ABCF->ab-angle a

Percentage Accurate: 19.1% → 49.1%

Time: 37.8s

Alternatives: 18

Speedup: 3.1×

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: 19.1% 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: 49.1% accurate, 0.4× speedup

Math

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

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0))))
        (t_1 (hypot B (- A C)))
        (t_2 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_2)))
   (if (<= t_3 0.0)
     (/ (* (sqrt t_0) (* (sqrt (+ A (+ C t_1))) (- (sqrt (* 2.0 F))))) t_2)
     (if (<= t_3 INFINITY)
       (/ (- (* (sqrt (+ C (+ A t_1))) (sqrt (* 2.0 (* F t_0))))) t_2)
       (* (sqrt (/ F B)) (- (sqrt 2.0)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = hypot(B, (A - C));
	double t_2 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = (sqrt(t_0) * (sqrt((A + (C + t_1))) * -sqrt((2.0 * F)))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = -(sqrt((C + (A + t_1))) * sqrt((2.0 * (F * t_0)))) / t_2;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[(A + N[(C + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[((-N[(N[Sqrt[N[(C + N[(A + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$2), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -0.0

   1. Initial program 25.2%

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

   3. Applied egg-rr 48.1%

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

      1. *-commutative 48.1%

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

      2. associate-*r* 48.1%

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

      3. +-commutative 48.1%

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

   5. Simplified 48.1%

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

      1. *-commutative 48.1%

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

      2. sqrt-prod 59.1%

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

      3. *-commutative 59.1%

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

      4. +-commutative 59.1%

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

      5. associate-+l+ 59.4%

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

      6. *-commutative 59.4%

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

   7. Applied egg-rr 59.4%

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

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

   1. Initial program 29.7%

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

   3. Applied egg-rr 71.7%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 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. Taylor expanded in C around 0 0.5%

