ABCF->ab-angle a

Percentage Accurate: 18.5% → 61.7%

Time: 14.9s

Alternatives: 18

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.5% 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: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(t\_0 \cdot F\right)\\ t_2 := -t\_0\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(A \cdot -4, C, B\_m \cdot B\_m\right)}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, 2 \cdot C\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \left(\sqrt{\mathsf{hypot}\left(B\_m, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1 (* 2.0 (* t_0 F)))
        (t_2 (- t_0))
        (t_3
         (/
          (sqrt (* t_1 (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_2)))
   (if (<= t_3 -1e-218)
     (*
      (* (sqrt (* F 2.0)) (- (sqrt (fma (* A -4.0) C (* B_m B_m)))))
      (/
       (sqrt (+ (+ (hypot B_m (- A C)) A) C))
       (fma -4.0 (* C A) (* B_m B_m))))
     (if (<= t_3 INFINITY)
       (/ (sqrt (* t_1 (fma (/ (* B_m B_m) A) -0.5 (* 2.0 C)))) t_2)
       (* (/ (- (sqrt 2.0)) B_m) (* (sqrt (+ (hypot B_m A) A)) (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = 2.0 * (t_0 * F);
	double t_2 = -t_0;
	double t_3 = sqrt((t_1 * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -1e-218) {
		tmp = (sqrt((F * 2.0)) * -sqrt(fma((A * -4.0), C, (B_m * B_m)))) * (sqrt(((hypot(B_m, (A - C)) + A) + C)) / fma(-4.0, (C * A), (B_m * B_m)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(((B_m * B_m) / A), -0.5, (2.0 * C)))) / t_2;
	} else {
		tmp = (-sqrt(2.0) / B_m) * (sqrt((hypot(B_m, A) + A)) * sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(2.0 * Float64(t_0 * F))
	t_2 = Float64(-t_0)
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= -1e-218)
		tmp = Float64(Float64(sqrt(Float64(F * 2.0)) * Float64(-sqrt(fma(Float64(A * -4.0), C, Float64(B_m * B_m))))) * Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(2.0 * C)))) / t_2);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * Float64(sqrt(Float64(hypot(B_m, A) + A)) * sqrt(F)));
	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[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-218], N[(N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(A * -4.0), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 49.8%

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

   4. Applied rewrites 70.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 78.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 78.5

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-*r* N/A

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

      23. lower-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 N/A

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

      26. lift-*.f64 78.5

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

   6. Applied rewrites 78.5%

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

   if -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.9%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 35.6

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

   5. Applied rewrites 35.6%

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

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

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 18.3

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

   7. Applied rewrites 18.3%

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

   9. Applied rewrites 27.5%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{\mathsf{hypot}\left(B, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \left(2 \cdot F\right) \cdot t\_0\\ t_2 := \left(-\sqrt{t\_1}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C}}{t\_0}\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := {B\_m}^{2} - t\_3\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_4}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right) \cdot t\_1}}{\mathsf{fma}\left(-B\_m, B\_m, t\_3\right)}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \left(\sqrt{\mathsf{hypot}\left(B\_m, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* (* 2.0 F) t_0))
        (t_2
         (*
          (- (sqrt t_1))
          (/ (sqrt (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) t_0)))
        (t_3 (* (* 4.0 A) C))
        (t_4 (- (pow B_m 2.0) t_3))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_4))))
   (if (<= t_5 (- INFINITY))
     t_2
     (if (<= t_5 -1e-218)
       (/ (sqrt (* (+ (+ (hypot B_m (- A C)) A) C) t_1)) (fma (- B_m) B_m t_3))
       (if (<= t_5 INFINITY)
         t_2
         (*
          (/ (- (sqrt 2.0)) B_m)
          (* (sqrt (+ (hypot B_m A) A)) (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = (2.0 * F) * t_0;
	double t_2 = -sqrt(t_1) * (sqrt(((C + (-0.5 * ((B_m * B_m) / A))) + C)) / t_0);
	double t_3 = (4.0 * A) * C;
	double t_4 = pow(B_m, 2.0) - t_3;
	double t_5 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_5 <= -1e-218) {
		tmp = sqrt((((hypot(B_m, (A - C)) + A) + C) * t_1)) / fma(-B_m, B_m, t_3);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (-sqrt(2.0) / B_m) * (sqrt((hypot(B_m, A) + A)) * sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(Float64(2.0 * F) * t_0)
	t_2 = Float64(Float64(-sqrt(t_1)) * Float64(sqrt(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) / t_0))
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64((B_m ^ 2.0) - t_3)
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_4))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_5 <= -1e-218)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C) * t_1)) / fma(Float64(-B_m), B_m, t_3));
	elseif (t_5 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * Float64(sqrt(Float64(hypot(B_m, A) + A)) * sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[t$95$1], $MachinePrecision]) * N[(N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], t$95$2, If[LessEqual[t$95$5, -1e-218], N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$2, N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 11.1%

