ABCF->ab-angle b

Percentage Accurate: 17.9% → 47.3%

Time: 16.2s

Alternatives: 11

Speedup: 8.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.9% 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: 47.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{1}{\sqrt{2}}\\ \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{t\_1 \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B\_m \leq 5.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\left(\left({B\_m}^{2} - t\_0\right) \cdot F\right) \cdot 2\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 1.55 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{\mathsf{fma}\left(A \cdot -4, C, B\_m \cdot B\_m\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B\_m}{F}} \cdot t\_1}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)) (t_1 (/ 1.0 (sqrt 2.0))))
   (if (<= B_m 3.2e-283)
     (/ -1.0 (* t_1 (sqrt (* (/ C F) -2.0))))
     (if (<= B_m 5.2e-60)
       (/
        (sqrt (* (+ A A) (* (* (- (pow B_m 2.0) t_0) F) 2.0)))
        (- t_0 (pow B_m 2.0)))
       (if (<= B_m 1.55e+149)
         (*
          (sqrt
           (*
            (/ (- (+ A C) (hypot (- A C) B_m)) (fma (* A -4.0) C (* B_m B_m)))
            F))
          (- (sqrt 2.0)))
         (/ -1.0 (* (sqrt (/ (- B_m) F)) t_1)))))))

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 = (4.0 * A) * C;
	double t_1 = 1.0 / sqrt(2.0);
	double tmp;
	if (B_m <= 3.2e-283) {
		tmp = -1.0 / (t_1 * sqrt(((C / F) * -2.0)));
	} else if (B_m <= 5.2e-60) {
		tmp = sqrt(((A + A) * (((pow(B_m, 2.0) - t_0) * F) * 2.0))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 1.55e+149) {
		tmp = sqrt(((((A + C) - hypot((A - C), B_m)) / fma((A * -4.0), C, (B_m * B_m))) * F)) * -sqrt(2.0);
	} else {
		tmp = -1.0 / (sqrt((-B_m / F)) * t_1);
	}
	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(Float64(4.0 * A) * C)
	t_1 = Float64(1.0 / sqrt(2.0))
	tmp = 0.0
	if (B_m <= 3.2e-283)
		tmp = Float64(-1.0 / Float64(t_1 * sqrt(Float64(Float64(C / F) * -2.0))));
	elseif (B_m <= 5.2e-60)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(Float64(Float64((B_m ^ 2.0) - t_0) * F) * 2.0))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 1.55e+149)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(A + C) - hypot(Float64(A - C), B_m)) / fma(Float64(A * -4.0), C, Float64(B_m * B_m))) * F)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(-1.0 / Float64(sqrt(Float64(Float64(-B_m) / F)) * t_1));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.2e-283], N[(-1.0 / N[(t$95$1 * N[Sqrt[N[(N[(C / F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5.2e-60], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.55e+149], N[(N[Sqrt[N[(N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(A * -4.0), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[N[((-B$95$m) / F), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.20000000000000012e-283

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 28.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 5.2%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 27.1%

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

   10. Taylor expanded in A around -inf

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 14.2%

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]

   if 3.20000000000000012e-283 < B < 5.1999999999999995e-60

   1. Initial program 13.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 26.7

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

   5. Applied rewrites 26.7%

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

   if 5.1999999999999995e-60 < B < 1.54999999999999993e149

   1. Initial program 21.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 50.7%

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

   if 1.54999999999999993e149 < B

   1. Initial program 0.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 0.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 7.3%

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

   10. Taylor expanded in B around inf

       \[\leadsto \frac{-1}{\sqrt{-1 \cdot \frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.9%

       \[\leadsto \frac{-1}{\sqrt{-\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{2}}\\ \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{t\_0 \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B\_m \leq 8 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}{\sqrt{\left(\left(\left(A \cdot C\right) \cdot -4\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right)}}}\\ \mathbf{elif}\;B\_m \leq 1.55 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{\mathsf{fma}\left(A \cdot -4, C, B\_m \cdot B\_m\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B\_m}{F}} \cdot t\_0}\\ \end{array} \end{array} \]

