ABCF->ab-angle a

Percentage Accurate: 18.7% → 60.4%

Time: 16.7s

Alternatives: 13

Speedup: 16.9×

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: 18.7% 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: 60.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\\ t_2 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_3 := t\_0 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot t\_1}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2}\right) \cdot \sqrt{C \cdot 2}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\left(\sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)} + \left(C + A\right)\right) \cdot t\_1}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(t\_2 \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{-t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot F} \cdot \sqrt{\left(C \cdot 2\right) \cdot 2}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* C (* A 4.0)))
        (t_1 (* (* F (- (pow B_m 2.0) t_0)) 2.0))
        (t_2 (fma (* -4.0 C) A (* B_m B_m)))
        (t_3 (- t_0 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt (* (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A)) t_1))
          t_3)))
   (if (<= t_4 -5e+263)
     (/
      (*
       (* (sqrt F) (sqrt (* (fma (* C A) -4.0 (* B_m B_m)) 2.0)))
       (sqrt (* C 2.0)))
      t_3)
     (if (<= t_4 -5e-189)
       (/
        (sqrt (* (+ (sqrt (fma (- A C) (- A C) (* B_m B_m))) (+ C A)) t_1))
        t_3)
       (if (<= t_4 0.0)
         (/
          (sqrt (* (* t_2 (* F 2.0)) (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))
          (- t_2))
         (if (<= t_4 INFINITY)
           (/ (* (sqrt (* t_2 F)) (sqrt (* (* C 2.0) 2.0))) t_3)
           (/ (sqrt (* F 2.0)) (- (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 = C * (A * 4.0);
	double t_1 = (F * (pow(B_m, 2.0) - t_0)) * 2.0;
	double t_2 = fma((-4.0 * C), A, (B_m * B_m));
	double t_3 = t_0 - pow(B_m, 2.0);
	double t_4 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * t_1)) / t_3;
	double tmp;
	if (t_4 <= -5e+263) {
		tmp = ((sqrt(F) * sqrt((fma((C * A), -4.0, (B_m * B_m)) * 2.0))) * sqrt((C * 2.0))) / t_3;
	} else if (t_4 <= -5e-189) {
		tmp = sqrt(((sqrt(fma((A - C), (A - C), (B_m * B_m))) + (C + A)) * t_1)) / t_3;
	} else if (t_4 <= 0.0) {
		tmp = sqrt(((t_2 * (F * 2.0)) * fma(-0.5, ((B_m * B_m) / A), (C * 2.0)))) / -t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((t_2 * F)) * sqrt(((C * 2.0) * 2.0))) / t_3;
	} else {
		tmp = sqrt((F * 2.0)) / -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 = Float64(C * Float64(A * 4.0))
	t_1 = Float64(Float64(F * Float64((B_m ^ 2.0) - t_0)) * 2.0)
	t_2 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_3 = Float64(t_0 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * t_1)) / t_3)
	tmp = 0.0
	if (t_4 <= -5e+263)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0))) * sqrt(Float64(C * 2.0))) / t_3);
	elseif (t_4 <= -5e-189)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))) + Float64(C + A)) * t_1)) / t_3);
	elseif (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(t_2 * Float64(F * 2.0)) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)))) / Float64(-t_2));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_2 * F)) * sqrt(Float64(Float64(C * 2.0) * 2.0))) / t_3);
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / 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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+263], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -5e-189], N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(N[(t$95$2 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(t$95$2 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $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))) < -5.00000000000000022e263

   1. Initial program 4.8%

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

   3. Taylor expanded in C around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 16.4

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

   5. Applied rewrites 16.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

   7. Applied rewrites 21.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-neg-in N/A

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

   9. Applied rewrites 40.7%

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

   if -5.00000000000000022e263 < (/.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))) < -4.9999999999999997e-189

   1. Initial program 97.1%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 97.1

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.1

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

   4. Applied rewrites 97.1%

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

   if -4.9999999999999997e-189 < (/.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))) < 0.0