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

   4. Applied rewrites 40.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 31.5

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

   7. Applied rewrites 31.5%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.5%

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

   4. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 18.3

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

   7. Applied rewrites 18.3%

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

   9. Applied rewrites 27.5%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{\mathsf{hypot}\left(B, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \left(-\sqrt{\left(2 \cdot F\right) \cdot t\_1}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C}}{t\_1}\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+143}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + C\right) + A\right) \cdot F}{t\_1}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{B\_m} \cdot \left(\sqrt{\mathsf{hypot}\left(B\_m, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4
         (*
          (- (sqrt (* (* 2.0 F) t_1)))
          (/ (sqrt (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) t_1))))
   (if (<= t_3 -4e+143)
     t_4
     (if (<= t_3 -5e-153)
       (* t_0 (sqrt (/ (* (+ (+ (hypot (- A C) B_m) C) A) F) t_1)))
       (if (<= t_3 INFINITY)
         t_4
         (* (/ t_0 B_m) (* (sqrt (+ (hypot B_m A) A)) (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = -sqrt(((2.0 * F) * t_1)) * (sqrt(((C + (-0.5 * ((B_m * B_m) / A))) + C)) / t_1);
	double tmp;
	if (t_3 <= -4e+143) {
		tmp = t_4;
	} else if (t_3 <= -5e-153) {
		tmp = t_0 * sqrt(((((hypot((A - C), B_m) + C) + A) * F) / t_1));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = (t_0 / B_m) * (sqrt((hypot(B_m, A) + A)) * sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * t_1))) * Float64(sqrt(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) / t_1))
	tmp = 0.0
	if (t_3 <= -4e+143)
		tmp = t_4;
	elseif (t_3 <= -5e-153)
		tmp = Float64(t_0 * sqrt(Float64(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + C) + A) * F) / t_1)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(t_0 / B_m) * Float64(sqrt(Float64(hypot(B_m, A) + A)) * sqrt(F)));
	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[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+143], t$95$4, If[LessEqual[t$95$3, -5e-153], N[(t$95$0 * N[Sqrt[N[(N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] + A), $MachinePrecision] * F), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e143 or -5.00000000000000033e-153 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 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

   4. Applied rewrites 44.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.1

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

   7. Applied rewrites 30.1%

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

   if -4.0000000000000001e143 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000033e-153

   1. Initial program 99.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 18.3

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

   7. Applied rewrites 18.3%

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

   9. Applied rewrites 27.5%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{+143}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\right) + C\right) + A\right) \cdot F}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{\mathsf{hypot}\left(B, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_1}\\ t_3 := \left(-\sqrt{\left(2 \cdot F\right) \cdot t\_0}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C}}{t\_0}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+143}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, C\right) + C\right) \cdot F\right) \cdot 2} \cdot B\_m}{-\mathsf{fma}\left(A \cdot -4, C, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \left(\sqrt{\mathsf{hypot}\left(B\_m, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1)))
        (t_3
         (*
          (- (sqrt (* (* 2.0 F) t_0)))
          (/ (sqrt (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) t_0))))
   (if (<= t_2 -4e+143)
     t_3
     (if (<= t_2 -1e-218)
       (/
        (* (sqrt (* (* (+ (hypot B_m C) C) F) 2.0)) B_m)
        (- (fma (* A -4.0) C (* B_m B_m))))
       (if (<= t_2 INFINITY)
         t_3
         (*
          (/ (- (sqrt 2.0)) B_m)
          (* (sqrt (+ (hypot B_m A) A)) (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double t_3 = -sqrt(((2.0 * F) * t_0)) * (sqrt(((C + (-0.5 * ((B_m * B_m) / A))) + C)) / t_0);
	double tmp;
	if (t_2 <= -4e+143) {
		tmp = t_3;
	} else if (t_2 <= -1e-218) {
		tmp = (sqrt((((hypot(B_m, C) + C) * F) * 2.0)) * B_m) / -fma((A * -4.0), C, (B_m * B_m));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (-sqrt(2.0) / B_m) * (sqrt((hypot(B_m, A) + A)) * sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_1))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * t_0))) * Float64(sqrt(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) / t_0))
	tmp = 0.0
	if (t_2 <= -4e+143)
		tmp = t_3;
	elseif (t_2 <= -1e-218)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(hypot(B_m, C) + C) * F) * 2.0)) * B_m) / Float64(-fma(Float64(A * -4.0), C, Float64(B_m * B_m))));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * Float64(sqrt(Float64(hypot(B_m, A) + A)) * sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+143], t$95$3, If[LessEqual[t$95$2, -1e-218], N[(N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * B$95$m), $MachinePrecision] / (-N[(N[(A * -4.0), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e143 or -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 16.0%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.7

