.29999999999999984e71 < B

   1. Initial program 10.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 54.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sqrt-div 69.2%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]

   7. Applied egg-rr 69.2%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 69.1%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F}}{\sqrt{B}}} \]

      2. pow1/2 69.1%

         \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F}}{\sqrt{B}} \]

      3. pow1/2 69.1%

         \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{F}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 69.4%

         \[\leadsto -\frac{\color{blue}{{\left(2 \cdot F\right)}^{0.5}}}{\sqrt{B}} \]

   9. Applied egg-rr 69.4%

      \[\leadsto -\color{blue}{\frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 69.4%

          \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}} \]

       2. *-commutative 69.4%

          \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}} \]

   11. Simplified 69.4%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   12. Step-by-step derivation

       1. sqrt-undiv 55.0%

          \[\leadsto -\color{blue}{\sqrt{\frac{F \cdot 2}{B}}} \]

       2. associate-*r/ 55.0%

          \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]

       3. *-commutative 55.0%

          \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]

       4. sqrt-prod 69.3%

          \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}} \]

   13. Applied egg-rr 69.3%

       \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.45 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{+71}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.55 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 1.45 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left|\frac{F}{B\_m}\right|}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.55e-168)
   (/ (sqrt (* (* 4.0 C) (* (* A -4.0) (* C F)))) (* (* 4.0 A) C))
   (if (<= B_m 1.45e+72)
     (- (sqrt (/ F (- A))))
     (- (sqrt (* 2.0 (fabs (/ F B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.55e-168) {
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 1.45e+72) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -sqrt((2.0 * fabs((F / B_m))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 2.55d-168) then
        tmp = sqrt(((4.0d0 * c) * ((a * (-4.0d0)) * (c * f)))) / ((4.0d0 * a) * c)
    else if (b_m <= 1.45d+72) then
        tmp = -sqrt((f / -a))
    else
        tmp = -sqrt((2.0d0 * abs((f / b_m))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.55e-168) {
		tmp = Math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 1.45e+72) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = -Math.sqrt((2.0 * Math.abs((F / B_m))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2.55e-168:
		tmp = math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C)
	elif B_m <= 1.45e+72:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = -math.sqrt((2.0 * math.fabs((F / B_m))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.55e-168)
		tmp = Float64(sqrt(Float64(Float64(4.0 * C) * Float64(Float64(A * -4.0) * Float64(C * F)))) / Float64(Float64(4.0 * A) * C));
	elseif (B_m <= 1.45e+72)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(2.0 * abs(Float64(F / B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.55e-168)
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	elseif (B_m <= 1.45e+72)
		tmp = -sqrt((F / -A));
	else
		tmp = -sqrt((2.0 * abs((F / B_m))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.55e-168], N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(N[(A * -4.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.45e+72], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(2.0 * N[Abs[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if B < 2.5499999999999998e-168

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 21.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Taylor expanded in B around 0 19.8%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 19.8%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   10. Simplified 19.8%

       \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   11. Taylor expanded in B around 0 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]
   12. Step-by-step derivation

       1. associate-*r* 19.6%

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   13. Simplified 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   if 2.5499999999999998e-168 < B < 1.45000000000000009e72

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 18.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around 0 15.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

   7. Taylor expanded in B around 0 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 15.1%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   9. Simplified 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 1.45000000000000009e72 < B

   1. Initial program 10.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 54.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. pow1/2 54.7%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 54.7%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 55.0%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 55.0%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 55.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   9. Simplified 55.0%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 54.9%

          \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{\frac{F}{B}}\right)} \cdot 2} \]

       2. pow1/2 54.9%

          \[\leadsto -\sqrt{\left(\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{\frac{F}{B}}\right) \cdot 2} \]

       3. pow1/2 54.9%

          \[\leadsto -\sqrt{\left({\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right) \cdot 2} \]

       4. pow-prod-down 38.8%

          \[\leadsto -\sqrt{\color{blue}{{\left(\frac{F}{B} \cdot \frac{F}{B}\right)}^{0.5}} \cdot 2} \]

       5. pow2 38.8%

          \[\leadsto -\sqrt{{\color{blue}{\left({\left(\frac{F}{B}\right)}^{2}\right)}}^{0.5} \cdot 2} \]

   11. Applied egg-rr 38.8%

       \[\leadsto -\sqrt{\color{blue}{{\left({\left(\frac{F}{B}\right)}^{2}\right)}^{0.5}} \cdot 2} \]
   12. Step-by-step derivation

       1. unpow1/2 38.8%

          \[\leadsto -\sqrt{\color{blue}{\sqrt{{\left(\frac{F}{B}\right)}^{2}}} \cdot 2} \]

       2. unpow2 38.8%

          \[\leadsto -\sqrt{\sqrt{\color{blue}{\frac{F}{B} \cdot \frac{F}{B}}} \cdot 2} \]