   1. Initial program 3.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.3

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

   5. Applied rewrites 27.3%

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

   7. Applied rewrites 27.3%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 38.9%

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

   3. Taylor expanded in C around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.0

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

   5. Applied rewrites 33.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

   7. Applied rewrites 58.5%

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

   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

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 17.4

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

   5. Applied rewrites 17.4%

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

   7. Applied rewrites 17.5%

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

   9. Applied rewrites 21.7%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2}\right) \cdot \sqrt{C \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F} \cdot \sqrt{\left(C \cdot 2\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_1}\\ t_3 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2}\right) \cdot \sqrt{C \cdot 2}}{t\_1}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{\frac{t\_4}{\sqrt{\left(t\_4 \cdot \left(F \cdot 2\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)} + \left(C + A\right)\right)}}}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(t\_3 \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{-t\_3}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot F} \cdot \sqrt{\left(C \cdot 2\right) \cdot 2}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* C (* A 4.0)))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_0)) 2.0)))
          t_1))
        (t_3 (fma (* -4.0 C) A (* B_m B_m)))
        (t_4 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_2 -5e+263)
     (/
      (*
       (* (sqrt F) (sqrt (* (fma (* C A) -4.0 (* B_m B_m)) 2.0)))
       (sqrt (* C 2.0)))
      t_1)
     (if (<= t_2 -5e-189)
       (/
        -1.0
        (/
         t_4
         (sqrt
          (*
           (* t_4 (* F 2.0))
           (+ (sqrt (fma (- A C) (- A C) (* B_m B_m))) (+ C A))))))
       (if (<= t_2 0.0)
         (/
          (sqrt (* (* t_3 (* F 2.0)) (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))
          (- t_3))
         (if (<= t_2 INFINITY)
           (/ (* (sqrt (* t_3 F)) (sqrt (* (* C 2.0) 2.0))) t_1)
           (/ (sqrt (* F 2.0)) (- (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 = C * (A * 4.0);
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_0)) * 2.0))) / t_1;
	double t_3 = fma((-4.0 * C), A, (B_m * B_m));
	double t_4 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_2 <= -5e+263) {
		tmp = ((sqrt(F) * sqrt((fma((C * A), -4.0, (B_m * B_m)) * 2.0))) * sqrt((C * 2.0))) / t_1;
	} else if (t_2 <= -5e-189) {
		tmp = -1.0 / (t_4 / sqrt(((t_4 * (F * 2.0)) * (sqrt(fma((A - C), (A - C), (B_m * B_m))) + (C + A)))));
	} else if (t_2 <= 0.0) {
		tmp = sqrt(((t_3 * (F * 2.0)) * fma(-0.5, ((B_m * B_m) / A), (C * 2.0)))) / -t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt((t_3 * F)) * sqrt(((C * 2.0) * 2.0))) / t_1;
	} else {
		tmp = sqrt((F * 2.0)) / -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 = Float64(C * Float64(A * 4.0))
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_0)) * 2.0))) / t_1)
	t_3 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_4 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_2 <= -5e+263)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0))) * sqrt(Float64(C * 2.0))) / t_1);
	elseif (t_2 <= -5e-189)
		tmp = Float64(-1.0 / Float64(t_4 / sqrt(Float64(Float64(t_4 * Float64(F * 2.0)) * Float64(sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))) + Float64(C + A))))));
	elseif (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(t_3 * Float64(F * 2.0)) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)))) / Float64(-t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_3 * F)) * sqrt(Float64(Float64(C * 2.0) * 2.0))) / t_1);
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / 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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+263], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -5e-189], N[(-1.0 / N[(t$95$4 / N[Sqrt[N[(N[(t$95$4 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(N[(t$95$3 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(t$95$3 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $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))) < -5.00000000000000022e263

   1. Initial program 4.8%

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

   3. Taylor expanded in C around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 16.4

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

   5. Applied rewrites 16.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

   7. Applied rewrites 21.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-neg-in N/A

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

   9. Applied rewrites 40.7%

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

   if -5.00000000000000022e263 < (/.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))) < -4.9999999999999997e-189

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 13.3%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 96.9

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

   6. Applied rewrites 96.9%

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

   if -4.9999999999999997e-189 < (/.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))) < 0.0