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

   7. Applied rewrites 30.7%

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

   if -4.0000000000000001e143 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 48.5

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

   7. Applied rewrites 48.5%

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

   9. Applied rewrites 48.5%

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

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

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 18.3

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

   7. Applied rewrites 18.3%

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

   9. Applied rewrites 27.5%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{+143}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F\right) \cdot 2} \cdot B}{-\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{\mathsf{hypot}\left(B, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_1}\\ t_3 := \left(-\sqrt{\left(2 \cdot F\right) \cdot t\_0}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C}}{t\_0}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+143}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F\right)} \cdot B\_m}{-B\_m \cdot B\_m}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \left(\sqrt{\mathsf{hypot}\left(B\_m, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1)))
        (t_3
         (*
          (- (sqrt (* (* 2.0 F) t_0)))
          (/ (sqrt (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) t_0))))
   (if (<= t_2 -4e+143)
     t_3
     (if (<= t_2 -1e-218)
       (/ (* (sqrt (* 2.0 (* (+ (hypot C B_m) C) F))) B_m) (- (* B_m B_m)))
       (if (<= t_2 INFINITY)
         t_3
         (*
          (/ (- (sqrt 2.0)) B_m)
          (* (sqrt (+ (hypot B_m A) A)) (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double t_3 = -sqrt(((2.0 * F) * t_0)) * (sqrt(((C + (-0.5 * ((B_m * B_m) / A))) + C)) / t_0);
	double tmp;
	if (t_2 <= -4e+143) {
		tmp = t_3;
	} else if (t_2 <= -1e-218) {
		tmp = (sqrt((2.0 * ((hypot(C, B_m) + C) * F))) * B_m) / -(B_m * B_m);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (-sqrt(2.0) / B_m) * (sqrt((hypot(B_m, A) + A)) * sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_1))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * t_0))) * Float64(sqrt(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) / t_0))
	tmp = 0.0
	if (t_2 <= -4e+143)
		tmp = t_3;
	elseif (t_2 <= -1e-218)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(hypot(C, B_m) + C) * F))) * B_m) / Float64(-Float64(B_m * B_m)));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * Float64(sqrt(Float64(hypot(B_m, A) + A)) * sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+143], t$95$3, If[LessEqual[t$95$2, -1e-218], N[(N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * B$95$m), $MachinePrecision] / (-N[(B$95$m * B$95$m), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e143 or -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 16.0%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.7

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

   7. Applied rewrites 30.7%

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

   if -4.0000000000000001e143 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 48.5

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

   7. Applied rewrites 48.5%

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

   8. Taylor expanded in A around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   12. Applied rewrites 48.3%

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

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

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 18.3

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

   7. Applied rewrites 18.3%

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

   9. Applied rewrites 27.5%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{+143}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right)} \cdot B}{-B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{\mathsf{hypot}\left(B, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_1}\\ t_3 := \left(-\sqrt{\left(2 \cdot F\right) \cdot t\_0}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C}}{t\_0}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+143}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F\right)} \cdot B\_m}{-B\_m \cdot B\_m}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1)))
        (t_3
         (*
          (- (sqrt (* (* 2.0 F) t_0)))
          (/ (sqrt (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) t_0))))
   (if (<= t_2 -4e+143)
     t_3
     (if (<= t_2 -1e-218)
       (/ (* (sqrt (* 2.0 (* (+ (hypot C B_m) C) F))) B_m) (- (* B_m B_m)))
       (if (<= t_2 INFINITY) t_3 (/ (sqrt (* F 2.0)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double t_3 = -sqrt(((2.0 * F) * t_0)) * (sqrt(((C + (-0.5 * ((B_m * B_m) / A))) + C)) / t_0);
	double tmp;
	if (t_2 <= -4e+143) {
		tmp = t_3;
	} else if (t_2 <= -1e-218) {
		tmp = (sqrt((2.0 * ((hypot(C, B_m) + C) * F))) * B_m) / -(B_m * B_m);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_1))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * t_0))) * Float64(sqrt(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) / t_0))
	tmp = 0.0
	if (t_2 <= -4e+143)
		tmp = t_3;
	elseif (t_2 <= -1e-218)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(hypot(C, B_m) + C) * F))) * B_m) / Float64(-Float64(B_m * B_m)));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+143], t$95$3, If[LessEqual[t$95$2, -1e-218], N[(N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * B$95$m), $MachinePrecision] / (-N[(B$95$m * B$95$m), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e143 or -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 16.0%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.7

