ABCF->ab-angle a

Percentage Accurate: 18.7% → 60.0%

Time: 20.5s

Alternatives: 11

Speedup: 16.9×

• could not determine a ground truth

  1. A = 2.9334609022579226e-249
  2. B = -3.755304623136173e-130
  3. C = -8.444212580369829e-9
  4. F = 5.571183955139176e-270

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: 60.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -2 \cdot \sqrt{C}\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := {B\_m}^{2} - t\_2\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_5 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\frac{\left(t\_0 \cdot \sqrt{F}\right) \cdot \sqrt{t\_1}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-187}:\\ \;\;\;\;\frac{-1}{\frac{t\_5}{\sqrt{\mathsf{fma}\left(A - C, \left(A + C\right) \cdot \frac{1}{A - C}, \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot \left(t\_5 \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t\_1 \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)\right)\right)}}{-t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{\sqrt{F \cdot t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* -2.0 (sqrt C)))
        (t_1 (fma -4.0 (* A C) (* B_m B_m)))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- (pow B_m 2.0) t_2))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0))))
        (t_5 (fma C (* A -4.0) (* B_m B_m))))
   (if (<= t_4 -5e+138)
     (/ (* (* t_0 (sqrt F)) (sqrt t_1)) t_3)
     (if (<= t_4 -1e-187)
       (/
        -1.0
        (/
         t_5
         (sqrt
          (*
           (fma
            (- A C)
            (* (+ A C) (/ 1.0 (- A C)))
            (sqrt (fma (- A C) (- A C) (* B_m B_m))))
           (* t_5 (* 2.0 F))))))
       (if (<= t_4 2e+166)
         (/
          (sqrt (* 2.0 (* t_1 (* F (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))))))
          (- t_1))
         (if (<= t_4 INFINITY)
           (* t_0 (/ (sqrt (* F t_1)) t_1))
           (- (/ (sqrt F) (sqrt (* B_m 0.5))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -2.0 * sqrt(C);
	double t_1 = fma(-4.0, (A * C), (B_m * B_m));
	double t_2 = (4.0 * A) * C;
	double t_3 = pow(B_m, 2.0) - t_2;
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double t_5 = fma(C, (A * -4.0), (B_m * B_m));
	double tmp;
	if (t_4 <= -5e+138) {
		tmp = ((t_0 * sqrt(F)) * sqrt(t_1)) / t_3;
	} else if (t_4 <= -1e-187) {
		tmp = -1.0 / (t_5 / sqrt((fma((A - C), ((A + C) * (1.0 / (A - C))), sqrt(fma((A - C), (A - C), (B_m * B_m)))) * (t_5 * (2.0 * F)))));
	} else if (t_4 <= 2e+166) {
		tmp = sqrt((2.0 * (t_1 * (F * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))))) / -t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_0 * (sqrt((F * t_1)) / t_1);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-2.0 * sqrt(C))
	t_1 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64((B_m ^ 2.0) - t_2)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	t_5 = fma(C, Float64(A * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if (t_4 <= -5e+138)
		tmp = Float64(Float64(Float64(t_0 * sqrt(F)) * sqrt(t_1)) / t_3);
	elseif (t_4 <= -1e-187)
		tmp = Float64(-1.0 / Float64(t_5 / sqrt(Float64(fma(Float64(A - C), Float64(Float64(A + C) * Float64(1.0 / Float64(A - C))), sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * Float64(t_5 * Float64(2.0 * F))))));
	elseif (t_4 <= 2e+166)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_1 * Float64(F * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))))) / Float64(-t_1));
	elseif (t_4 <= Inf)
		tmp = Float64(t_0 * Float64(sqrt(Float64(F * t_1)) / t_1));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-2.0 * N[Sqrt[C], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+138], N[(N[(N[(t$95$0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -1e-187], N[(-1.0 / N[(t$95$5 / N[Sqrt[N[(N[(N[(A - C), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] * N[(1.0 / N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$5 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+166], N[(N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$0 * N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $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))) < -5.00000000000000016e138

   1. Initial program 12.1%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 16.1

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

   5. Simplified 16.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 20.8%

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

   8. Taylor expanded in B around 0

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

      1. *-lowering-*.f64 24.7

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

   10. Simplified 24.7%

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

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

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

       2. pow1/2 N/A

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

       3. unpow-prod-down N/A

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

       4. associate-*r* N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. metadata-eval N/A