   1. Initial program 3.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.3

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

   5. Applied rewrites 27.3%

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

   7. Applied rewrites 27.3%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 38.9%

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

   3. Taylor expanded in C around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.0

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

   5. Applied rewrites 33.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

   7. Applied rewrites 58.5%

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

   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

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 17.4

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

   5. Applied rewrites 17.4%

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

   7. Applied rewrites 17.5%

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

   9. Applied rewrites 21.7%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2}\right) \cdot \sqrt{C \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(C + A\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F} \cdot \sqrt{\left(C \cdot 2\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_3 := C \cdot \left(A \cdot 4\right)\\ t_4 := t\_3 - {B\_m}^{2}\\ t_5 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_3\right)\right) \cdot 2\right)}}{t\_4}\\ t_6 := \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\left(\frac{-\sqrt{F}}{t\_1} \cdot \sqrt{t\_6}\right) \cdot \sqrt{t\_1 \cdot 2}\\ \mathbf{elif}\;t\_5 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)} + \left(C + A\right)\right)}}}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(t\_2 \cdot \left(F \cdot 2\right)\right) \cdot t\_6}}{-t\_2}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot F} \cdot \sqrt{\left(C \cdot 2\right) \cdot 2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 -4.0 (* C A) (* B_m B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m)))
        (t_2 (fma (* -4.0 C) A (* B_m B_m)))
        (t_3 (* C (* A 4.0)))
        (t_4 (- t_3 (pow B_m 2.0)))
        (t_5
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_3)) 2.0)))
          t_4))
        (t_6 (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))
   (if (<= t_5 (- INFINITY))
     (* (* (/ (- (sqrt F)) t_1) (sqrt t_6)) (sqrt (* t_1 2.0)))
     (if (<= t_5 -5e-189)
       (/
        -1.0
        (/
         t_0
         (sqrt
          (*
           (* t_0 (* F 2.0))
           (+ (sqrt (fma (- A C) (- A C) (* B_m B_m))) (+ C A))))))
       (if (<= t_5 0.0)
         (/ (sqrt (* (* t_2 (* F 2.0)) t_6)) (- t_2))
         (if (<= t_5 INFINITY)
           (/ (* (sqrt (* t_2 F)) (sqrt (* (* C 2.0) 2.0))) t_4)
           (/ (sqrt (* F 2.0)) (- (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(-4.0, (C * A), (B_m * B_m));
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double t_2 = fma((-4.0 * C), A, (B_m * B_m));
	double t_3 = C * (A * 4.0);
	double t_4 = t_3 - pow(B_m, 2.0);
	double t_5 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_3)) * 2.0))) / t_4;
	double t_6 = fma(-0.5, ((B_m * B_m) / A), (C * 2.0));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = ((-sqrt(F) / t_1) * sqrt(t_6)) * sqrt((t_1 * 2.0));
	} else if (t_5 <= -5e-189) {
		tmp = -1.0 / (t_0 / sqrt(((t_0 * (F * 2.0)) * (sqrt(fma((A - C), (A - C), (B_m * B_m))) + (C + A)))));
	} else if (t_5 <= 0.0) {
		tmp = sqrt(((t_2 * (F * 2.0)) * t_6)) / -t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (sqrt((t_2 * F)) * sqrt(((C * 2.0) * 2.0))) / t_4;
	} else {
		tmp = sqrt((F * 2.0)) / -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(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_2 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_3 = Float64(C * Float64(A * 4.0))
	t_4 = Float64(t_3 - (B_m ^ 2.0))
	t_5 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_3)) * 2.0))) / t_4)
	t_6 = fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(-sqrt(F)) / t_1) * sqrt(t_6)) * sqrt(Float64(t_1 * 2.0)));
	elseif (t_5 <= -5e-189)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * Float64(sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))) + Float64(C + A))))));
	elseif (t_5 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(t_2 * Float64(F * 2.0)) * t_6)) / Float64(-t_2));
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_2 * F)) * sqrt(Float64(Float64(C * 2.0) * 2.0))) / t_4);
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / 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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[((-N[Sqrt[F], $MachinePrecision]) / t$95$1), $MachinePrecision] * N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -5e-189], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[Sqrt[N[(N[(t$95$2 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(N[Sqrt[N[(t$95$2 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $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 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 17.8

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

   5. Applied rewrites 17.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

   7. Applied rewrites 21.4%

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

   9. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. pow1/2 N/A

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

       4. lift-*.f64 N/A

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

       5. unpow-prod-down N/A

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

   11. Applied rewrites 36.5%

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

   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))) < -4.9999999999999997e-189

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 13.5%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 96.9