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

   7. Applied rewrites 30.7%

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

   if -4.0000000000000001e143 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 48.5

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

   7. Applied rewrites 48.5%

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

   8. Taylor expanded in A around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   12. Applied rewrites 48.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 16.8

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

   5. Applied rewrites 16.8%

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

   7. Applied rewrites 16.8%

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

   9. Applied rewrites 24.8%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{+143}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right)} \cdot B}{-B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_1}\\ t_3 := \left(-\sqrt{\left(2 \cdot F\right) \cdot t\_0}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C}}{t\_0}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+143}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(B\_m, C\right) + C\right) \cdot F}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1)))
        (t_3
         (*
          (- (sqrt (* (* 2.0 F) t_0)))
          (/ (sqrt (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) t_0))))
   (if (<= t_2 -4e+143)
     t_3
     (if (<= t_2 -1e-218)
       (* (/ (- (sqrt 2.0)) B_m) (sqrt (* (+ (hypot B_m C) C) F)))
       (if (<= t_2 INFINITY) t_3 (/ (sqrt (* F 2.0)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double t_3 = -sqrt(((2.0 * F) * t_0)) * (sqrt(((C + (-0.5 * ((B_m * B_m) / A))) + C)) / t_0);
	double tmp;
	if (t_2 <= -4e+143) {
		tmp = t_3;
	} else if (t_2 <= -1e-218) {
		tmp = (-sqrt(2.0) / B_m) * sqrt(((hypot(B_m, C) + C) * F));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_1))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * t_0))) * Float64(sqrt(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) / t_0))
	tmp = 0.0
	if (t_2 <= -4e+143)
		tmp = t_3;
	elseif (t_2 <= -1e-218)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(Float64(hypot(B_m, C) + C) * F)));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+143], t$95$3, If[LessEqual[t$95$2, -1e-218], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e143 or -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 16.0%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.7

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

   7. Applied rewrites 30.7%

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

   if -4.0000000000000001e143 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.5%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 48.4

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

   5. Applied rewrites 48.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 16.8

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

   5. Applied rewrites 16.8%

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

   7. Applied rewrites 16.8%

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

   9. Applied rewrites 24.8%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{+143}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ t_2 := \frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{-0.5 \cdot \frac{B\_m \cdot B\_m}{A}} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(B\_m + A \cdot \left(1 + 0.5 \cdot \frac{A}{B\_m}\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-133}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0)))
        (t_2
         (*
          (/ (sqrt 2.0) B_m)
          (* (sqrt (* -0.5 (/ (* B_m B_m) A))) (- (sqrt F))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -1e-218)
       (*
        (/ (- (sqrt 2.0)) B_m)
        (sqrt (* F (+ B_m (* A (+ 1.0 (* 0.5 (/ A B_m))))))))
       (if (<= t_1 4e-133) t_2 (/ (sqrt (* F 2.0)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double t_2 = (sqrt(2.0) / B_m) * (sqrt((-0.5 * ((B_m * B_m) / A))) * -sqrt(F));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -1e-218) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (B_m + (A * (1.0 + (0.5 * (A / B_m)))))));
	} else if (t_1 <= 4e-133) {
		tmp = t_2;
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

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 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_0;
	double t_2 = (Math.sqrt(2.0) / B_m) * (Math.sqrt((-0.5 * ((B_m * B_m) / A))) * -Math.sqrt(F));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= -1e-218) {
		tmp = (-Math.sqrt(2.0) / B_m) * Math.sqrt((F * (B_m + (A * (1.0 + (0.5 * (A / B_m)))))));
	} else if (t_1 <= 4e-133) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_1 = math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_0
	t_2 = (math.sqrt(2.0) / B_m) * (math.sqrt((-0.5 * ((B_m * B_m) / A))) * -math.sqrt(F))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= -1e-218:
		tmp = (-math.sqrt(2.0) / B_m) * math.sqrt((F * (B_m + (A * (1.0 + (0.5 * (A / B_m)))))))
	elif t_1 <= 4e-133:
		tmp = t_2
	else:
		tmp = math.sqrt((F * 2.0)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	t_2 = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) * Float64(-sqrt(F))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -1e-218)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(B_m + Float64(A * Float64(1.0 + Float64(0.5 * Float64(A / B_m))))))));
	elseif (t_1 <= 4e-133)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_0;
	t_2 = (sqrt(2.0) / B_m) * (sqrt((-0.5 * ((B_m * B_m) / A))) * -sqrt(F));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= -1e-218)
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (B_m + (A * (1.0 + (0.5 * (A / B_m)))))));
	elseif (t_1 <= 4e-133)
		tmp = t_2;
	else
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -1e-218], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(B$95$m + N[(A * N[(1.0 + N[(0.5 * N[(A / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-133], t$95$2, N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 4.0000000000000003e-133

