ABCF->ab-angle a

Percentage Accurate: 19.0% → 57.7%

Time: 16.3s

Alternatives: 14

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: 19.0% 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: 57.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := \left(F \cdot 2\right) \cdot t\_1\\ t_3 := \left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\\ t_4 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ t_5 := t\_0 - {B\_m}^{2}\\ t_6 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot t\_3}}{t\_5}\\ t_7 := \frac{B\_m \cdot B\_m}{A}\\ \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot F} \cdot \sqrt{\left(\mathsf{fma}\left(-0.5, t\_7, C\right) + C\right) \cdot 2}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot t\_4}}{-t\_1}\\ \mathbf{elif}\;t\_6 \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(t\_7, -0.5, C \cdot 2\right) \cdot t\_3}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2}}{t\_1} \cdot \frac{\sqrt{t\_4}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* A 4.0)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (* (* F 2.0) t_1))
        (t_3 (* (* F (- (pow B_m 2.0) t_0)) 2.0))
        (t_4 (+ (+ (hypot (- A C) B_m) A) C))
        (t_5 (- t_0 (pow B_m 2.0)))
        (t_6
         (/
          (sqrt (* (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A)) t_3))
          t_5))
        (t_7 (/ (* B_m B_m) A)))
   (if (<= t_6 (- INFINITY))
     (/ (* (sqrt (* t_1 F)) (sqrt (* (+ (fma -0.5 t_7 C) C) 2.0))) t_5)
     (if (<= t_6 -4e-203)
       (/ (sqrt (* t_2 t_4)) (- t_1))
       (if (<= t_6 4e-48)
         (/ (sqrt (* (fma t_7 -0.5 (* C 2.0)) t_3)) t_5)
         (if (<= t_6 INFINITY)
           (* (/ (sqrt t_2) t_1) (/ (sqrt t_4) -1.0))
           (* (/ (* (- 2.0) F) (sqrt (* F 2.0))) (pow B_m -0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (A * 4.0);
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = (F * 2.0) * t_1;
	double t_3 = (F * (pow(B_m, 2.0) - t_0)) * 2.0;
	double t_4 = (hypot((A - C), B_m) + A) + C;
	double t_5 = t_0 - pow(B_m, 2.0);
	double t_6 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * t_3)) / t_5;
	double t_7 = (B_m * B_m) / A;
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = (sqrt((t_1 * F)) * sqrt(((fma(-0.5, t_7, C) + C) * 2.0))) / t_5;
	} else if (t_6 <= -4e-203) {
		tmp = sqrt((t_2 * t_4)) / -t_1;
	} else if (t_6 <= 4e-48) {
		tmp = sqrt((fma(t_7, -0.5, (C * 2.0)) * t_3)) / t_5;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = (sqrt(t_2) / t_1) * (sqrt(t_4) / -1.0);
	} else {
		tmp = ((-2.0 * F) / sqrt((F * 2.0))) * pow(B_m, -0.5);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(A * 4.0))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = Float64(Float64(F * 2.0) * t_1)
	t_3 = Float64(Float64(F * Float64((B_m ^ 2.0) - t_0)) * 2.0)
	t_4 = Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)
	t_5 = Float64(t_0 - (B_m ^ 2.0))
	t_6 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * t_3)) / t_5)
	t_7 = Float64(Float64(B_m * B_m) / A)
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(t_1 * F)) * sqrt(Float64(Float64(fma(-0.5, t_7, C) + C) * 2.0))) / t_5);
	elseif (t_6 <= -4e-203)
		tmp = Float64(sqrt(Float64(t_2 * t_4)) / Float64(-t_1));
	elseif (t_6 <= 4e-48)
		tmp = Float64(sqrt(Float64(fma(t_7, -0.5, Float64(C * 2.0)) * t_3)) / t_5);
	elseif (t_6 <= Inf)
		tmp = Float64(Float64(sqrt(t_2) / t_1) * Float64(sqrt(t_4) / -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0) * F) / sqrt(Float64(F * 2.0))) * (B_m ^ -0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * 4.0), $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[(N[(F * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]}, If[LessEqual[t$95$6, (-Infinity)], N[(N[(N[Sqrt[N[(t$95$1 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(-0.5 * t$95$7 + C), $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, -4e-203], N[(N[Sqrt[N[(t$95$2 * t$95$4), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$6, 4e-48], N[(N[Sqrt[N[(N[(t$95$7 * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[(N[Sqrt[t$95$2], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[Sqrt[t$95$4], $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-2.0) * F), $MachinePrecision] / N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 42.5%

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

   5. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.4

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

   7. Applied rewrites 26.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e-203

   1. Initial program 99.3%

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

   4. Applied rewrites 99.4%

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

   if -4.0000000000000001e-203 < (/.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))) < 3.9999999999999999e-48