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

   6. Applied rewrites 96.9%

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

   if -4.9999999999999997e-189 < (/.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))) < 0.0

   1. Initial program 3.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.3

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

   5. Applied rewrites 27.3%

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

   7. Applied rewrites 27.3%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 38.9%

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

   3. Taylor expanded in C around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.0

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

   5. Applied rewrites 33.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

   7. Applied rewrites 58.5%

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

   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

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 17.4

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

   5. Applied rewrites 17.4%

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

   7. Applied rewrites 17.5%

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

   9. Applied rewrites 21.7%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}\right) \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(C + A\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F} \cdot \sqrt{\left(C \cdot 2\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := C \cdot \left(A \cdot 4\right)\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_3\right)\right) \cdot 2\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(\frac{-\sqrt{F}}{t\_1} \cdot \sqrt{t\_5}\right) \cdot \sqrt{t\_1 \cdot 2}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{\frac{t\_2}{\sqrt{\left(t\_2 \cdot \left(F \cdot 2\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)} + \left(C + A\right)\right)}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot t\_5}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* -4.0 C) A (* B_m B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m)))
        (t_3 (* C (* A 4.0)))
        (t_4
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_3)) 2.0)))
          (- t_3 (pow B_m 2.0))))
        (t_5 (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))
   (if (<= t_4 (- INFINITY))
     (* (* (/ (- (sqrt F)) t_1) (sqrt t_5)) (sqrt (* t_1 2.0)))
     (if (<= t_4 -5e-189)
       (/
        -1.0
        (/
         t_2
         (sqrt
          (*
           (* t_2 (* F 2.0))
           (+ (sqrt (fma (- A C) (- A C) (* B_m B_m))) (+ C A))))))
       (if (<= t_4 INFINITY)
         (/ (sqrt (* (* t_0 (* F 2.0)) t_5)) (- t_0))
         (/ (sqrt (* F 2.0)) (- (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((-4.0 * C), A, (B_m * B_m));
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = C * (A * 4.0);
	double t_4 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_3)) * 2.0))) / (t_3 - pow(B_m, 2.0));
	double t_5 = fma(-0.5, ((B_m * B_m) / A), (C * 2.0));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((-sqrt(F) / t_1) * sqrt(t_5)) * sqrt((t_1 * 2.0));
	} else if (t_4 <= -5e-189) {
		tmp = -1.0 / (t_2 / sqrt(((t_2 * (F * 2.0)) * (sqrt(fma((A - C), (A - C), (B_m * B_m))) + (C + A)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((t_0 * (F * 2.0)) * t_5)) / -t_0;
	} else {
		tmp = sqrt((F * 2.0)) / -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(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_3 = Float64(C * Float64(A * 4.0))
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_3)) * 2.0))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(-sqrt(F)) / t_1) * sqrt(t_5)) * sqrt(Float64(t_1 * 2.0)));
	elseif (t_4 <= -5e-189)
		tmp = Float64(-1.0 / Float64(t_2 / sqrt(Float64(Float64(t_2 * Float64(F * 2.0)) * Float64(sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))) + Float64(C + A))))));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * t_5)) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / 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[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[((-N[Sqrt[F], $MachinePrecision]) / t$95$1), $MachinePrecision] * N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-189], N[(-1.0 / N[(t$95$2 / N[Sqrt[N[(N[(t$95$2 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 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 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 17.8

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

   5. Applied rewrites 17.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

   7. Applied rewrites 21.4%

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

   9. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. pow1/2 N/A

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

       4. lift-*.f64 N/A

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

       5. unpow-prod-down N/A

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

   11. Applied rewrites 36.5%

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

   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))) < -4.9999999999999997e-189

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 13.5%

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

      1. lift-hypot.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 96.9

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

   6. Applied rewrites 96.9%

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

   if -4.9999999999999997e-189 < (/.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 20.6%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 30.1

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

   5. Applied rewrites 30.1%

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

   7. Applied rewrites 30.1%

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

   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

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 17.4

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

   5. Applied rewrites 17.4%

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

   7. Applied rewrites 17.5%

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

   9. Applied rewrites 21.7%

      \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}\right) \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(C + A\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* -4.0 C) A (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-32)
     (/
      (sqrt (* (* t_0 (* F 2.0)) (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))
      (- t_0))
     (/ (sqrt (* F 2.0)) (- (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((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = sqrt(((t_0 * (F * 2.0)) * fma(-0.5, ((B_m * B_m) / A), (C * 2.0)))) / -t_0;
	} else {
		tmp = sqrt((F * 2.0)) / -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(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / 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[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 24.1%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 25.4