   1. Initial program 4.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 28.0%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 7.1

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

   7. Applied rewrites 7.1%

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

   9. Applied rewrites 8.2%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 18.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 44.2

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

   7. Applied rewrites 44.2%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 34.4%

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

   if 4.0000000000000003e-133 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 5.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 14.3

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

   5. Applied rewrites 14.3%

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

   7. Applied rewrites 14.3%

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

   9. Applied rewrites 20.9%

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

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{-0.5 \cdot \frac{B \cdot B}{A}} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A \cdot \left(1 + 0.5 \cdot \frac{A}{B}\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 4 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{-0.5 \cdot \frac{B \cdot B}{A}} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq -1 \cdot 10^{-218} \lor \neg \left(t\_1 \leq \infty\right)\right):\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0))))
   (if (or (<= t_1 (- INFINITY))
           (not (or (<= t_1 -1e-218) (not (<= t_1 INFINITY)))))
     (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))))
     (/ (sqrt (* F 2.0)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !((t_1 <= -1e-218) || !(t_1 <= ((double) INFINITY)))) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

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 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !((t_1 <= -1e-218) || !(t_1 <= Double.POSITIVE_INFINITY))) {
		tmp = (-Math.sqrt(2.0) / B_m) * Math.sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	} else {
		tmp = Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_1 = math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_0
	tmp = 0
	if (t_1 <= -math.inf) or not ((t_1 <= -1e-218) or not (t_1 <= math.inf)):
		tmp = (-math.sqrt(2.0) / B_m) * math.sqrt((F * (-0.5 * ((B_m * B_m) / A))))
	else:
		tmp = math.sqrt((F * 2.0)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !((t_1 <= -1e-218) || !(t_1 <= Inf)))
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_0;
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~(((t_1 <= -1e-218) || ~((t_1 <= Inf)))))
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	else
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[Or[LessEqual[t$95$1, -1e-218], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]]], $MachinePrecision]], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 11.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 5.8

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

   7. Applied rewrites 5.8%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 14.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218 or +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 30.4%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 21.1

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

   5. Applied rewrites 21.1%

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

   7. Applied rewrites 21.1%

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

   9. Applied rewrites 27.8%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty \lor \neg \left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218} \lor \neg \left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty\right)\right):\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ t_2 := \frac{-\sqrt{2}}{B\_m}\\ t_3 := t\_2 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;t\_2 \cdot \sqrt{F \cdot \left(B\_m + A \cdot \left(1 + 0.5 \cdot \frac{A}{B\_m}\right)\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0)))
        (t_2 (/ (- (sqrt 2.0)) B_m))
        (t_3 (* t_2 (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 -1e-218)
       (* t_2 (sqrt (* F (+ B_m (* A (+ 1.0 (* 0.5 (/ A B_m))))))))
       (if (<= t_1 INFINITY) t_3 (/ (sqrt (* F 2.0)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double t_2 = -sqrt(2.0) / B_m;
	double t_3 = t_2 * sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= -1e-218) {
		tmp = t_2 * sqrt((F * (B_m + (A * (1.0 + (0.5 * (A / B_m)))))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

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 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_0;
	double t_2 = -Math.sqrt(2.0) / B_m;
	double t_3 = t_2 * Math.sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_1 <= -1e-218) {
		tmp = t_2 * Math.sqrt((F * (B_m + (A * (1.0 + (0.5 * (A / B_m)))))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_1 = math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_0
	t_2 = -math.sqrt(2.0) / B_m
	t_3 = t_2 * math.sqrt((F * (-0.5 * ((B_m * B_m) / A))))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_3
	elif t_1 <= -1e-218:
		tmp = t_2 * math.sqrt((F * (B_m + (A * (1.0 + (0.5 * (A / B_m)))))))
	elif t_1 <= math.inf:
		tmp = t_3
	else:
		tmp = math.sqrt((F * 2.0)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	t_2 = Float64(Float64(-sqrt(2.0)) / B_m)
	t_3 = Float64(t_2 * sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= -1e-218)
		tmp = Float64(t_2 * sqrt(Float64(F * Float64(B_m + Float64(A * Float64(1.0 + Float64(0.5 * Float64(A / B_m))))))));
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_0;
	t_2 = -sqrt(2.0) / B_m;
	t_3 = t_2 * sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_3;
	elseif (t_1 <= -1e-218)
		tmp = t_2 * sqrt((F * (B_m + (A * (1.0 + (0.5 * (A / B_m)))))));
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, -1e-218], N[(t$95$2 * N[Sqrt[N[(F * N[(B$95$m + N[(A * N[(1.0 + N[(0.5 * N[(A / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 11.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 5.8

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

   7. Applied rewrites 5.8%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 14.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 44.2