   1. Initial program 7.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   if 3.9999999999999999e-48 < (/.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 42.9%

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

   4. Applied rewrites 93.9%

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

   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. 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 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 23.9%

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

   9. Applied rewrites 24.1%

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

   11. Applied rewrites 24.1%

       \[\leadsto {B}^{-0.5} \cdot \frac{0 - F \cdot 2}{\color{blue}{0 + \sqrt{F \cdot 2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := \left(F \cdot 2\right) \cdot t\_2\\ t_4 := C \cdot \left(A \cdot 4\right)\\ t_5 := t\_4 - {B\_m}^{2}\\ t_6 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_4\right)\right) \cdot 2\right)}}{t\_5}\\ t_7 := \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C\\ \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot F} \cdot \sqrt{t\_7 \cdot 2}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot t\_0}}{-t\_2}\\ \mathbf{elif}\;t\_6 \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{t\_1}{\sqrt{\left(t\_7 \cdot F\right) \cdot t\_1}}}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3}}{t\_2} \cdot \frac{\sqrt{t\_0}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ (hypot (- A C) B_m) A) C))
        (t_1 (fma (* -4.0 C) A (* B_m B_m)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m)))
        (t_3 (* (* F 2.0) t_2))
        (t_4 (* C (* A 4.0)))
        (t_5 (- t_4 (pow B_m 2.0)))
        (t_6
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_4)) 2.0)))
          t_5))
        (t_7 (+ (fma -0.5 (/ (* B_m B_m) A) C) C)))
   (if (<= t_6 (- INFINITY))
     (/ (* (sqrt (* t_2 F)) (sqrt (* t_7 2.0))) t_5)
     (if (<= t_6 -4e-203)
       (/ (sqrt (* t_3 t_0)) (- t_2))
       (if (<= t_6 4e-48)
         (/ (- (sqrt 2.0)) (/ t_1 (sqrt (* (* t_7 F) t_1))))
         (if (<= t_6 INFINITY)
           (* (/ (sqrt t_3) t_2) (/ (sqrt t_0) -1.0))
           (* (/ (* (- 2.0) F) (sqrt (* F 2.0))) (pow B_m -0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (hypot((A - C), B_m) + A) + C;
	double t_1 = fma((-4.0 * C), A, (B_m * B_m));
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = (F * 2.0) * t_2;
	double t_4 = C * (A * 4.0);
	double t_5 = t_4 - pow(B_m, 2.0);
	double t_6 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_4)) * 2.0))) / t_5;
	double t_7 = fma(-0.5, ((B_m * B_m) / A), C) + C;
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = (sqrt((t_2 * F)) * sqrt((t_7 * 2.0))) / t_5;
	} else if (t_6 <= -4e-203) {
		tmp = sqrt((t_3 * t_0)) / -t_2;
	} else if (t_6 <= 4e-48) {
		tmp = -sqrt(2.0) / (t_1 / sqrt(((t_7 * F) * t_1)));
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = (sqrt(t_3) / t_2) * (sqrt(t_0) / -1.0);
	} else {
		tmp = ((-2.0 * F) / sqrt((F * 2.0))) * pow(B_m, -0.5);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)
	t_1 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_3 = Float64(Float64(F * 2.0) * t_2)
	t_4 = Float64(C * Float64(A * 4.0))
	t_5 = Float64(t_4 - (B_m ^ 2.0))
	t_6 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_4)) * 2.0))) / t_5)
	t_7 = Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(t_2 * F)) * sqrt(Float64(t_7 * 2.0))) / t_5);
	elseif (t_6 <= -4e-203)
		tmp = Float64(sqrt(Float64(t_3 * t_0)) / Float64(-t_2));
	elseif (t_6 <= 4e-48)
		tmp = Float64(Float64(-sqrt(2.0)) / Float64(t_1 / sqrt(Float64(Float64(t_7 * F) * t_1))));
	elseif (t_6 <= Inf)
		tmp = Float64(Float64(sqrt(t_3) / t_2) * Float64(sqrt(t_0) / -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0) * F) / sqrt(Float64(F * 2.0))) * (B_m ^ -0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(F * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]}, If[LessEqual[t$95$6, (-Infinity)], N[(N[(N[Sqrt[N[(t$95$2 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$7 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, -4e-203], N[(N[Sqrt[N[(t$95$3 * t$95$0), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[t$95$6, 4e-48], N[((-N[Sqrt[2.0], $MachinePrecision]) / N[(t$95$1 / N[Sqrt[N[(N[(t$95$7 * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[(N[Sqrt[t$95$3], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-2.0) * F), $MachinePrecision] / N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 42.5%