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

      1. associate-*l* 0.5%

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

      2. +-commutative 0.5%

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

   4. Simplified 0.5%

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

   5. Taylor expanded in A around 0 12.4%

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

      1. mul-1-neg 12.4%

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

   7. Simplified 12.4%

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

4. Final simplification 41.5%

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

Alternative 2: 37.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \sqrt{2 \cdot F}\\ t_2 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_3 := \sqrt{t_2}\\ t_4 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot t_0\right) \cdot t_2}}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(t_1 \cdot \left(-\sqrt{A + A}\right)\right)}{t_4}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-92}:\\ \;\;\;\;\frac{B \cdot \left(t_3 \cdot \left(-t_1\right)\right)}{t_4}\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;\frac{\sqrt{A \cdot 0 + 2 \cdot C} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_0\right)}\right)}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;\frac{\left(t_3 \cdot t_1\right) \cdot \left(\left(-B\right) - -2 \cdot \frac{A \cdot C}{B}\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0))))
        (t_1 (sqrt (* 2.0 F)))
        (t_2 (+ A (+ C (hypot B (- A C)))))
        (t_3 (sqrt t_2))
        (t_4 (- (pow B 2.0) (* (* 4.0 A) C))))
   (if (<= (pow B 2.0) 5e-307)
     (/ (- (sqrt (* (* (* 2.0 F) t_0) t_2))) t_0)
     (if (<= (pow B 2.0) 2e-259)
       (/ (* (sqrt (fma B B (* A (* C -4.0)))) (* t_1 (- (sqrt (+ A A))))) t_4)
       (if (<= (pow B 2.0) 2e-92)
         (/ (* B (* t_3 (- t_1))) t_4)
         (if (<= (pow B 2.0) 0.005)
           (/
            (* (sqrt (+ (* A 0.0) (* 2.0 C))) (- (sqrt (* 2.0 (* F t_0)))))
            t_0)
           (if (<= (pow B 2.0) 1e+271)
             (/ (* (* t_3 t_1) (- (- B) (* -2.0 (/ (* A C) B)))) t_4)
             (* (sqrt (/ F B)) (- (sqrt 2.0))))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double t_1 = sqrt((2.0 * F));
	double t_2 = A + (C + hypot(B, (A - C)));
	double t_3 = sqrt(t_2);
	double t_4 = pow(B, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (pow(B, 2.0) <= 5e-307) {
		tmp = -sqrt((((2.0 * F) * t_0) * t_2)) / t_0;
	} else if (pow(B, 2.0) <= 2e-259) {
		tmp = (sqrt(fma(B, B, (A * (C * -4.0)))) * (t_1 * -sqrt((A + A)))) / t_4;
	} else if (pow(B, 2.0) <= 2e-92) {
		tmp = (B * (t_3 * -t_1)) / t_4;
	} else if (pow(B, 2.0) <= 0.005) {
		tmp = (sqrt(((A * 0.0) + (2.0 * C))) * -sqrt((2.0 * (F * t_0)))) / t_0;
	} else if (pow(B, 2.0) <= 1e+271) {
		tmp = ((t_3 * t_1) * (-B - (-2.0 * ((A * C) / B)))) / t_4;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_1 = sqrt(Float64(2.0 * F))
	t_2 = Float64(A + Float64(C + hypot(B, Float64(A - C))))
	t_3 = sqrt(t_2)
	t_4 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e-307)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * F) * t_0) * t_2))) / t_0);
	elseif ((B ^ 2.0) <= 2e-259)
		tmp = Float64(Float64(sqrt(fma(B, B, Float64(A * Float64(C * -4.0)))) * Float64(t_1 * Float64(-sqrt(Float64(A + A))))) / t_4);
	elseif ((B ^ 2.0) <= 2e-92)
		tmp = Float64(Float64(B * Float64(t_3 * Float64(-t_1))) / t_4);
	elseif ((B ^ 2.0) <= 0.005)
		tmp = Float64(Float64(sqrt(Float64(Float64(A * 0.0) + Float64(2.0 * C))) * Float64(-sqrt(Float64(2.0 * Float64(F * t_0))))) / t_0);
	elseif ((B ^ 2.0) <= 1e+271)
		tmp = Float64(Float64(Float64(t_3 * t_1) * Float64(Float64(-B) - Float64(-2.0 * Float64(Float64(A * C) / B)))) / t_4);
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-307], N[((-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-259], N[(N[(N[Sqrt[N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * (-N[Sqrt[N[(A + A), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-92], N[(N[(B * N[(t$95$3 * (-t$95$1)), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 0.005], N[(N[(N[Sqrt[N[(N[(A * 0.0), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+271], N[(N[(N[(t$95$3 * t$95$1), $MachinePrecision] * N[((-B) - N[(-2.0 * N[(N[(A * C), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (pow.f64 B 2) < 5.00000000000000014e-307