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

   7. Applied rewrites 44.2%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 34.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 16.8

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

   5. Applied rewrites 16.8%

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

   7. Applied rewrites 16.8%

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

   9. Applied rewrites 24.8%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A \cdot \left(1 + 0.5 \cdot \frac{A}{B}\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(A \cdot -4, C, B\_m \cdot B\_m\right)}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{t\_2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot t\_2}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \left(\sqrt{\mathsf{hypot}\left(B\_m, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_1 -1e-218)
     (*
      (* (sqrt (* F 2.0)) (- (sqrt (fma (* A -4.0) C (* B_m B_m)))))
      (/ (sqrt (+ (+ (hypot B_m (- A C)) A) C)) t_2))
     (if (<= t_1 INFINITY)
       (*
        (- (sqrt (* (* 2.0 F) t_2)))
        (/ (sqrt (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) t_2))
       (* (/ (- (sqrt 2.0)) B_m) (* (sqrt (+ (hypot B_m A) A)) (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_1 <= -1e-218) {
		tmp = (sqrt((F * 2.0)) * -sqrt(fma((A * -4.0), C, (B_m * B_m)))) * (sqrt(((hypot(B_m, (A - C)) + A) + C)) / t_2);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = -sqrt(((2.0 * F) * t_2)) * (sqrt(((C + (-0.5 * ((B_m * B_m) / A))) + C)) / t_2);
	} else {
		tmp = (-sqrt(2.0) / B_m) * (sqrt((hypot(B_m, A) + A)) * sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_1 <= -1e-218)
		tmp = Float64(Float64(sqrt(Float64(F * 2.0)) * Float64(-sqrt(fma(Float64(A * -4.0), C, Float64(B_m * B_m))))) * Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) / t_2));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * t_2))) * Float64(sqrt(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) / t_2));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * Float64(sqrt(Float64(hypot(B_m, A) + A)) * sqrt(F)));
	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[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-218], N[(N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(A * -4.0), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 49.8%

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

   4. Applied rewrites 70.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 78.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 78.5

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-*r* N/A

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

      23. lower-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 N/A

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

      26. lift-*.f64 78.5

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

   6. Applied rewrites 78.5%

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

   if -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.9%

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

   4. Applied rewrites 37.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.3

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

   7. Applied rewrites 32.3%

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

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

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 18.3

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

   7. Applied rewrites 18.3%

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

   9. Applied rewrites 27.5%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{\mathsf{hypot}\left(B, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(A \cdot -4, C, B\_m \cdot B\_m\right) \cdot 2} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{t\_2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot t\_2}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \left(\sqrt{\mathsf{hypot}\left(B\_m, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_1 -1e-218)
     (*
      (* (sqrt (* (fma (* A -4.0) C (* B_m B_m)) 2.0)) (- (sqrt F)))
      (/ (sqrt (+ (+ (hypot B_m (- A C)) A) C)) t_2))
     (if (<= t_1 INFINITY)
       (*
        (- (sqrt (* (* 2.0 F) t_2)))
        (/ (sqrt (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) t_2))
       (* (/ (- (sqrt 2.0)) B_m) (* (sqrt (+ (hypot B_m A) A)) (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_1 <= -1e-218) {
		tmp = (sqrt((fma((A * -4.0), C, (B_m * B_m)) * 2.0)) * -sqrt(F)) * (sqrt(((hypot(B_m, (A - C)) + A) + C)) / t_2);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = -sqrt(((2.0 * F) * t_2)) * (sqrt(((C + (-0.5 * ((B_m * B_m) / A))) + C)) / t_2);
	} else {
		tmp = (-sqrt(2.0) / B_m) * (sqrt((hypot(B_m, A) + A)) * sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_1 <= -1e-218)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(A * -4.0), C, Float64(B_m * B_m)) * 2.0)) * Float64(-sqrt(F))) * Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) / t_2));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * t_2))) * Float64(sqrt(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) / t_2));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * Float64(sqrt(Float64(hypot(B_m, A) + A)) * sqrt(F)));
	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[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-218], N[(N[(N[Sqrt[N[(N[(N[(A * -4.0), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 49.8%