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

   5. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.4

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

   7. Applied rewrites 26.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e-203

   1. Initial program 99.3%

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

   4. Applied rewrites 99.4%

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

   if -4.0000000000000001e-203 < (/.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))) < 3.9999999999999999e-48

   1. Initial program 7.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 8.6%

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

   6. Applied rewrites 11.4%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 31.0

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

   9. Applied rewrites 31.0%

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

   if 3.9999999999999999e-48 < (/.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 42.9%

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

   4. Applied rewrites 93.9%

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

   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. 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 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 23.9%

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

   9. Applied rewrites 24.1%

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

   11. Applied rewrites 24.1%

       \[\leadsto {B}^{-0.5} \cdot \frac{0 - F \cdot 2}{\color{blue}{0 + \sqrt{F \cdot 2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{hypot}\left(A - C, B\_m\right)\\ t_2 := \left(t\_1 + A\right) + C\\ t_3 := C \cdot \left(A \cdot 4\right)\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_3\right)\right) \cdot 2\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := -\sqrt{2}\\ t_6 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(\left(t\_1 + C\right) + A\right) \cdot F}{\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}} \cdot t\_5\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot t\_2}}{-t\_0}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{t\_5}{\frac{t\_6}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C\right) \cdot F\right) \cdot t\_6}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot F} \cdot \left(-\sqrt{t\_2 \cdot 2}\right)}{\left(C \cdot A\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (hypot (- A C) B_m))
        (t_2 (+ (+ t_1 A) C))
        (t_3 (* C (* A 4.0)))
        (t_4
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_3)) 2.0)))
          (- t_3 (pow B_m 2.0))))
        (t_5 (- (sqrt 2.0)))
        (t_6 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= t_4 (- INFINITY))
     (* (sqrt (/ (* (+ (+ t_1 C) A) F) (fma (* -4.0 A) C (* B_m B_m)))) t_5)
     (if (<= t_4 -4e-203)
       (/ (sqrt (* (* (* F 2.0) t_0) t_2)) (- t_0))
       (if (<= t_4 0.0)
         (/
          t_5
          (/ t_6 (sqrt (* (* (+ (fma -0.5 (/ (* B_m B_m) A) C) C) F) t_6))))
         (if (<= t_4 INFINITY)
           (/ (* (sqrt (* t_0 F)) (- (sqrt (* t_2 2.0)))) (* (* C A) -4.0))
           (* (/ (* (- 2.0) F) (sqrt (* F 2.0))) (pow B_m -0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = hypot((A - C), B_m);
	double t_2 = (t_1 + A) + C;
	double t_3 = C * (A * 4.0);
	double t_4 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_3)) * 2.0))) / (t_3 - pow(B_m, 2.0));
	double t_5 = -sqrt(2.0);
	double t_6 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = sqrt(((((t_1 + C) + A) * F) / fma((-4.0 * A), C, (B_m * B_m)))) * t_5;
	} else if (t_4 <= -4e-203) {
		tmp = sqrt((((F * 2.0) * t_0) * t_2)) / -t_0;
	} else if (t_4 <= 0.0) {
		tmp = t_5 / (t_6 / sqrt((((fma(-0.5, ((B_m * B_m) / A), C) + C) * F) * t_6)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((t_0 * F)) * -sqrt((t_2 * 2.0))) / ((C * A) * -4.0);
	} else {
		tmp = ((-2.0 * F) / sqrt((F * 2.0))) * pow(B_m, -0.5);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = hypot(Float64(A - C), B_m)
	t_2 = Float64(Float64(t_1 + A) + C)
	t_3 = Float64(C * Float64(A * 4.0))
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_3)) * 2.0))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = Float64(-sqrt(2.0))
	t_6 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(t_1 + C) + A) * F) / fma(Float64(-4.0 * A), C, Float64(B_m * B_m)))) * t_5);
	elseif (t_4 <= -4e-203)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * t_2)) / Float64(-t_0));
	elseif (t_4 <= 0.0)
		tmp = Float64(t_5 / Float64(t_6 / sqrt(Float64(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C) * F) * t_6))));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_0 * F)) * Float64(-sqrt(Float64(t_2 * 2.0)))) / Float64(Float64(C * A) * -4.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0) * F) / sqrt(Float64(F * 2.0))) * (B_m ^ -0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + A), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$6 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[Sqrt[N[(N[(N[(N[(t$95$1 + C), $MachinePrecision] + A), $MachinePrecision] * F), $MachinePrecision] / N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$4, -4e-203], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(t$95$5 / N[(t$95$6 / N[Sqrt[N[(N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(t$95$2 * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-2.0) * F), $MachinePrecision] / N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 42.6%

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

   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))) < -4.0000000000000001e-203

   1. Initial program 99.3%

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

   4. Applied rewrites 99.4%

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

   if -4.0000000000000001e-203 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 4.7%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.2