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

       9. pow1/2 N/A

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

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

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

       11. *-lowering-*.f64 N/A

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

       12. metadata-eval N/A

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

       13. pow1/2 N/A

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

       14. sqrt-lowering-sqrt.f64 N/A

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

       15. pow1/2 N/A

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

       16. sqrt-lowering-sqrt.f64 N/A

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

       17. pow1/2 N/A

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

       18. sqrt-lowering-sqrt.f64 N/A

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

       19. accelerator-lowering-fma.f64 N/A

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

       20. *-lowering-*.f64 N/A

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

       21. *-lowering-*.f64 31.2

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

   12. Applied egg-rr 31.2%

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

   if -5.00000000000000016e138 < (/.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))) < -1e-187

   1. Initial program 99.5%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.4%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. sqrt-lowering-sqrt.f64 N/A

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

      14. +-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. --lowering--.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. *-lowering-*.f64 99.5

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

   6. Applied egg-rr 99.5%

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

   if -1e-187 < (/.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))) < 1.99999999999999988e166

   1. Initial program 16.0%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 37.2

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

   5. Simplified 37.2%

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

      1. pow2 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 37.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 37.1%

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

   if 1.99999999999999988e166 < (/.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 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 A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 9.4

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

   5. Simplified 9.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 26.5%

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

   8. Taylor expanded in B around 0

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

      1. *-lowering-*.f64 26.5

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

   10. Simplified 26.5%

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

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

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

       2. pow2 N/A

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

       3. associate-*l* N/A

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

       4. cancel-sign-sub-inv N/A

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. *-lowering-*.f64 N/A

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

       9. sqrt-prod N/A

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

       10. metadata-eval N/A

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

       11. pow1/2 N/A

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

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

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

       13. *-lowering-*.f64 N/A

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

       14. metadata-eval N/A

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

       15. pow1/2 N/A

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

       16. sqrt-lowering-sqrt.f64 N/A

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

   12. Applied egg-rr 26.6%

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

   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. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 20.7

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

   5. Simplified 20.7%

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

      1. distribute-rgt-neg-out N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 20.8

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

   7. Applied egg-rr 20.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 20.8

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

   9. Applied egg-rr 20.8%

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

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot \frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

       2. unpow-prod-down N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}} \]

       4. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \color{blue}{\sqrt{\frac{2}{B}}} \]

       5. clear-num N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{B}{2}}}} \]

       6. metadata-eval N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{\frac{B}{2}}} \]

       9. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{B}{2}}} \]

       10. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       12. neg-lowering-neg.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}}{\sqrt{\frac{B}{2}}} \]

       13. pow1/2 N/A

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

       14. sqrt-lowering-sqrt.f64 N/A

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

       15. sqrt-lowering-sqrt.f64 N/A

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

       16. div-inv N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{\color{blue}{B \cdot \frac{1}{2}}}} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{B \cdot \color{blue}{\frac{1}{2}}}} \]

       18. *-lowering-*.f64 32.9

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

   11. Applied egg-rr 32.9%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\frac{\left(\left(-2 \cdot \sqrt{C}\right) \cdot \sqrt{F}\right) \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-187}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A - C, \left(A + C\right) \cdot \frac{1}{A - C}, \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\left(-2 \cdot \sqrt{C}\right) \cdot \frac{\sqrt{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-55)
   (/ (sqrt (* -16.0 (* F (* A (* C C))))) (- (fma -4.0 (* A C) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 4e+105)
     (* (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C)))))) (/ (sqrt 2.0) (- B_m)))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-55) {
		tmp = sqrt((-16.0 * (F * (A * (C * C))))) / -fma(-4.0, (A * C), (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 4e+105) {
		tmp = sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-55)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(A * Float64(C * C))))) / Float64(-fma(-4.0, Float64(A * C), Float64(B_m * B_m))));
	elseif ((B_m ^ 2.0) <= 4e+105)
		tmp = Float64(sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-55], N[(N[Sqrt[N[(-16.0 * N[(F * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+105], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999999e-55

   1. Initial program 21.9%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 28.5

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

   5. Simplified 28.5%

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

      1. pow2 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 28.5%

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

   8. Taylor expanded in A around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.9

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

   10. Simplified 18.9%

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

   if 1.99999999999999999e-55 < (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999998e105

   1. Initial program 40.7%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 40.8%

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

   5. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 7.6

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

   7. Simplified 7.6%

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

   if 3.9999999999999998e105 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.2%