   1. Initial program 16.3%

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

   3. Simplified 28.7%

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

   if 5.00000000000000014e-307 < (pow.f64 B 2) < 2.0000000000000001e-259

   1. Initial program 11.3%

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

   3. Applied egg-rr 7.0%

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

      1. *-commutative 7.0%

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

      2. associate-*r* 7.0%

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

      3. +-commutative 7.0%

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

   5. Simplified 7.0%

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

      1. *-commutative 7.0%

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

      2. sqrt-prod 31.6%

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

      3. *-commutative 31.6%

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

      4. +-commutative 31.6%

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

      5. associate-+l+ 32.8%

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

      6. *-commutative 32.8%

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

   7. Applied egg-rr 32.8%

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

   8. Taylor expanded in B around 0 48.7%

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

   if 2.0000000000000001e-259 < (pow.f64 B 2) < 1.99999999999999998e-92

   1. Initial program 24.3%

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

   3. Applied egg-rr 36.2%

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

      1. *-commutative 36.2%

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

      2. associate-*r* 36.2%

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

      3. +-commutative 36.2%

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

   5. Simplified 36.2%

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

      1. *-commutative 36.2%

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

      2. sqrt-prod 45.8%

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

      3. *-commutative 45.8%

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

      4. +-commutative 45.8%

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

      5. associate-+l+ 47.0%

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

      6. *-commutative 47.0%

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

   7. Applied egg-rr 47.0%

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

   8. Taylor expanded in B around inf 21.8%

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

   if 1.99999999999999998e-92 < (pow.f64 B 2) < 0.0050000000000000001

   1. Initial program 12.1%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in C around inf 12.7%

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

      1. sqrt-prod 30.2%

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

      2. *-commutative 30.2%

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

      3. associate-+r+ 44.9%

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

      4. +-commutative 44.9%

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

      5. *-commutative 44.9%

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

      6. *-un-lft-identity 44.9%

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

      7. metadata-eval 44.9%

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

      8. distribute-rgt-out 44.9%

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

      9. metadata-eval 44.9%

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

      10. metadata-eval 44.9%

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

      11. associate-*r* 44.9%

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

      12. *-commutative 44.9%

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

      13. *-commutative 44.9%

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

   6. Applied egg-rr 44.9%

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

   if 0.0050000000000000001 < (pow.f64 B 2) < 9.99999999999999953e270

   1. Initial program 26.8%

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

   3. Applied egg-rr 48.4%

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

      1. *-commutative 48.4%

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

      2. associate-*r* 48.4%

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

      3. +-commutative 48.4%

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

   5. Simplified 48.4%

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

      1. *-commutative 48.4%

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

      2. sqrt-prod 62.1%

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

      3. *-commutative 62.1%

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

      4. +-commutative 62.1%

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

      5. associate-+l+ 61.9%

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

      6. *-commutative 61.9%

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

   7. Applied egg-rr 61.9%

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

   8. Taylor expanded in B around inf 32.2%

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

   if 9.99999999999999953e270 < (pow.f64 B 2)

   1. Initial program 3.1%

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

   2. Taylor expanded in C around 0 3.1%

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

      1. associate-*l* 3.1%

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

      2. +-commutative 3.1%

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

   4. Simplified 3.1%

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

   5. Taylor expanded in A around 0 21.9%

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

      1. mul-1-neg 21.9%

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

   7. Simplified 21.9%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(\sqrt{2 \cdot F} \cdot \left(-\sqrt{A + A}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-92}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;\frac{\sqrt{A \cdot 0 + 2 \cdot C} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;\frac{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot F}\right) \cdot \left(\left(-B\right) - -2 \cdot \frac{A \cdot C}{B}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 3: 37.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot F}\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_4 := \sqrt{t_2}\\ \mathbf{if}\;{B}^{2} \leq 10^{-270}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot t_3\right) \cdot t_2}}{t_3}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-92}:\\ \;\;\;\;\frac{B \cdot \left(t_4 \cdot \left(-t_0\right)\right)}{t_1}\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;\frac{\sqrt{A \cdot 0 + 2 \cdot C} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_3\right)}\right)}{t_3}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;\frac{\left(t_4 \cdot t_0\right) \cdot \left(\left(-B\right) - -2 \cdot \frac{A \cdot C}{B}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 F)))
        (t_1 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_2 (+ A (+ C (hypot B (- A C)))))
        (t_3 (fma B B (* C (* A -4.0))))
        (t_4 (sqrt t_2)))
   (if (<= (pow B 2.0) 1e-270)
     (/ (- (sqrt (* (* (* 2.0 F) t_3) t_2))) t_3)
     (if (<= (pow B 2.0) 2e-92)
       (/ (* B (* t_4 (- t_0))) t_1)
       (if (<= (pow B 2.0) 0.005)
         (/
          (* (sqrt (+ (* A 0.0) (* 2.0 C))) (- (sqrt (* 2.0 (* F t_3)))))
          t_3)
         (if (<= (pow B 2.0) 1e+271)
           (/ (* (* t_4 t_0) (- (- B) (* -2.0 (/ (* A C) B)))) t_1)
           (* (sqrt (/ F B)) (- (sqrt 2.0)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((2.0 * F));
	double t_1 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_2 = A + (C + hypot(B, (A - C)));
	double t_3 = fma(B, B, (C * (A * -4.0)));
	double t_4 = sqrt(t_2);
	double tmp;
	if (pow(B, 2.0) <= 1e-270) {
		tmp = -sqrt((((2.0 * F) * t_3) * t_2)) / t_3;
	} else if (pow(B, 2.0) <= 2e-92) {
		tmp = (B * (t_4 * -t_0)) / t_1;
	} else if (pow(B, 2.0) <= 0.005) {
		tmp = (sqrt(((A * 0.0) + (2.0 * C))) * -sqrt((2.0 * (F * t_3)))) / t_3;
	} else if (pow(B, 2.0) <= 1e+271) {
		tmp = ((t_4 * t_0) * (-B - (-2.0 * ((A * C) / B)))) / t_1;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = sqrt(Float64(2.0 * F))
	t_1 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(A + Float64(C + hypot(B, Float64(A - C))))
	t_3 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_4 = sqrt(t_2)
	tmp = 0.0
	if ((B ^ 2.0) <= 1e-270)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * F) * t_3) * t_2))) / t_3);
	elseif ((B ^ 2.0) <= 2e-92)
		tmp = Float64(Float64(B * Float64(t_4 * Float64(-t_0))) / t_1);
	elseif ((B ^ 2.0) <= 0.005)
		tmp = Float64(Float64(sqrt(Float64(Float64(A * 0.0) + Float64(2.0 * C))) * Float64(-sqrt(Float64(2.0 * Float64(F * t_3))))) / t_3);
	elseif ((B ^ 2.0) <= 1e+271)
		tmp = Float64(Float64(Float64(t_4 * t_0) * Float64(Float64(-B) - Float64(-2.0 * Float64(Float64(A * C) / B)))) / t_1);
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$2], $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-270], N[((-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-92], N[(N[(B * N[(t$95$4 * (-t$95$0)), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 0.005], N[(N[(N[Sqrt[N[(N[(A * 0.0), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+271], N[(N[(N[(t$95$4 * t$95$0), $MachinePrecision] * N[((-B) - N[(-2.0 * N[(N[(A * C), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B 2) < 1e-270