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

   9. Applied rewrites 32.2%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 48.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 94.3%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 94.3

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

   7. Applied rewrites 94.3%

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

   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. 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 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 23.9%

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

   9. Applied rewrites 24.1%

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

   11. Applied rewrites 24.1%

       \[\leadsto {B}^{-0.5} \cdot \frac{0 - F \cdot 2}{\color{blue}{0 + \sqrt{F \cdot 2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\right) + C\right) + A\right) \cdot F}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \left(-\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}\right)}{\left(C \cdot A\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\\ t_3 := t\_1 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot t\_2}}{t\_3}\\ t_5 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{t\_0 \cdot 2}\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C \cdot 2\right) \cdot t\_2}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot t\_5}}{t\_5} \cdot \frac{\sqrt{t\_0}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ (hypot (- A C) B_m) A) C))
        (t_1 (* C (* A 4.0)))
        (t_2 (* (* F (- (pow B_m 2.0) t_1)) 2.0))
        (t_3 (- t_1 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt (* (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A)) t_2))
          t_3))
        (t_5 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_4 -4e-203)
     (/
      (* (* (sqrt F) (sqrt (* t_0 2.0))) (sqrt (fma (* -4.0 C) A (* B_m B_m))))
      t_3)
     (if (<= t_4 4e-48)
       (/ (sqrt (* (fma (/ (* B_m B_m) A) -0.5 (* C 2.0)) t_2)) t_3)
       (if (<= t_4 INFINITY)
         (* (/ (sqrt (* (* F 2.0) t_5)) t_5) (/ (sqrt t_0) -1.0))
         (* (/ (* (- 2.0) F) (sqrt (* F 2.0))) (pow B_m -0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (hypot((A - C), B_m) + A) + C;
	double t_1 = C * (A * 4.0);
	double t_2 = (F * (pow(B_m, 2.0) - t_1)) * 2.0;
	double t_3 = t_1 - pow(B_m, 2.0);
	double t_4 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * t_2)) / t_3;
	double t_5 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_4 <= -4e-203) {
		tmp = ((sqrt(F) * sqrt((t_0 * 2.0))) * sqrt(fma((-4.0 * C), A, (B_m * B_m)))) / t_3;
	} else if (t_4 <= 4e-48) {
		tmp = sqrt((fma(((B_m * B_m) / A), -0.5, (C * 2.0)) * t_2)) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(((F * 2.0) * t_5)) / t_5) * (sqrt(t_0) / -1.0);
	} else {
		tmp = ((-2.0 * F) / sqrt((F * 2.0))) * pow(B_m, -0.5);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(Float64(F * Float64((B_m ^ 2.0) - t_1)) * 2.0)
	t_3 = Float64(t_1 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * t_2)) / t_3)
	t_5 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_4 <= -4e-203)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(Float64(t_0 * 2.0))) * sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)))) / t_3);
	elseif (t_4 <= 4e-48)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(C * 2.0)) * t_2)) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * t_5)) / t_5) * Float64(sqrt(t_0) / -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0) * F) / sqrt(Float64(F * 2.0))) * (B_m ^ -0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e-203], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 4e-48], N[(N[Sqrt[N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-2.0) * F), $MachinePrecision] / N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e-203

   1. Initial program 46.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 68.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 72.0%

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

   if -4.0000000000000001e-203 < (/.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))) < 3.9999999999999999e-48

   1. Initial program 7.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   if 3.9999999999999999e-48 < (/.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 42.9%

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

   4. Applied rewrites 93.9%

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

   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. 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 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 23.9%