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 (/ 1.0 (sqrt 2.0))))
   (if (<= B_m 3.2e-283)
     (/ -1.0 (* t_0 (sqrt (* (/ C F) -2.0))))
     (if (<= B_m 8e-61)
       (/
        -1.0
        (/
         (fma (* -4.0 C) A (* B_m B_m))
         (sqrt
          (*
           (* (* (* A C) -4.0) (* 2.0 F))
           (+ (fma -0.5 (/ (* B_m B_m) C) A) A)))))
       (if (<= B_m 1.55e+149)
         (*
          (sqrt
           (*
            (/ (- (+ A C) (hypot (- A C) B_m)) (fma (* A -4.0) C (* B_m B_m)))
            F))
          (- (sqrt 2.0)))
         (/ -1.0 (* (sqrt (/ (- B_m) F)) t_0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 1.0 / sqrt(2.0);
	double tmp;
	if (B_m <= 3.2e-283) {
		tmp = -1.0 / (t_0 * sqrt(((C / F) * -2.0)));
	} else if (B_m <= 8e-61) {
		tmp = -1.0 / (fma((-4.0 * C), A, (B_m * B_m)) / sqrt(((((A * C) * -4.0) * (2.0 * F)) * (fma(-0.5, ((B_m * B_m) / C), A) + A))));
	} else if (B_m <= 1.55e+149) {
		tmp = sqrt(((((A + C) - hypot((A - C), B_m)) / fma((A * -4.0), C, (B_m * B_m))) * F)) * -sqrt(2.0);
	} else {
		tmp = -1.0 / (sqrt((-B_m / F)) * t_0);
	}
	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(1.0 / sqrt(2.0))
	tmp = 0.0
	if (B_m <= 3.2e-283)
		tmp = Float64(-1.0 / Float64(t_0 * sqrt(Float64(Float64(C / F) * -2.0))));
	elseif (B_m <= 8e-61)
		tmp = Float64(-1.0 / Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) / sqrt(Float64(Float64(Float64(Float64(A * C) * -4.0) * Float64(2.0 * F)) * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A)))));
	elseif (B_m <= 1.55e+149)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(A + C) - hypot(Float64(A - C), B_m)) / fma(Float64(A * -4.0), C, Float64(B_m * B_m))) * F)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(-1.0 / Float64(sqrt(Float64(Float64(-B_m) / F)) * t_0));
	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[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.2e-283], N[(-1.0 / N[(t$95$0 * N[Sqrt[N[(N[(C / F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8e-61], N[(-1.0 / N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.55e+149], N[(N[Sqrt[N[(N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(A * -4.0), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[N[((-B$95$m) / F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.20000000000000012e-283

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 28.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 5.2%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 27.1%

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

   10. Taylor expanded in A around -inf

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 14.2%

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]

   if 3.20000000000000012e-283 < B < 8.0000000000000003e-61

   1. Initial program 13.9%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 26.7

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

   5. Applied rewrites 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 26.7

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

   10. Applied rewrites 26.7%

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

   if 8.0000000000000003e-61 < B < 1.54999999999999993e149

   1. Initial program 21.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 50.7%

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

   if 1.54999999999999993e149 < B

   1. Initial program 0.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 0.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 7.3%

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

   10. Taylor expanded in B around inf

       \[\leadsto \frac{-1}{\sqrt{-1 \cdot \frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.9%