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

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

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 27.8

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

   5. Simplified 27.8%

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

      1. distribute-rgt-neg-out N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 28.0

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

   7. Applied egg-rr 28.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 28.0

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

   9. Applied egg-rr 28.0%

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

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot \frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

       2. unpow-prod-down N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}} \]

       4. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \color{blue}{\sqrt{\frac{2}{B}}} \]

       5. clear-num N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{B}{2}}}} \]

       6. metadata-eval N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{\frac{B}{2}}} \]

       9. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{B}{2}}} \]

       10. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       12. neg-lowering-neg.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}}{\sqrt{\frac{B}{2}}} \]

       13. pow1/2 N/A

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

       14. sqrt-lowering-sqrt.f64 N/A

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

       15. sqrt-lowering-sqrt.f64 N/A

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

       16. div-inv N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{\color{blue}{B \cdot \frac{1}{2}}}} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{B \cdot \color{blue}{\frac{1}{2}}}} \]

       18. *-lowering-*.f64 39.3

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

   11. Applied egg-rr 39.3%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right) \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 4e+133)
     (/
      (sqrt (* 2.0 (* (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)) (* F t_0))))
      (- t_0))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 4e+133) {
		tmp = sqrt((2.0 * (fma(-0.5, ((B_m * B_m) / A), (2.0 * C)) * (F * t_0)))) / -t_0;
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+133)
		tmp = Float64(sqrt(Float64(2.0 * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)) * Float64(F * t_0)))) / Float64(-t_0));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+133], N[(N[Sqrt[N[(2.0 * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000001e133

   1. Initial program 26.8%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 25.1

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

   5. Simplified 25.1%

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

      1. pow2 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 25.1%

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

   if 4.0000000000000001e133 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 7.3%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 27.6

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

   5. Simplified 27.6%

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

      1. distribute-rgt-neg-out N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 27.8

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

   7. Applied egg-rr 27.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 27.8

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

   9. Applied egg-rr 27.8%

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

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot \frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

       2. unpow-prod-down N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}} \]

       4. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \color{blue}{\sqrt{\frac{2}{B}}} \]

       5. clear-num N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{B}{2}}}} \]

       6. metadata-eval N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{\frac{B}{2}}} \]

       9. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{B}{2}}} \]

       10. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       12. neg-lowering-neg.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}}{\sqrt{\frac{B}{2}}} \]

       13. pow1/2 N/A

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

       14. sqrt-lowering-sqrt.f64 N/A

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

       15. sqrt-lowering-sqrt.f64 N/A

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

       16. div-inv N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{\color{blue}{B \cdot \frac{1}{2}}}} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{B \cdot \color{blue}{\frac{1}{2}}}} \]

       18. *-lowering-*.f64 39.4

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

   11. Applied egg-rr 39.4%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+105}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot \left(F \cdot t\_0\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 4e+105)
     (* -2.0 (/ (sqrt (* C (* F t_0))) t_0))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 4e+105) {
		tmp = -2.0 * (sqrt((C * (F * t_0))) / t_0);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+105)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * Float64(F * t_0))) / t_0));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+105], N[(-2.0 * N[(N[Sqrt[N[(C * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999998e105

   1. Initial program 26.6%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 25.5

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

   5. Simplified 25.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 19.9%

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

   8. Taylor expanded in B around 0

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

      1. *-lowering-*.f64 20.0

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

   10. Simplified 20.0%

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

   11. Taylor expanded in F around 0

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

       1. *-lowering-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. cancel-sign-sub-inv N/A

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

       7. /-lowering-/.f64 N/A

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

   13. Simplified 25.6%

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

   if 3.9999999999999998e105 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.2%

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

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

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 27.8

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

   5. Simplified 27.8%

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

      1. distribute-rgt-neg-out N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 28.0

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

   7. Applied egg-rr 28.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 28.0

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

   9. Applied egg-rr 28.0%

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

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot \frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

       2. unpow-prod-down N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}} \]

       4. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \color{blue}{\sqrt{\frac{2}{B}}} \]

       5. clear-num N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{B}{2}}}} \]

       6. metadata-eval N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{\frac{B}{2}}} \]

       9. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{B}{2}}} \]

       10. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       12. neg-lowering-neg.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}}{\sqrt{\frac{B}{2}}} \]