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

   5. Applied rewrites 25.4%

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

   7. Applied rewrites 25.4%

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

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 26.9

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

   5. Applied rewrites 26.9%

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

   7. Applied rewrites 27.0%

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

   9. Applied rewrites 31.8%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot 2\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* -4.0 C) A (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e+54)
     (/ (sqrt (* (* C 2.0) (* t_0 (* F 2.0)))) (- t_0))
     (/ (sqrt (* F 2.0)) (- (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((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+54) {
		tmp = sqrt(((C * 2.0) * (t_0 * (F * 2.0)))) / -t_0;
	} else {
		tmp = sqrt((F * 2.0)) / -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(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+54)
		tmp = Float64(sqrt(Float64(Float64(C * 2.0) * Float64(t_0 * Float64(F * 2.0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / 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[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+54], N[(N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000005e54

   1. Initial program 25.2%

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

   3. Taylor expanded in C around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.9

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

   5. Applied rewrites 24.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 24.9%

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

   if 5.00000000000000005e54 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 28.8%

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

   9. Applied rewrites 34.5%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot 2\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\left(\left(C \cdot C\right) \cdot A\right) \cdot -16} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 5e-112)
   (*
    (sqrt (* (* (* C C) A) -16.0))
    (/ (- (sqrt F)) (fma (* C A) -4.0 (* B_m B_m))))
   (/ (sqrt (* F 2.0)) (- (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 (pow(B_m, 2.0) <= 5e-112) {
		tmp = sqrt((((C * C) * A) * -16.0)) * (-sqrt(F) / fma((C * A), -4.0, (B_m * B_m)));
	} else {
		tmp = sqrt((F * 2.0)) / -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 ^ 2.0) <= 5e-112)
		tmp = Float64(sqrt(Float64(Float64(Float64(C * C) * A) * -16.0)) * Float64(Float64(-sqrt(F)) / fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / 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_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-112], N[(N[Sqrt[N[(N[(N[(C * C), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000044e-112

   1. Initial program 22.2%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 25.4

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

   5. Applied rewrites 25.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

   7. Applied rewrites 19.8%

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

   9. Applied rewrites 19.7%

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

   10. Taylor expanded in C around inf

       \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot {C}^{2}\right)}} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot {C}^{2}\right)}} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(A \cdot {C}^{2}\right)}} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. unpow2 N/A

          \[\leadsto \sqrt{-16 \cdot \left(A \cdot \color{blue}{\left(C \cdot C\right)}\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. lower-*.f64 16.6

          \[\leadsto \sqrt{-16 \cdot \left(A \cdot \color{blue}{\left(C \cdot C\right)}\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   12. Applied rewrites 16.6%

       \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)}} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 5.00000000000000044e-112 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 24.5

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

   5. Applied rewrites 24.5%

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

   7. Applied rewrites 24.6%

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

   9. Applied rewrites 28.6%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\left(\left(C \cdot C\right) \cdot A\right) \cdot -16} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\left(C \cdot A\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 5e-112)
   (/ (sqrt (* (* (* (* C C) F) A) -16.0)) (- (* (* C A) -4.0)))
   (/ (sqrt (* F 2.0)) (- (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 (pow(B_m, 2.0) <= 5e-112) {
		tmp = sqrt(((((C * C) * F) * A) * -16.0)) / -((C * A) * -4.0);
	} else {
		tmp = sqrt((F * 2.0)) / -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 ** 2.0d0) <= 5d-112) then
        tmp = sqrt(((((c * c) * f) * a) * (-16.0d0))) / -((c * a) * (-4.0d0))
    else
        tmp = sqrt((f * 2.0d0)) / -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 (Math.pow(B_m, 2.0) <= 5e-112) {
		tmp = Math.sqrt(((((C * C) * F) * A) * -16.0)) / -((C * A) * -4.0);
	} else {
		tmp = Math.sqrt((F * 2.0)) / -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 math.pow(B_m, 2.0) <= 5e-112:
		tmp = math.sqrt(((((C * C) * F) * A) * -16.0)) / -((C * A) * -4.0)
	else:
		tmp = math.sqrt((F * 2.0)) / -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 ^ 2.0) <= 5e-112)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * F) * A) * -16.0)) / Float64(-Float64(Float64(C * A) * -4.0)));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / 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 ^ 2.0) <= 5e-112)
		tmp = sqrt(((((C * C) * F) * A) * -16.0)) / -((C * A) * -4.0);
	else
		tmp = sqrt((F * 2.0)) / -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[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-112], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000044e-112