       \[\leadsto \frac{-1}{\sqrt{-\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\left(A \cdot C\right) \cdot -4\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_1 := \frac{1}{\sqrt{2}}\\ \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{t\_1 \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B\_m \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(\left(2 \cdot F\right) \cdot t\_0\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\left(\frac{-1}{F} - \frac{\frac{A}{F} + \frac{C}{F}}{B\_m}\right) \cdot B\_m} \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))) (t_1 (/ 1.0 (sqrt 2.0))))
   (if (<= B_m 3.2e-283)
     (/ -1.0 (* t_1 (sqrt (* (/ C F) -2.0))))
     (if (<= B_m 3.6e+47)
       (/
        -1.0
        (/
         t_0
         (sqrt (* (* (* 2.0 F) t_0) (+ (fma -0.5 (/ (* B_m B_m) C) A) A)))))
       (/
        -1.0
        (* (sqrt (* (- (/ -1.0 F) (/ (+ (/ A F) (/ C F)) B_m)) B_m)) t_1))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double t_1 = 1.0 / sqrt(2.0);
	double tmp;
	if (B_m <= 3.2e-283) {
		tmp = -1.0 / (t_1 * sqrt(((C / F) * -2.0)));
	} else if (B_m <= 3.6e+47) {
		tmp = -1.0 / (t_0 / sqrt((((2.0 * F) * t_0) * (fma(-0.5, ((B_m * B_m) / C), A) + A))));
	} else {
		tmp = -1.0 / (sqrt((((-1.0 / F) - (((A / F) + (C / F)) / B_m)) * B_m)) * t_1);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_1 = Float64(1.0 / sqrt(2.0))
	tmp = 0.0
	if (B_m <= 3.2e-283)
		tmp = Float64(-1.0 / Float64(t_1 * sqrt(Float64(Float64(C / F) * -2.0))));
	elseif (B_m <= 3.6e+47)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(Float64(Float64(2.0 * F) * t_0) * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A)))));
	else
		tmp = Float64(-1.0 / Float64(sqrt(Float64(Float64(Float64(-1.0 / F) - Float64(Float64(Float64(A / F) + Float64(C / F)) / B_m)) * B_m)) * t_1));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.2e-283], N[(-1.0 / N[(t$95$1 * N[Sqrt[N[(N[(C / F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.6e+47], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[N[(N[(N[(-1.0 / F), $MachinePrecision] - N[(N[(N[(A / F), $MachinePrecision] + N[(C / F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.20000000000000012e-283

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 28.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 5.2%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 27.1%

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

   10. Taylor expanded in A around -inf

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 14.2%

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]

   if 3.20000000000000012e-283 < B < 3.60000000000000008e47

   1. Initial program 18.3%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   7. Applied rewrites 28.4%

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

   if 3.60000000000000008e47 < B

   1. Initial program 6.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 6.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 0.1%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 22.1%

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

   10. Taylor expanded in B around inf

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

   12. Applied rewrites 54.1%

       \[\leadsto \frac{-1}{\sqrt{B \cdot \left(\left(-\frac{\frac{A}{F} + \frac{C}{F}}{B}\right) - \frac{1}{F}\right)} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\left(\frac{-1}{F} - \frac{\frac{A}{F} + \frac{C}{F}}{B}\right) \cdot B} \cdot \frac{1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{2}}\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{t\_0 \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B\_m \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot t\_1\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\left(\frac{-1}{F} - \frac{\frac{A}{F} + \frac{C}{F}}{B\_m}\right) \cdot B\_m} \cdot t\_0}\\ \end{array} \end{array} \]

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 (/ 1.0 (sqrt 2.0))) (t_1 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 3.2e-283)
     (/ -1.0 (* t_0 (sqrt (* (/ C F) -2.0))))
     (if (<= B_m 3.6e+47)
       (/
        (sqrt (* (* (* 2.0 F) t_1) (+ (fma -0.5 (/ (* B_m B_m) C) A) A)))
        (- t_1))
       (/
        -1.0
        (* (sqrt (* (- (/ -1.0 F) (/ (+ (/ A F) (/ C F)) B_m)) B_m)) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 1.0 / sqrt(2.0);
	double t_1 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 3.2e-283) {
		tmp = -1.0 / (t_0 * sqrt(((C / F) * -2.0)));
	} else if (B_m <= 3.6e+47) {
		tmp = sqrt((((2.0 * F) * t_1) * (fma(-0.5, ((B_m * B_m) / C), A) + A))) / -t_1;
	} else {
		tmp = -1.0 / (sqrt((((-1.0 / F) - (((A / F) + (C / F)) / B_m)) * B_m)) * t_0);
	}
	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(1.0 / sqrt(2.0))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.2e-283)
		tmp = Float64(-1.0 / Float64(t_0 * sqrt(Float64(Float64(C / F) * -2.0))));
	elseif (B_m <= 3.6e+47)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * F) * t_1) * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A))) / Float64(-t_1));
	else
		tmp = Float64(-1.0 / Float64(sqrt(Float64(Float64(Float64(-1.0 / F) - Float64(Float64(Float64(A / F) + Float64(C / F)) / B_m)) * B_m)) * t_0));
	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[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.2e-283], N[(-1.0 / N[(t$95$0 * N[Sqrt[N[(N[(C / F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.6e+47], N[(N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[N[(N[(N[(-1.0 / F), $MachinePrecision] - N[(N[(N[(A / F), $MachinePrecision] + N[(C / F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.20000000000000012e-283