       3. rem-sqrt-square 55.0%

          \[\leadsto -\sqrt{\color{blue}{\left|\frac{F}{B}\right|} \cdot 2} \]

   13. Simplified 55.0%

       \[\leadsto -\sqrt{\color{blue}{\left|\frac{F}{B}\right|} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.55 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left|\frac{F}{B}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.6 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 1.1 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.6e-168)
   (/ (sqrt (* (* 4.0 C) (* (* A -4.0) (* C F)))) (* (* 4.0 A) C))
   (if (<= B_m 1.1e+72) (- (sqrt (/ F (- A)))) (- (sqrt (* 2.0 (/ F B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.6e-168) {
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 1.1e+72) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -sqrt((2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.6d-168) then
        tmp = sqrt(((4.0d0 * c) * ((a * (-4.0d0)) * (c * f)))) / ((4.0d0 * a) * c)
    else if (b_m <= 1.1d+72) then
        tmp = -sqrt((f / -a))
    else
        tmp = -sqrt((2.0d0 * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.6e-168) {
		tmp = Math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	} else if (B_m <= 1.1e+72) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = -Math.sqrt((2.0 * (F / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.6e-168:
		tmp = math.sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C)
	elif B_m <= 1.1e+72:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = -math.sqrt((2.0 * (F / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.6e-168)
		tmp = Float64(sqrt(Float64(Float64(4.0 * C) * Float64(Float64(A * -4.0) * Float64(C * F)))) / Float64(Float64(4.0 * A) * C));
	elseif (B_m <= 1.1e+72)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.6e-168)
		tmp = sqrt(((4.0 * C) * ((A * -4.0) * (C * F)))) / ((4.0 * A) * C);
	elseif (B_m <= 1.1e+72)
		tmp = -sqrt((F / -A));
	else
		tmp = -sqrt((2.0 * (F / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.6e-168], N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(N[(A * -4.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.1e+72], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if B < 1.60000000000000003e-168

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 21.3%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Simplified 21.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(C \cdot 4\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Taylor expanded in B around 0 19.8%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 19.8%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   10. Simplified 19.8%

       \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   11. Taylor expanded in B around 0 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]
   12. Step-by-step derivation

       1. associate-*r* 19.6%

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   13. Simplified 19.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(C \cdot F\right)\right)} \cdot \left(C \cdot 4\right)}}{\left(4 \cdot A\right) \cdot C} \]

   if 1.60000000000000003e-168 < B < 1.1e72

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 18.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around 0 15.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

   7. Taylor expanded in B around 0 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 15.1%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   9. Simplified 15.1%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 1.1e72 < B

   1. Initial program 10.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 54.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. pow1/2 54.7%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 54.7%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 55.0%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 55.0%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 55.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   9. Simplified 55.0%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.6 \cdot 10^{-168}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.05 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.05e+72) (- (sqrt (/ F (- A)))) (- (sqrt (* 2.0 (/ F B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.05e+72) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -sqrt((2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 3.05d+72) then
        tmp = -sqrt((f / -a))
    else
        tmp = -sqrt((2.0d0 * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.05e+72) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = -Math.sqrt((2.0 * (F / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.05e+72:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = -math.sqrt((2.0 * (F / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.05e+72)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.05e+72)
		tmp = -sqrt((F / -A));
	else
		tmp = -sqrt((2.0 * (F / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.05e+72], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 3.04999999999999996e72

   1. Initial program 18.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 25.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf 19.4%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around 0 12.7%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{A} + 4 \cdot C\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

   7. Taylor expanded in B around 0 17.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 17.8%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   9. Simplified 17.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 3.04999999999999996e72 < B

   1. Initial program 10.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 54.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative 54.7%

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. pow1/2 54.7%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 54.7%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 55.0%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 55.0%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 55.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   9. Simplified 55.0%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.05 \cdot 10^{+72}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* F (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F * Float64(2.0 / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 17.1%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf 14.6%

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg 14.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   2. *-commutative 14.6%

      \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

5. Simplified 14.6%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. *-commutative 14.6%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   2. pow1/2 14.9%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

   3. pow1/2 14.9%

      \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

   4. pow-prod-down 14.9%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

7. Applied egg-rr 14.9%

   \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
8. Step-by-step derivation

   1. unpow1/2 14.7%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

9. Simplified 14.7%

   \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

10. Taylor expanded in F around 0 14.7%

    \[\leadsto -\sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]
11. Step-by-step derivation

    1. associate-*r/ 14.7%

       \[\leadsto -\sqrt{\color{blue}{\frac{2 \cdot F}{B}}} \]

    2. *-commutative 14.7%

       \[\leadsto -\sqrt{\frac{\color{blue}{F \cdot 2}}{B}} \]

    3. associate-/l* 14.7%

       \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]

12. Simplified 14.7%

    \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))