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

   4. Applied rewrites 70.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 78.3%

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

   if -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.9%

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

   4. Applied rewrites 37.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.3

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

   7. Applied rewrites 32.3%

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

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

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 18.3

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

   7. Applied rewrites 18.3%

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

   9. Applied rewrites 27.5%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right) \cdot 2} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{\mathsf{hypot}\left(B, A\right) + A} \cdot \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{\left(2 \cdot F\right) \cdot t\_0}\\ \mathbf{if}\;B\_m \leq 2.65 \cdot 10^{-108}:\\ \;\;\;\;\left(-t\_1\right) \cdot \left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{{C}^{-1}}\right)\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{C + C}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))) (t_1 (sqrt (* (* 2.0 F) t_0))))
   (if (<= B_m 2.65e-108)
     (* (- t_1) (* (* -0.25 (/ (sqrt 2.0) A)) (sqrt (pow C -1.0))))
     (if (<= B_m 1.05e+38)
       (* t_1 (/ (sqrt (+ C C)) (- t_0)))
       (if (<= B_m 6.5e+97)
         (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))))
         (/ (sqrt (* F 2.0)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = sqrt(((2.0 * F) * t_0));
	double tmp;
	if (B_m <= 2.65e-108) {
		tmp = -t_1 * ((-0.25 * (sqrt(2.0) / A)) * sqrt(pow(C, -1.0)));
	} else if (B_m <= 1.05e+38) {
		tmp = t_1 * (sqrt((C + C)) / -t_0);
	} else if (B_m <= 6.5e+97) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = sqrt(Float64(Float64(2.0 * F) * t_0))
	tmp = 0.0
	if (B_m <= 2.65e-108)
		tmp = Float64(Float64(-t_1) * Float64(Float64(-0.25 * Float64(sqrt(2.0) / A)) * sqrt((C ^ -1.0))));
	elseif (B_m <= 1.05e+38)
		tmp = Float64(t_1 * Float64(sqrt(Float64(C + C)) / Float64(-t_0)));
	elseif (B_m <= 6.5e+97)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 2.65e-108], N[((-t$95$1) * N[(N[(-0.25 * N[(N[Sqrt[2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[C, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.05e+38], N[(t$95$1 * N[(N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.5e+97], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.64999999999999994e-108

   1. Initial program 18.5%

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 11.6

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{\color{blue}{\frac{1}{C}}}\right) \]

   7. Applied rewrites 11.6%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \color{blue}{\left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{\frac{1}{C}}\right)} \]

   if 2.64999999999999994e-108 < B < 1.05e38

   1. Initial program 53.0%

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(1 + \color{blue}{0} \cdot \frac{A}{C}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(1 + \color{blue}{0 \cdot \frac{A}{C}}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 34.0

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(1 + 0 \cdot \color{blue}{\frac{A}{C}}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 34.0%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(1 + 0 \cdot \frac{A}{C}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 1.05e38 < B < 6.4999999999999999e97

   1. Initial program 35.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 36.6

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

   7. Applied rewrites 36.6%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 24.2%

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

   if 6.4999999999999999e97 < B

   1. Initial program 3.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 46.8

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

   5. Applied rewrites 46.8%

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

   7. Applied rewrites 46.9%

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

   9. Applied rewrites 74.6%

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

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.65 \cdot 10^{-108}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{{C}^{-1}}\right)\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.25 \cdot 10^{-56}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\right) \cdot \left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{{C}^{-1}}\right)\\ \mathbf{elif}\;B\_m \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{-0.5 \cdot \frac{B\_m \cdot B\_m}{A}} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.25e-56)
   (*
    (- (sqrt (* (* 2.0 F) (fma -4.0 (* C A) (* B_m B_m)))))
    (* (* -0.25 (/ (sqrt 2.0) A)) (sqrt (pow C -1.0))))
   (if (<= B_m 6.5e+97)
     (* (/ (sqrt 2.0) B_m) (* (sqrt (* -0.5 (/ (* B_m B_m) A))) (- (sqrt F))))
     (/ (sqrt (* F 2.0)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.25e-56) {
		tmp = -sqrt(((2.0 * F) * fma(-4.0, (C * A), (B_m * B_m)))) * ((-0.25 * (sqrt(2.0) / A)) * sqrt(pow(C, -1.0)));
	} else if (B_m <= 6.5e+97) {
		tmp = (sqrt(2.0) / B_m) * (sqrt((-0.5 * ((B_m * B_m) / A))) * -sqrt(F));
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.25e-56)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * fma(-4.0, Float64(C * A), Float64(B_m * B_m))))) * Float64(Float64(-0.25 * Float64(sqrt(2.0) / A)) * sqrt((C ^ -1.0))));
	elseif (B_m <= 6.5e+97)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) * Float64(-sqrt(F))));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.25e-56], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * N[(N[(-0.25 * N[(N[Sqrt[2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[C, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.5e+97], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.24999999999999999e-56

   1. Initial program 19.9%

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 11.9

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{\color{blue}{\frac{1}{C}}}\right) \]