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

   9. Applied rewrites 24.1%

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

   11. Applied rewrites 24.1%

       \[\leadsto {B}^{-0.5} \cdot \frac{0 - F \cdot 2}{\color{blue}{0 + \sqrt{F \cdot 2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := C \cdot \left(A \cdot 4\right)\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_3\right)\right) \cdot 2\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot F}}{t\_0} \cdot \left(-\sqrt{t\_5 \cdot 2}\right)\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{t\_1}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C\right) \cdot F\right) \cdot t\_1}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot t\_2}}{t\_2} \cdot \frac{\sqrt{t\_5}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (fma (* -4.0 C) A (* B_m B_m)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m)))
        (t_3 (* C (* A 4.0)))
        (t_4
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_3)) 2.0)))
          (- t_3 (pow B_m 2.0))))
        (t_5 (+ (+ (hypot (- A C) B_m) A) C)))
   (if (<= t_4 -4e-203)
     (* (/ (sqrt (* t_0 F)) t_0) (- (sqrt (* t_5 2.0))))
     (if (<= t_4 4e-48)
       (/
        (- (sqrt 2.0))
        (/ t_1 (sqrt (* (* (+ (fma -0.5 (/ (* B_m B_m) A) C) C) F) t_1))))
       (if (<= t_4 INFINITY)
         (* (/ (sqrt (* (* F 2.0) t_2)) t_2) (/ (sqrt t_5) -1.0))
         (* (/ (* (- 2.0) F) (sqrt (* F 2.0))) (pow B_m -0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = fma((-4.0 * C), A, (B_m * B_m));
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = C * (A * 4.0);
	double t_4 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_3)) * 2.0))) / (t_3 - pow(B_m, 2.0));
	double t_5 = (hypot((A - C), B_m) + A) + C;
	double tmp;
	if (t_4 <= -4e-203) {
		tmp = (sqrt((t_0 * F)) / t_0) * -sqrt((t_5 * 2.0));
	} else if (t_4 <= 4e-48) {
		tmp = -sqrt(2.0) / (t_1 / sqrt((((fma(-0.5, ((B_m * B_m) / A), C) + C) * F) * t_1)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(((F * 2.0) * t_2)) / t_2) * (sqrt(t_5) / -1.0);
	} else {
		tmp = ((-2.0 * F) / sqrt((F * 2.0))) * pow(B_m, -0.5);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_3 = Float64(C * Float64(A * 4.0))
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_3)) * 2.0))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)
	tmp = 0.0
	if (t_4 <= -4e-203)
		tmp = Float64(Float64(sqrt(Float64(t_0 * F)) / t_0) * Float64(-sqrt(Float64(t_5 * 2.0))));
	elseif (t_4 <= 4e-48)
		tmp = Float64(Float64(-sqrt(2.0)) / Float64(t_1 / sqrt(Float64(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C) * F) * t_1))));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * t_2)) / t_2) * Float64(sqrt(t_5) / -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0) * F) / sqrt(Float64(F * 2.0))) * (B_m ^ -0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, If[LessEqual[t$95$4, -4e-203], N[(N[(N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * (-N[Sqrt[N[(t$95$5 * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, 4e-48], N[((-N[Sqrt[2.0], $MachinePrecision]) / N[(t$95$1 / N[Sqrt[N[(N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[Sqrt[t$95$5], $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-2.0) * F), $MachinePrecision] / N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e-203

   1. Initial program 46.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 68.3%

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

   6. Applied rewrites 50.1%

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

   8. Applied rewrites 68.2%

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

   if -4.0000000000000001e-203 < (/.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))) < 3.9999999999999999e-48

   1. Initial program 7.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 8.6%

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

   6. Applied rewrites 11.4%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 31.0

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

   9. Applied rewrites 31.0%

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

   if 3.9999999999999999e-48 < (/.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 42.9%

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

   4. Applied rewrites 93.9%

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

   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. 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 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 23.9%

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

   9. Applied rewrites 24.1%

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

   11. Applied rewrites 24.1%

       \[\leadsto {B}^{-0.5} \cdot \frac{0 - F \cdot 2}{\color{blue}{0 + \sqrt{F \cdot 2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \left(-\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_2 := -\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot 2}\\ t_3 := C \cdot \left(A \cdot 4\right)\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_3\right)\right) \cdot 2\right)}}{t\_3 - {B\_m}^{2}}\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot F}}{t\_0} \cdot t\_2\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{t\_1}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C\right) \cdot F\right) \cdot t\_1}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot F}}{t\_1} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (fma (* -4.0 C) A (* B_m B_m)))
        (t_2 (- (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) 2.0))))
        (t_3 (* C (* A 4.0)))
        (t_4
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_3)) 2.0)))
          (- t_3 (pow B_m 2.0)))))
   (if (<= t_4 -4e-203)
     (* (/ (sqrt (* t_0 F)) t_0) t_2)
     (if (<= t_4 0.0)
       (/
        (- (sqrt 2.0))
        (/ t_1 (sqrt (* (* (+ (fma -0.5 (/ (* B_m B_m) A) C) C) F) t_1))))
       (if (<= t_4 INFINITY)
         (* (/ (sqrt (* t_1 F)) t_1) t_2)
         (* (/ (* (- 2.0) F) (sqrt (* F 2.0))) (pow B_m -0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = fma((-4.0 * C), A, (B_m * B_m));
	double t_2 = -sqrt((((hypot((A - C), B_m) + A) + C) * 2.0));
	double t_3 = C * (A * 4.0);
	double t_4 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_3)) * 2.0))) / (t_3 - pow(B_m, 2.0));
	double tmp;
	if (t_4 <= -4e-203) {
		tmp = (sqrt((t_0 * F)) / t_0) * t_2;
	} else if (t_4 <= 0.0) {
		tmp = -sqrt(2.0) / (t_1 / sqrt((((fma(-0.5, ((B_m * B_m) / A), C) + C) * F) * t_1)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((t_1 * F)) / t_1) * t_2;
	} else {
		tmp = ((-2.0 * F) / sqrt((F * 2.0))) * pow(B_m, -0.5);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_2 = Float64(-sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * 2.0)))
	t_3 = Float64(C * Float64(A * 4.0))
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_3)) * 2.0))) / Float64(t_3 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_4 <= -4e-203)
		tmp = Float64(Float64(sqrt(Float64(t_0 * F)) / t_0) * t_2);
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(-sqrt(2.0)) / Float64(t_1 / sqrt(Float64(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C) * F) * t_1))));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_1 * F)) / t_1) * t_2);
	else
		tmp = Float64(Float64(Float64(Float64(-2.0) * F) / sqrt(Float64(F * 2.0))) * (B_m ^ -0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$3 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e-203], N[(N[(N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[((-N[Sqrt[2.0], $MachinePrecision]) / N[(t$95$1 / N[Sqrt[N[(N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(t$95$1 * F), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[((-2.0) * F), $MachinePrecision] / N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e-203