   1. Initial program 22.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. clear-num N/A

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

      10. flip-+ N/A

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

      11. lift-+.f64 N/A

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

   4. Applied rewrites 7.1%

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

   5. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 15.5

         \[\leadsto \frac{-\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 15.5%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

      2. lower-*.f64 15.4

         \[\leadsto \frac{-\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-4 \cdot \color{blue}{\left(A \cdot C\right)}} \]

   10. Applied rewrites 15.4%

       \[\leadsto \frac{-\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

   if 5.00000000000000044e-112 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 24.5

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

   5. Applied rewrites 24.5%

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

   7. Applied rewrites 24.6%

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

   9. Applied rewrites 28.6%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\left(C \cdot A\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.2% accurate, 12.6× speedup

Math

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

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)) (- (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) {
	return sqrt((F * 2.0)) / -sqrt(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)) / -sqrt(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)) / -Math.sqrt(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)) / -math.sqrt(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 * 2.0)) / Float64(-sqrt(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)) / -sqrt(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[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 15.3

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

5. Applied rewrites 15.3%

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

7. Applied rewrites 15.4%

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

9. Applied rewrites 18.1%

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

10. Final simplification 18.1%

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

Alternative 10: 35.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}} \end{array} \]

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)) (sqrt (* 0.5 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) / sqrt((0.5 * 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) / sqrt((0.5d0 * 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) / Math.sqrt((0.5 * 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) / math.sqrt((0.5 * 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(Float64(-sqrt(F)) / sqrt(Float64(0.5 * 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) / sqrt((0.5 * 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[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 15.3

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

5. Applied rewrites 15.3%

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

7. Applied rewrites 15.4%

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

9. Applied rewrites 18.1%

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

11. Applied rewrites 18.1%

    \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]

12. Final simplification 18.1%

    \[\leadsto \frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}} \]
13. Add Preprocessing

Alternative 11: 35.2% accurate, 12.6× speedup

Math

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

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 (/ 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) {
	return sqrt((2.0 / B_m)) * -sqrt(F);
}

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((2.0d0 / b_m)) * -sqrt(f)
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((2.0 / B_m)) * -Math.sqrt(F);
}

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((2.0 / B_m)) * -math.sqrt(F)

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(2.0 / B_m)) * Float64(-sqrt(F)))
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((2.0 / B_m)) * -sqrt(F);
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[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 15.3

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

5. Applied rewrites 15.3%

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

7. Applied rewrites 15.4%

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

9. Applied rewrites 18.1%

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

10. Final simplification 18.1%

    \[\leadsto \sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right) \]
11. Add Preprocessing

Alternative 12: 27.0% accurate, 16.9× speedup

Math

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

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 B_m) 2.0))))

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 / B_m) * 2.0));
}

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 / b_m) * 2.0d0))
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 / B_m) * 2.0));
}

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 / B_m) * 2.0))

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(Float64(F / B_m) * 2.0)))
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 / B_m) * 2.0));
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[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 18.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 15.3

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

5. Applied rewrites 15.3%

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

7. Applied rewrites 15.4%

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

Alternative 13: 27.0% accurate, 16.9× speedup

Math

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

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 (* (/ 2.0 B_m) 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) {
	return -sqrt(((2.0 / B_m) * F));
}

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(((2.0d0 / b_m) * f))
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(((2.0 / B_m) * F));
}

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(((2.0 / B_m) * F))

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(Float64(2.0 / B_m) * F)))
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(((2.0 / B_m) * F));
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[(N[(2.0 / B$95$m), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 18.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 15.3

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

5. Applied rewrites 15.3%

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

7. Applied rewrites 15.4%

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

9. Applied rewrites 15.4%

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

10. Final simplification 15.4%

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

Reproduce

herbie shell --seed 2024249 
(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))))