   7. Applied rewrites 11.9%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \color{blue}{\left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{\frac{1}{C}}\right)} \]

   if 1.24999999999999999e-56 < B < 6.4999999999999999e97

   1. Initial program 49.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 40.4

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

   7. Applied rewrites 40.4%

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

   9. Applied rewrites 42.5%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 23.8%

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

   if 6.4999999999999999e97 < B

   1. Initial program 3.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 46.8

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

   5. Applied rewrites 46.8%

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

   7. Applied rewrites 46.9%

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

   9. Applied rewrites 74.6%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.25 \cdot 10^{-56}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{{C}^{-1}}\right)\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{-0.5 \cdot \frac{B \cdot B}{A}} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := \sqrt{\left(2 \cdot F\right) \cdot t\_1}\\ \mathbf{if}\;B\_m \leq 1.35 \cdot 10^{-59}:\\ \;\;\;\;\left(-t\_2\right) \cdot \frac{\sqrt{\left(C + t\_0\right) + C}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;t\_2 \cdot \frac{\sqrt{C + C}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (* B_m B_m) A)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (sqrt (* (* 2.0 F) t_1))))
   (if (<= B_m 1.35e-59)
     (* (- t_2) (/ (sqrt (+ (+ C t_0) C)) t_1))
     (if (<= B_m 1.05e+38)
       (* t_2 (/ (sqrt (+ C C)) (- t_1)))
       (if (<= B_m 6.5e+97)
         (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F t_0)))
         (/ (sqrt (* F 2.0)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -0.5 * ((B_m * B_m) / A);
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = sqrt(((2.0 * F) * t_1));
	double tmp;
	if (B_m <= 1.35e-59) {
		tmp = -t_2 * (sqrt(((C + t_0) + C)) / t_1);
	} else if (B_m <= 1.05e+38) {
		tmp = t_2 * (sqrt((C + C)) / -t_1);
	} else if (B_m <= 6.5e+97) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * t_0));
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-0.5 * Float64(Float64(B_m * B_m) / A))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = sqrt(Float64(Float64(2.0 * F) * t_1))
	tmp = 0.0
	if (B_m <= 1.35e-59)
		tmp = Float64(Float64(-t_2) * Float64(sqrt(Float64(Float64(C + t_0) + C)) / t_1));
	elseif (B_m <= 1.05e+38)
		tmp = Float64(t_2 * Float64(sqrt(Float64(C + C)) / Float64(-t_1)));
	elseif (B_m <= 6.5e+97)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * t_0)));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 1.35e-59], N[((-t$95$2) * N[(N[Sqrt[N[(N[(C + t$95$0), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.05e+38], N[(t$95$2 * N[(N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.5e+97], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.3499999999999999e-59

   1. Initial program 19.4%

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 15.2

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

   7. Applied rewrites 15.2%

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

   if 1.3499999999999999e-59 < B < 1.05e38

   1. Initial program 65.8%

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(1 + \color{blue}{0} \cdot \frac{A}{C}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(1 + \color{blue}{0 \cdot \frac{A}{C}}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 39.8

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(1 + 0 \cdot \color{blue}{\frac{A}{C}}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 39.8%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(1 + 0 \cdot \frac{A}{C}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 1.05e38 < B < 6.4999999999999999e97

   1. Initial program 35.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. unpow2 N/A

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\sqrt{\color{blue}{A \cdot A} + {B}^{2}} + A\right)} \]

      11. unpow2 N/A

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\sqrt{A \cdot A + \color{blue}{B \cdot B}} + A\right)} \]

      12. lower-hypot.f64 36.6

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\color{blue}{\mathsf{hypot}\left(A, B\right)} + A\right)} \]

   7. Applied rewrites 36.6%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\mathsf{hypot}\left(A, B\right) + A\right)}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 24.2%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \]

   if 6.4999999999999999e97 < B

   1. Initial program 3.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 46.8

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.9%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.6%

      \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.35 \cdot 10^{-59}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (/ (sqrt (* F 2.0)) (- (sqrt B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * 2.0)) / -sqrt(B_m);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((f * 2.0d0)) / -sqrt(b_m)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((F * 2.0)) / -math.sqrt(B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F * 2.0)) / -sqrt(B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 22.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 14.5

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.5%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 14.5%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 18.5%

   \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]

10. Final simplification 18.5%

    \[\leadsto \frac{\sqrt{F \cdot 2}}{-\sqrt{B}} \]
11. Add Preprocessing

Alternative 17: 36.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt(F) * sqrt((2.0 / B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(f) * sqrt((2.0d0 / b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(F) * Math.sqrt((2.0 / B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(F) * math.sqrt((2.0 / B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(F) * sqrt((2.0 / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 14.5

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.5%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 14.5%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 18.5%

   \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]

10. Final simplification 18.5%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 18: 27.3% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* F (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F * Float64(2.0 / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 22.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 14.5

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.5%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 14.5%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 14.5%

   \[\leadsto -\sqrt{F \cdot \frac{2}{B}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))