   1. Initial program 46.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 68.3%

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

   6. Applied rewrites 50.1%

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

   8. Applied rewrites 68.2%

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

   if -4.0000000000000001e-203 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 4.7%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.2

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

   9. Applied rewrites 32.2%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 48.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 94.3%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 94.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 23.9%

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

   9. Applied rewrites 24.1%

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

   11. Applied rewrites 24.1%

       \[\leadsto {B}^{-0.5} \cdot \frac{0 - F \cdot 2}{\color{blue}{0 + \sqrt{F \cdot 2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \left(-\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \frac{\sqrt{t\_0 \cdot F}}{t\_0} \cdot \left(-\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot 2}\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-203}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{t\_0}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C\right) \cdot F\right) \cdot t\_0}}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m)))
        (t_1 (* C (* A 4.0)))
        (t_2
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_1)) 2.0)))
          (- t_1 (pow B_m 2.0))))
        (t_3
         (*
          (/ (sqrt (* t_0 F)) t_0)
          (- (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) 2.0))))))
   (if (<= t_2 -4e-203)
     t_3
     (if (<= t_2 0.0)
       (/
        (- (sqrt 2.0))
        (/ t_0 (sqrt (* (* (+ (fma -0.5 (/ (* B_m B_m) A) C) C) F) t_0))))
       (if (<= t_2 INFINITY)
         t_3
         (* (/ (* (- 2.0) F) (sqrt (* F 2.0))) (pow B_m -0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double t_1 = C * (A * 4.0);
	double t_2 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_1)) * 2.0))) / (t_1 - pow(B_m, 2.0));
	double t_3 = (sqrt((t_0 * F)) / t_0) * -sqrt((((hypot((A - C), B_m) + A) + C) * 2.0));
	double tmp;
	if (t_2 <= -4e-203) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = -sqrt(2.0) / (t_0 / sqrt((((fma(-0.5, ((B_m * B_m) / A), C) + C) * F) * t_0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = ((-2.0 * F) / sqrt((F * 2.0))) * pow(B_m, -0.5);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_1)) * 2.0))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = Float64(Float64(sqrt(Float64(t_0 * F)) / t_0) * Float64(-sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * 2.0))))
	tmp = 0.0
	if (t_2 <= -4e-203)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-sqrt(2.0)) / Float64(t_0 / sqrt(Float64(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C) * F) * t_0))));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(Float64(-2.0) * F) / sqrt(Float64(F * 2.0))) * (B_m ^ -0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * (-N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-203], t$95$3, If[LessEqual[t$95$2, 0.0], N[((-N[Sqrt[2.0], $MachinePrecision]) / N[(t$95$0 / N[Sqrt[N[(N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[(N[((-2.0) * F), $MachinePrecision] / N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[B$95$m, -0.5], $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))) < -4.0000000000000001e-203 or -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 47.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 72.8%

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

   if -4.0000000000000001e-203 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 4.7%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.2