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 28.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 5.2%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 27.1%

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

   10. Taylor expanded in A around -inf

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 14.2%

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]

   if 3.20000000000000012e-283 < B < 3.60000000000000008e47

   1. Initial program 18.3%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   7. Applied rewrites 28.4%

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

   if 3.60000000000000008e47 < B

   1. Initial program 6.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 6.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 0.1%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 22.1%

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

   10. Taylor expanded in B around inf

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

   12. Applied rewrites 54.1%

       \[\leadsto \frac{-1}{\sqrt{B \cdot \left(\left(-\frac{\frac{A}{F} + \frac{C}{F}}{B}\right) - \frac{1}{F}\right)} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\left(\frac{-1}{F} - \frac{\frac{A}{F} + \frac{C}{F}}{B}\right) \cdot B} \cdot \frac{1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{2}}\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{t\_0 \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B\_m \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot t\_1\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B\_m}{F}} \cdot t\_0}\\ \end{array} \end{array} \]

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 (/ 1.0 (sqrt 2.0))) (t_1 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 3.2e-283)
     (/ -1.0 (* t_0 (sqrt (* (/ C F) -2.0))))
     (if (<= B_m 3.6e+47)
       (/
        (sqrt (* (* (* 2.0 F) t_1) (+ (fma -0.5 (/ (* B_m B_m) C) A) A)))
        (- t_1))
       (/ -1.0 (* (sqrt (/ (- B_m) F)) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 1.0 / sqrt(2.0);
	double t_1 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 3.2e-283) {
		tmp = -1.0 / (t_0 * sqrt(((C / F) * -2.0)));
	} else if (B_m <= 3.6e+47) {
		tmp = sqrt((((2.0 * F) * t_1) * (fma(-0.5, ((B_m * B_m) / C), A) + A))) / -t_1;
	} else {
		tmp = -1.0 / (sqrt((-B_m / F)) * t_0);
	}
	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(1.0 / sqrt(2.0))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.2e-283)
		tmp = Float64(-1.0 / Float64(t_0 * sqrt(Float64(Float64(C / F) * -2.0))));
	elseif (B_m <= 3.6e+47)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * F) * t_1) * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A))) / Float64(-t_1));
	else
		tmp = Float64(-1.0 / Float64(sqrt(Float64(Float64(-B_m) / F)) * t_0));
	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[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.2e-283], N[(-1.0 / N[(t$95$0 * N[Sqrt[N[(N[(C / F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.6e+47], N[(N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[N[((-B$95$m) / F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.20000000000000012e-283

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 28.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 5.2%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 27.1%

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

   10. Taylor expanded in A around -inf

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 14.2%

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]

   if 3.20000000000000012e-283 < B < 3.60000000000000008e47

   1. Initial program 18.3%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   7. Applied rewrites 28.4%

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

   if 3.60000000000000008e47 < B

   1. Initial program 6.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 6.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 0.1%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 22.1%

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

   10. Taylor expanded in B around inf

       \[\leadsto \frac{-1}{\sqrt{-1 \cdot \frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.2%

       \[\leadsto \frac{-1}{\sqrt{-\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{2}}\\ \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{t\_0 \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}{\sqrt{\left(\left(\left(A \cdot C\right) \cdot F\right) \cdot -8\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B\_m}{F}} \cdot t\_0}\\ \end{array} \end{array} \]