       13. pow1/2 N/A

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

       14. sqrt-lowering-sqrt.f64 N/A

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

       15. sqrt-lowering-sqrt.f64 N/A

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

       16. div-inv N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{\color{blue}{B \cdot \frac{1}{2}}}} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{B \cdot \color{blue}{\frac{1}{2}}}} \]

       18. *-lowering-*.f64 39.3

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

   11. Applied egg-rr 39.3%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+105}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e+76)
   (/ (sqrt (* -16.0 (* F (* A (* C C))))) (- (fma -4.0 (* A C) (* B_m B_m))))
   (- (/ (sqrt F) (sqrt (* B_m 0.5))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e+76) {
		tmp = sqrt((-16.0 * (F * (A * (C * C))))) / -fma(-4.0, (A * C), (B_m * B_m));
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+76)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(A * Float64(C * C))))) / Float64(-fma(-4.0, Float64(A * C), Float64(B_m * B_m))));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+76], N[(N[Sqrt[N[(-16.0 * N[(F * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e76

   1. Initial program 26.8%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 26.9

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

   5. Simplified 26.9%

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

      1. pow2 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 26.9%

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

   8. Taylor expanded in A around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.1

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

   10. Simplified 18.1%

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

   if 2.0000000000000001e76 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 25.6

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

   5. Simplified 25.6%

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

      1. distribute-rgt-neg-out N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 25.8

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

   7. Applied egg-rr 25.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 25.8

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

   9. Applied egg-rr 25.8%

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

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot \frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

       2. unpow-prod-down N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}} \]

       4. pow1/2 N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \color{blue}{\sqrt{\frac{2}{B}}} \]

       5. clear-num N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{B}{2}}}} \]

       6. metadata-eval N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{\frac{B}{2}}} \]

       9. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{B}{2}}} \]

       10. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

       12. neg-lowering-neg.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}}{\sqrt{\frac{B}{2}}} \]

       13. pow1/2 N/A

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

       14. sqrt-lowering-sqrt.f64 N/A

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

       15. sqrt-lowering-sqrt.f64 N/A

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

       16. div-inv N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{\color{blue}{B \cdot \frac{1}{2}}}} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{B \cdot \color{blue}{\frac{1}{2}}}} \]

       18. *-lowering-*.f64 36.0

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

   11. Applied egg-rr 36.0%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.0% accurate, 12.6× speedup

Math

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

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 (* B_m 0.5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -(sqrt(F) / sqrt((B_m * 0.5)));
}

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((b_m * 0.5d0)))
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((B_m * 0.5)));
}

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((B_m * 0.5)))

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(B_m * 0.5))))
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((B_m * 0.5)));
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[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 19.9%

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 12.2

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

5. Simplified 12.2%

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

   1. distribute-rgt-neg-out N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. sqrt-unprod N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 12.3

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

7. Applied egg-rr 12.3%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 12.3

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

9. Applied egg-rr 12.3%

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

    1. pow1/2 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot \frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

    2. unpow-prod-down N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}}\right) \]

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

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot {\left(\frac{2}{B}\right)}^{\frac{1}{2}}} \]

    4. pow1/2 N/A

       \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \color{blue}{\sqrt{\frac{2}{B}}} \]

    5. clear-num N/A

       \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{B}{2}}}} \]

    6. metadata-eval N/A

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

    7. sqrt-div N/A

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

    8. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{\frac{B}{2}}} \]

    9. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left({F}^{\frac{1}{2}}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{B}{2}}} \]

    10. un-div-inv N/A

        \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}{\sqrt{\frac{B}{2}}}} \]

    12. neg-lowering-neg.f64 N/A

        \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({F}^{\frac{1}{2}}\right)}}{\sqrt{\frac{B}{2}}} \]

    13. pow1/2 N/A

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

    14. sqrt-lowering-sqrt.f64 N/A

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

    15. sqrt-lowering-sqrt.f64 N/A

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

    16. div-inv N/A

        \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{\color{blue}{B \cdot \frac{1}{2}}}} \]

    17. metadata-eval N/A

        \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{B \cdot \color{blue}{\frac{1}{2}}}} \]

    18. *-lowering-*.f64 16.8

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

11. Applied egg-rr 16.8%

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

12. Final simplification 16.8%

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

Alternative 7: 35.0% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.9%

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 12.2

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

5. Simplified 12.2%

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

   1. distribute-rgt-neg-out N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. sqrt-unprod N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 12.3