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

   9. Applied rewrites 32.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 23.9%

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

   9. Applied rewrites 24.1%

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

   11. Applied rewrites 24.1%

       \[\leadsto {B}^{-0.5} \cdot \frac{0 - F \cdot 2}{\color{blue}{0 + \sqrt{F \cdot 2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{-\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -8}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{\left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+29)
   (/
    (sqrt 2.0)
    (/ (- (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* (* (* (* C C) F) A) -8.0))))
   (if (<= (pow B_m 2.0) 5e+184)
     (* (/ (- (sqrt 2.0)) B_m) (sqrt (* (* -0.5 (/ (* B_m B_m) A)) F)))
     (/ (- (sqrt F)) (sqrt (* 0.5 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e+29) {
		tmp = sqrt(2.0) / (-fma((-4.0 * C), A, (B_m * B_m)) / sqrt(((((C * C) * F) * A) * -8.0)));
	} else if (pow(B_m, 2.0) <= 5e+184) {
		tmp = (-sqrt(2.0) / B_m) * sqrt(((-0.5 * ((B_m * B_m) / A)) * F));
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+29)
		tmp = Float64(sqrt(2.0) / Float64(Float64(-fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) / sqrt(Float64(Float64(Float64(Float64(C * C) * F) * A) * -8.0))));
	elseif ((B_m ^ 2.0) <= 5e+184)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(Float64(-0.5 * Float64(Float64(B_m * B_m) / A)) * F)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+29], N[(N[Sqrt[2.0], $MachinePrecision] / N[((-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]) / N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+184], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000001e29

   1. Initial program 28.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 41.2%

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

   6. Applied rewrites 33.4%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.3

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

   9. Applied rewrites 17.3%

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

   if 5.0000000000000001e29 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e184

   1. Initial program 14.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.1%

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

   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 15.0

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

   7. Applied rewrites 15.0%

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

   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 24.8%

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

   if 4.9999999999999999e184 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 24.5

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

   5. Applied rewrites 24.5%

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

   7. Applied rewrites 24.6%

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

   9. Applied rewrites 24.6%

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

   11. Applied rewrites 31.7%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -8}}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{\left(-0.5 \cdot \frac{B \cdot B}{A}\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{\left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+184)
   (* (/ (- (sqrt 2.0)) B_m) (sqrt (* (* -0.5 (/ (* B_m B_m) A)) F)))
   (/ (- (sqrt F)) (sqrt (* 0.5 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e+184) {
		tmp = (-sqrt(2.0) / B_m) * sqrt(((-0.5 * ((B_m * B_m) / A)) * F));
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 5d+184) then
        tmp = (-sqrt(2.0d0) / b_m) * sqrt((((-0.5d0) * ((b_m * b_m) / a)) * f))
    else
        tmp = -sqrt(f) / sqrt((0.5d0 * b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e+184) {
		tmp = (-Math.sqrt(2.0) / B_m) * Math.sqrt(((-0.5 * ((B_m * B_m) / A)) * F));
	} else {
		tmp = -Math.sqrt(F) / Math.sqrt((0.5 * B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e+184:
		tmp = (-math.sqrt(2.0) / B_m) * math.sqrt(((-0.5 * ((B_m * B_m) / A)) * F))
	else:
		tmp = -math.sqrt(F) / math.sqrt((0.5 * B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+184)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(Float64(-0.5 * Float64(Float64(B_m * B_m) / A)) * F)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e+184)
		tmp = (-sqrt(2.0) / B_m) * sqrt(((-0.5 * ((B_m * B_m) / A)) * F));
	else
		tmp = -sqrt(F) / sqrt((0.5 * B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+184], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e184

   1. Initial program 25.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 39.1%

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

   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 7.5

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

   7. Applied rewrites 7.5%

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

   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 10.9%

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

   if 4.9999999999999999e184 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 24.5

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

   5. Applied rewrites 24.5%

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

   7. Applied rewrites 24.6%

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

   9. Applied rewrites 24.6%

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

   11. Applied rewrites 31.7%

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

4. Final simplification 18.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{\left(-0.5 \cdot \frac{B \cdot B}{A}\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{t\_0}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C\right) \cdot F\right) \cdot t\_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B\_m}^{-0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 3.1e+83)
     (/
      (- (sqrt 2.0))
      (/ t_0 (sqrt (* (* (+ (fma -0.5 (/ (* B_m B_m) A) C) C) F) t_0))))
     (* (/ (* (- 2.0) F) (sqrt (* F 2.0))) (pow B_m -0.5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 3.1e+83) {
		tmp = -sqrt(2.0) / (t_0 / sqrt((((fma(-0.5, ((B_m * B_m) / A), C) + C) * F) * t_0)));
	} else {
		tmp = ((-2.0 * F) / sqrt((F * 2.0))) * pow(B_m, -0.5);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.1e+83)
		tmp = Float64(Float64(-sqrt(2.0)) / Float64(t_0 / sqrt(Float64(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C) * F) * t_0))));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0) * F) / sqrt(Float64(F * 2.0))) * (B_m ^ -0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.1e+83], N[((-N[Sqrt[2.0], $MachinePrecision]) / N[(t$95$0 / N[Sqrt[N[(N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-2.0) * F), $MachinePrecision] / N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.09999999999999992e83

   1. Initial program 21.2%

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

      1. lift-sqrt.f64 N/A

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

      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 33.8%

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

   6. Applied rewrites 25.5%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.8

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

   9. Applied rewrites 17.8%

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

   if 3.09999999999999992e83 < B

   1. Initial program 15.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-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 55.1