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 (/ 1.0 (sqrt 2.0))))
   (if (<= B_m 3.2e-283)
     (/ -1.0 (* t_0 (sqrt (* (/ C F) -2.0))))
     (if (<= B_m 2.6e+47)
       (/
        -1.0
        (/
         (fma (* -4.0 C) A (* B_m B_m))
         (sqrt
          (* (* (* (* A C) F) -8.0) (+ (fma -0.5 (/ (* B_m B_m) C) A) A)))))
       (/ -1.0 (* (sqrt (/ (- B_m) F)) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 1.0 / sqrt(2.0);
	double tmp;
	if (B_m <= 3.2e-283) {
		tmp = -1.0 / (t_0 * sqrt(((C / F) * -2.0)));
	} else if (B_m <= 2.6e+47) {
		tmp = -1.0 / (fma((-4.0 * C), A, (B_m * B_m)) / sqrt(((((A * C) * F) * -8.0) * (fma(-0.5, ((B_m * B_m) / C), A) + A))));
	} else {
		tmp = -1.0 / (sqrt((-B_m / F)) * t_0);
	}
	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(1.0 / sqrt(2.0))
	tmp = 0.0
	if (B_m <= 3.2e-283)
		tmp = Float64(-1.0 / Float64(t_0 * sqrt(Float64(Float64(C / F) * -2.0))));
	elseif (B_m <= 2.6e+47)
		tmp = Float64(-1.0 / Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) / sqrt(Float64(Float64(Float64(Float64(A * C) * F) * -8.0) * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A)))));
	else
		tmp = Float64(-1.0 / Float64(sqrt(Float64(Float64(-B_m) / F)) * t_0));
	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[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.2e-283], N[(-1.0 / N[(t$95$0 * N[Sqrt[N[(N[(C / F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.6e+47], N[(-1.0 / N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(A * C), $MachinePrecision] * F), $MachinePrecision] * -8.0), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[N[((-B$95$m) / F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.20000000000000012e-283

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 28.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 5.2%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 27.1%

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

   10. Taylor expanded in A around -inf

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 14.2%

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]

   if 3.20000000000000012e-283 < B < 2.60000000000000003e47

   1. Initial program 18.3%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   7. Applied rewrites 28.4%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 25.9

         \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(-8 \cdot \left(\color{blue}{\left(A \cdot C\right)} \cdot F\right)\right)}}} \]

   10. Applied rewrites 25.9%

       \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \color{blue}{\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}}}} \]

   if 2.60000000000000003e47 < B

   1. Initial program 6.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 6.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 0.1%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 22.1%

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

   10. Taylor expanded in B around inf

       \[\leadsto \frac{-1}{\sqrt{-1 \cdot \frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.2%

       \[\leadsto \frac{-1}{\sqrt{-\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\left(A \cdot C\right) \cdot F\right) \cdot -8\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{2}}\\ \mathbf{if}\;B\_m \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\frac{-1}{t\_0 \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B\_m}{F}} \cdot t\_0}\\ \end{array} \end{array} \]

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 (/ 1.0 (sqrt 2.0))))
   (if (<= B_m 1.55e+90)
     (/ -1.0 (* t_0 (sqrt (* (/ C F) -2.0))))
     (/ -1.0 (* (sqrt (/ (- B_m) F)) t_0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 1.0 / sqrt(2.0);
	double tmp;
	if (B_m <= 1.55e+90) {
		tmp = -1.0 / (t_0 * sqrt(((C / F) * -2.0)));
	} else {
		tmp = -1.0 / (sqrt((-B_m / F)) * t_0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / sqrt(2.0d0)
    if (b_m <= 1.55d+90) then
        tmp = (-1.0d0) / (t_0 * sqrt(((c / f) * (-2.0d0))))
    else
        tmp = (-1.0d0) / (sqrt((-b_m / f)) * t_0)
    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 t_0 = 1.0 / Math.sqrt(2.0);
	double tmp;
	if (B_m <= 1.55e+90) {
		tmp = -1.0 / (t_0 * Math.sqrt(((C / F) * -2.0)));
	} else {
		tmp = -1.0 / (Math.sqrt((-B_m / F)) * t_0);
	}
	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):
	t_0 = 1.0 / math.sqrt(2.0)
	tmp = 0
	if B_m <= 1.55e+90:
		tmp = -1.0 / (t_0 * math.sqrt(((C / F) * -2.0)))
	else:
		tmp = -1.0 / (math.sqrt((-B_m / F)) * t_0)
	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(1.0 / sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.55e+90)
		tmp = Float64(-1.0 / Float64(t_0 * sqrt(Float64(Float64(C / F) * -2.0))));
	else
		tmp = Float64(-1.0 / Float64(sqrt(Float64(Float64(-B_m) / F)) * t_0));
	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)
	t_0 = 1.0 / sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 1.55e+90)
		tmp = -1.0 / (t_0 * sqrt(((C / F) * -2.0)));
	else
		tmp = -1.0 / (sqrt((-B_m / F)) * t_0);
	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_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.55e+90], N[(-1.0 / N[(t$95$0 * N[Sqrt[N[(N[(C / F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[N[((-B$95$m) / F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.54999999999999994e90