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

7. Applied egg-rr 12.3%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}}} \cdot \sqrt{\frac{2}{B}}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \sqrt{\frac{2}{B}}}\right) \]

   6. pow1/2 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. /-lowering-/.f64 16.8

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

9. Applied egg-rr 16.8%

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

10. Final simplification 16.8%

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

Alternative 8: 27.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 3.1 \cdot 10^{+84}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 3.1e+84)
   (- (sqrt (/ F (* B_m 0.5))))
   (* -2.0 (/ (sqrt (* C F)) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 3.1e+84) {
		tmp = -sqrt((F / (B_m * 0.5)));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 3.1d+84) then
        tmp = -sqrt((f / (b_m * 0.5d0)))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 3.1e+84) {
		tmp = -Math.sqrt((F / (B_m * 0.5)));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 3.1e+84:
		tmp = -math.sqrt((F / (B_m * 0.5)))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 3.1e+84)
		tmp = Float64(-sqrt(Float64(F / Float64(B_m * 0.5))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 3.1e+84)
		tmp = -sqrt((F / (B_m * 0.5)));
	else
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 3.1e+84], (-N[Sqrt[N[(F / N[(B$95$m * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 3.10000000000000003e84

   1. Initial program 19.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 12.1

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

   5. Simplified 12.1%

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

      1. distribute-rgt-neg-out N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 12.1

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

   7. Applied egg-rr 12.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 12.1

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

   9. Applied egg-rr 12.1%

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

       1. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F \cdot \color{blue}{\frac{1}{\frac{B}{2}}}}\right) \]

       2. un-div-inv N/A

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

       3. /-lowering-/.f64 N/A

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

       4. div-inv N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{\color{blue}{B \cdot \frac{1}{2}}}}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B \cdot \color{blue}{\frac{1}{2}}}}\right) \]

       6. *-lowering-*.f64 12.2

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

   11. Applied egg-rr 12.2%

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

   if 3.10000000000000003e84 < C

   1. Initial program 22.7%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 30.2

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

   5. Simplified 30.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 40.2%

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

   8. Taylor expanded in B around 0

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

      1. *-lowering-*.f64 52.9

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

   10. Simplified 52.9%

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

   11. Taylor expanded in C around 0

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. sqrt-lowering-sqrt.f64 N/A

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

       6. *-lowering-*.f64 9.9

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

   13. Simplified 9.9%

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

Alternative 9: 26.8% accurate, 16.9× speedup

Math

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

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 0.5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((F / (B_m * 0.5)));
}

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 * 0.5d0)))
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 * 0.5)));
}

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 * 0.5)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F / Float64(B_m * 0.5))))
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 * 0.5)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(F / N[(B$95$m * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.9%

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 12.2

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

5. Simplified 12.2%

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

   1. distribute-rgt-neg-out N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. sqrt-unprod N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 12.3

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

7. Applied egg-rr 12.3%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 12.3

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

9. Applied egg-rr 12.3%

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

    1. clear-num N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F \cdot \color{blue}{\frac{1}{\frac{B}{2}}}}\right) \]

    2. un-div-inv N/A

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

    3. /-lowering-/.f64 N/A

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

    4. div-inv N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{\color{blue}{B \cdot \frac{1}{2}}}}\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B \cdot \color{blue}{\frac{1}{2}}}}\right) \]

    6. *-lowering-*.f64 12.3

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

11. Applied egg-rr 12.3%

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

Alternative 10: 26.8% accurate, 16.9× speedup

Math

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

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 (* 2.0 (/ F B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((2.0 * (F / 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((2.0d0 * (f / 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((2.0 * (F / 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((2.0 * (F / 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(2.0 * Float64(F / 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((2.0 * (F / B_m)));
end

Wolfram

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

Derivation

1. Initial program 19.9%

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 12.2

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

5. Simplified 12.2%

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

   1. distribute-rgt-neg-out N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. sqrt-unprod N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 12.3

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

7. Applied egg-rr 12.3%

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

8. Final simplification 12.3%

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

Alternative 11: 26.8% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.9%

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 12.2

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

5. Simplified 12.2%

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

   1. distribute-rgt-neg-out N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. sqrt-unprod N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 12.3

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

7. Applied egg-rr 12.3%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 12.3

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

9. Applied egg-rr 12.3%

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

Reproduce

herbie shell --seed 2024204 
(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))))