   1. Initial program 15.7%

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

   3. Simplified 26.8%

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

   if 1e-270 < (pow.f64 B 2) < 1.99999999999999998e-92

   1. Initial program 23.8%

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

   3. Applied egg-rr 35.2%

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

      1. *-commutative 35.2%

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

      2. associate-*r* 35.3%

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

      3. +-commutative 35.3%

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

   5. Simplified 35.3%

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

      1. *-commutative 35.3%

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

      2. sqrt-prod 44.5%

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

      3. *-commutative 44.5%

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

      4. +-commutative 44.5%

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

      5. associate-+l+ 46.1%

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

      6. *-commutative 46.1%

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

   7. Applied egg-rr 46.1%

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

   8. Taylor expanded in B around inf 21.3%

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

   if 1.99999999999999998e-92 < (pow.f64 B 2) < 0.0050000000000000001

   1. Initial program 12.1%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in C around inf 12.7%

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

      1. sqrt-prod 30.2%

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

      2. *-commutative 30.2%

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

      3. associate-+r+ 44.9%

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

      4. +-commutative 44.9%

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

      5. *-commutative 44.9%

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

      6. *-un-lft-identity 44.9%

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

      7. metadata-eval 44.9%

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

      8. distribute-rgt-out 44.9%

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

      9. metadata-eval 44.9%

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

      10. metadata-eval 44.9%

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

      11. associate-*r* 44.9%

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

      12. *-commutative 44.9%

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

      13. *-commutative 44.9%

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

   6. Applied egg-rr 44.9%

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

   if 0.0050000000000000001 < (pow.f64 B 2) < 9.99999999999999953e270

   1. Initial program 26.8%

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

   3. Applied egg-rr 48.4%

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

      1. *-commutative 48.4%

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

      2. associate-*r* 48.4%

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

      3. +-commutative 48.4%

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

   5. Simplified 48.4%

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

      1. *-commutative 48.4%

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

      2. sqrt-prod 62.1%

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

      3. *-commutative 62.1%

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

      4. +-commutative 62.1%

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

      5. associate-+l+ 61.9%

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

      6. *-commutative 61.9%

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

   7. Applied egg-rr 61.9%

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

   8. Taylor expanded in B around inf 32.2%

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

   if 9.99999999999999953e270 < (pow.f64 B 2)