   1. Initial program 20.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 28.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 5.7%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 23.9%

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

   10. Taylor expanded in A around -inf

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.9%

       \[\leadsto \frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}} \]

   if 1.54999999999999994e90 < B

   1. Initial program 5.3%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 5.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 0.1%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 22.5%

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

   10. Taylor expanded in B around inf

       \[\leadsto \frac{-1}{\sqrt{-1 \cdot \frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 58.5%

       \[\leadsto \frac{-1}{\sqrt{-\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{C}{F} \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-B}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-1}{\sqrt{\frac{-B\_m}{F}} \cdot \frac{1}{\sqrt{2}}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/ -1.0 (* (sqrt (/ (- B_m) F)) (/ 1.0 (sqrt 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -1.0 / (sqrt((-B_m / F)) * (1.0 / sqrt(2.0)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-1.0d0) / (sqrt((-b_m / f)) * (1.0d0 / sqrt(2.0d0)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -1.0 / (Math.sqrt((-B_m / F)) * (1.0 / Math.sqrt(2.0)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -1.0 / (math.sqrt((-B_m / F)) * (1.0 / math.sqrt(2.0)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-1.0 / Float64(sqrt(Float64(Float64(-B_m) / F)) * Float64(1.0 / sqrt(2.0))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -1.0 / (sqrt((-B_m / F)) * (1.0 / sqrt(2.0)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(-1.0 / N[(N[Sqrt[N[((-B$95$m) / F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. neg-mul-1 N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

4. Applied rewrites 24.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift--.f64 N/A

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

   17. *-commutative N/A

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

6. Applied rewrites 4.7%

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

7. Taylor expanded in F around 0

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

   1. lower-*.f64 N/A

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

9. Applied rewrites 23.7%

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

10. Taylor expanded in B around inf

    \[\leadsto \frac{-1}{\sqrt{-1 \cdot \frac{B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
11. Step-by-step derivation

12. Applied rewrites 13.1%

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

13. Final simplification 13.1%

    \[\leadsto \frac{-1}{\sqrt{\frac{-B}{F}} \cdot \frac{1}{\sqrt{2}}} \]
14. Add Preprocessing

Alternative 9: 1.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{1}{\sqrt{\frac{B\_m}{2 \cdot F}}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (/ 1.0 (sqrt (/ B_m (* 2.0 F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return 1.0 / sqrt((B_m / (2.0 * F)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = 1.0d0 / sqrt((b_m / (2.0d0 * f)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return 1.0 / Math.sqrt((B_m / (2.0 * F)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return 1.0 / math.sqrt((B_m / (2.0 * F)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(1.0 / sqrt(Float64(B_m / Float64(2.0 * F))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = 1.0 / sqrt((B_m / (2.0 * F)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(1.0 / N[Sqrt[N[(B$95$m / N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 1.7

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

5. Applied rewrites 1.7%

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

7. Applied rewrites 1.7%

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

9. Applied rewrites 1.7%

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

10. Final simplification 1.7%

    \[\leadsto \frac{1}{\sqrt{\frac{B}{2 \cdot F}}} \]
11. Add Preprocessing

Alternative 10: 1.6% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 1.7

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

5. Applied rewrites 1.7%

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

7. Applied rewrites 1.7%

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

9. Applied rewrites 1.7%

   \[\leadsto \sqrt{\frac{2}{\frac{B}{F}}} \]
10. Add Preprocessing

Alternative 11: 1.6% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 1.7

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

5. Applied rewrites 1.7%

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

7. Applied rewrites 1.7%

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

9. Applied rewrites 1.7%

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

10. Final simplification 1.7%

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

Reproduce

herbie shell --seed 2024284 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))