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

   5. Applied rewrites 55.1%

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

   7. Applied rewrites 73.6%

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

   9. Applied rewrites 74.1%

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

   11. Applied rewrites 74.1%

       \[\leadsto {B}^{-0.5} \cdot \frac{0 - F \cdot 2}{\color{blue}{0 + \sqrt{F \cdot 2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2\right) \cdot F}{\sqrt{F \cdot 2}} \cdot {B}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{t\_0}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C\right) \cdot F\right) \cdot t\_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 3.1e+83)
     (/
      (- (sqrt 2.0))
      (/ t_0 (sqrt (* (* (+ (fma -0.5 (/ (* B_m B_m) A) C) C) F) t_0))))
     (/ (- (sqrt F)) (sqrt (* 0.5 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 3.1e+83) {
		tmp = -sqrt(2.0) / (t_0 / sqrt((((fma(-0.5, ((B_m * B_m) / A), C) + C) * F) * t_0)));
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.1e+83)
		tmp = Float64(Float64(-sqrt(2.0)) / Float64(t_0 / sqrt(Float64(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C) * F) * t_0))));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.1e+83], N[((-N[Sqrt[2.0], $MachinePrecision]) / N[(t$95$0 / N[Sqrt[N[(N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.09999999999999992e83

   1. Initial program 21.2%

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

      1. lift-sqrt.f64 N/A

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

      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 33.8%

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

   6. Applied rewrites 25.5%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.8

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

   9. Applied rewrites 17.8%

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

   if 3.09999999999999992e83 < B

   1. Initial program 15.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-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 55.1

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

   5. Applied rewrites 55.1%

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

   7. Applied rewrites 55.4%

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

   9. Applied rewrites 55.4%

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

   11. Applied rewrites 74.0%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_0}{\frac{t\_1}{\sqrt{\left(\left(\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C\right) \cdot F\right) \cdot t\_1}}}\\ \mathbf{elif}\;B\_m \leq 2.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{t\_0}{B\_m} \cdot \sqrt{\left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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 (- (sqrt 2.0))) (t_1 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 3e+15)
     (/ t_0 (/ t_1 (sqrt (* (* (+ (* (fma 0.0 (/ A C) 1.0) C) C) F) t_1))))
     (if (<= B_m 2.3e+92)
       (* (/ t_0 B_m) (sqrt (* (* -0.5 (/ (* B_m B_m) A)) F)))
       (/ (- (sqrt F)) (sqrt (* 0.5 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 3e+15) {
		tmp = t_0 / (t_1 / sqrt(((((fma(0.0, (A / C), 1.0) * C) + C) * F) * t_1)));
	} else if (B_m <= 2.3e+92) {
		tmp = (t_0 / B_m) * sqrt(((-0.5 * ((B_m * B_m) / A)) * F));
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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(-sqrt(2.0))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3e+15)
		tmp = Float64(t_0 / Float64(t_1 / sqrt(Float64(Float64(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C) * F) * t_1))));
	elseif (B_m <= 2.3e+92)
		tmp = Float64(Float64(t_0 / B_m) * sqrt(Float64(Float64(-0.5 * Float64(Float64(B_m * B_m) / A)) * F)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * 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[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3e+15], N[(t$95$0 / N[(t$95$1 / N[Sqrt[N[(N[(N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.3e+92], N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3e15

   1. Initial program 21.6%

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

      1. lift-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 34.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 14.6

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

   9. Applied rewrites 14.6%

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

   if 3e15 < B < 2.29999999999999998e92

   1. Initial program 21.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.4%

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

   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 22.7

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

   7. Applied rewrites 22.7%

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

   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 29.5%

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

   if 2.29999999999999998e92 < B

   1. Initial program 13.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.5%

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

   9. Applied rewrites 55.5%

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

   11. Applied rewrites 75.1%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{\left(-0.5 \cdot \frac{B \cdot B}{A}\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.5% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(f) / sqrt((0.5d0 * b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(F) / Math.sqrt((0.5 * B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(F) / math.sqrt((0.5 * B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(F) / sqrt((0.5 * B_m));
end

Wolfram

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

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-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 11.7

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

5. Applied rewrites 11.7%

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

7. Applied rewrites 11.7%

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

9. Applied rewrites 11.7%

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

11. Applied rewrites 14.9%

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

12. Final simplification 14.9%

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

Alternative 14: 26.9% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-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 11.7

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

5. Applied rewrites 11.7%

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

7. Applied rewrites 11.7%

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

9. Applied rewrites 11.7%

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

10. Final simplification 11.7%

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

Reproduce

herbie shell --seed 2024268 
(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))))