   1. Initial program 3.1%

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

   2. Taylor expanded in C around 0 3.1%

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

      1. associate-*l* 3.1%

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

      2. +-commutative 3.1%

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

   4. Simplified 3.1%

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

   5. Taylor expanded in A around 0 21.9%

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

      1. mul-1-neg 21.9%

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

   7. Simplified 21.9%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-270}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-92}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;\frac{\sqrt{A \cdot 0 + 2 \cdot C} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;\frac{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot F}\right) \cdot \left(\left(-B\right) - -2 \cdot \frac{A \cdot C}{B}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 4: 36.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{A \cdot 0 + 2 \cdot C} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_0\right)}\right)}{t_0}\\ t_2 := \frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;t_1\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0))))
        (t_1
         (/
          (* (sqrt (+ (* A 0.0) (* 2.0 C))) (- (sqrt (* 2.0 (* F t_0)))))
          t_0))
        (t_2
         (/
          (* B (* (sqrt (+ A (+ C (hypot B (- A C))))) (- (sqrt (* 2.0 F)))))
          (- (pow B 2.0) (* (* 4.0 A) C)))))
   (if (<= (pow B 2.0) 4e-289)
     t_1
     (if (<= (pow B 2.0) 2e-92)
       t_2
       (if (<= (pow B 2.0) 0.005)
         t_1
         (if (<= (pow B 2.0) 1e+271)
           t_2
           (* (sqrt (/ F B)) (- (sqrt 2.0)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double t_1 = (sqrt(((A * 0.0) + (2.0 * C))) * -sqrt((2.0 * (F * t_0)))) / t_0;
	double t_2 = (B * (sqrt((A + (C + hypot(B, (A - C))))) * -sqrt((2.0 * F)))) / (pow(B, 2.0) - ((4.0 * A) * C));
	double tmp;
	if (pow(B, 2.0) <= 4e-289) {
		tmp = t_1;
	} else if (pow(B, 2.0) <= 2e-92) {
		tmp = t_2;
	} else if (pow(B, 2.0) <= 0.005) {
		tmp = t_1;
	} else if (pow(B, 2.0) <= 1e+271) {
		tmp = t_2;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(Float64(sqrt(Float64(Float64(A * 0.0) + Float64(2.0 * C))) * Float64(-sqrt(Float64(2.0 * Float64(F * t_0))))) / t_0)
	t_2 = Float64(Float64(B * Float64(sqrt(Float64(A + Float64(C + hypot(B, Float64(A - C))))) * Float64(-sqrt(Float64(2.0 * F))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)))
	tmp = 0.0
	if ((B ^ 2.0) <= 4e-289)
		tmp = t_1;
	elseif ((B ^ 2.0) <= 2e-92)
		tmp = t_2;
	elseif ((B ^ 2.0) <= 0.005)
		tmp = t_1;
	elseif ((B ^ 2.0) <= 1e+271)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(A * 0.0), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 4e-289], t$95$1, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-92], t$95$2, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 0.005], t$95$1, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+271], t$95$2, N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 4e-289 or 1.99999999999999998e-92 < (pow.f64 B 2) < 0.0050000000000000001

   1. Initial program 15.3%

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

   3. Simplified 26.4%

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

   4. Taylor expanded in C around inf 16.6%

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

      1. sqrt-prod 21.6%

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

      2. *-commutative 21.6%

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

      3. associate-+r+ 30.7%

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

      4. +-commutative 30.7%

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

      5. *-commutative 30.7%

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

      6. *-un-lft-identity 30.7%

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

      7. metadata-eval 30.7%

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

      8. distribute-rgt-out 30.7%

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

      9. metadata-eval 30.7%

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

      10. metadata-eval 30.7%

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

      11. associate-*r* 30.7%

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

      12. *-commutative 30.7%

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

      13. *-commutative 30.7%

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

   6. Applied egg-rr 30.7%

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

   if 4e-289 < (pow.f64 B 2) < 1.99999999999999998e-92 or 0.0050000000000000001 < (pow.f64 B 2) < 9.99999999999999953e270

   1. Initial program 25.0%

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

   3. Applied egg-rr 42.2%

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

      1. *-commutative 42.2%

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

      2. associate-*r* 42.2%

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

      3. +-commutative 42.2%

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

   5. Simplified 42.2%

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

      1. *-commutative 42.2%

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

      2. sqrt-prod 54.9%

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

      3. *-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. associate-+l+ 55.6%

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

      6. *-commutative 55.6%

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

   7. Applied egg-rr 55.6%

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

   8. Taylor expanded in B around inf 27.0%

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

   if 9.99999999999999953e270 < (pow.f64 B 2)

   1. Initial program 3.1%

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

   2. Taylor expanded in C around 0 3.1%

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

      1. associate-*l* 3.1%

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

      2. +-commutative 3.1%

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

   4. Simplified 3.1%

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

   5. Taylor expanded in A around 0 21.9%

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

      1. mul-1-neg 21.9%

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

   7. Simplified 21.9%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sqrt{A \cdot 0 + 2 \cdot C} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-92}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;\frac{\sqrt{A \cdot 0 + 2 \cdot C} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 5: 37.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_2 := \frac{B \cdot \left(\sqrt{t_1} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{if}\;{B}^{2} \leq 10^{-270}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot t_0\right) \cdot t_1}}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;\frac{\sqrt{A \cdot 0 + 2 \cdot C} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_0\right)}\right)}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0))))
        (t_1 (+ A (+ C (hypot B (- A C)))))
        (t_2
         (/
          (* B (* (sqrt t_1) (- (sqrt (* 2.0 F)))))
          (- (pow B 2.0) (* (* 4.0 A) C)))))
   (if (<= (pow B 2.0) 1e-270)
     (/ (- (sqrt (* (* (* 2.0 F) t_0) t_1))) t_0)
     (if (<= (pow B 2.0) 2e-92)
       t_2
       (if (<= (pow B 2.0) 0.005)
         (/
          (* (sqrt (+ (* A 0.0) (* 2.0 C))) (- (sqrt (* 2.0 (* F t_0)))))
          t_0)
         (if (<= (pow B 2.0) 1e+271)
           t_2
           (* (sqrt (/ F B)) (- (sqrt 2.0)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double t_1 = A + (C + hypot(B, (A - C)));
	double t_2 = (B * (sqrt(t_1) * -sqrt((2.0 * F)))) / (pow(B, 2.0) - ((4.0 * A) * C));
	double tmp;
	if (pow(B, 2.0) <= 1e-270) {
		tmp = -sqrt((((2.0 * F) * t_0) * t_1)) / t_0;
	} else if (pow(B, 2.0) <= 2e-92) {
		tmp = t_2;
	} else if (pow(B, 2.0) <= 0.005) {
		tmp = (sqrt(((A * 0.0) + (2.0 * C))) * -sqrt((2.0 * (F * t_0)))) / t_0;
	} else if (pow(B, 2.0) <= 1e+271) {
		tmp = t_2;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(A + Float64(C + hypot(B, Float64(A - C))))
	t_2 = Float64(Float64(B * Float64(sqrt(t_1) * Float64(-sqrt(Float64(2.0 * F))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)))
	tmp = 0.0
	if ((B ^ 2.0) <= 1e-270)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * F) * t_0) * t_1))) / t_0);
	elseif ((B ^ 2.0) <= 2e-92)
		tmp = t_2;
	elseif ((B ^ 2.0) <= 0.005)
		tmp = Float64(Float64(sqrt(Float64(Float64(A * 0.0) + Float64(2.0 * C))) * Float64(-sqrt(Float64(2.0 * Float64(F * t_0))))) / t_0);
	elseif ((B ^ 2.0) <= 1e+271)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-270], N[((-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-92], t$95$2, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 0.005], N[(N[(N[Sqrt[N[(N[(A * 0.0), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+271], t$95$2, N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 1e-270

   1. Initial program 15.7%

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

   3. Simplified 26.8%

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

   if 1e-270 < (pow.f64 B 2) < 1.99999999999999998e-92 or 0.0050000000000000001 < (pow.f64 B 2) < 9.99999999999999953e270

   1. Initial program 25.7%

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

   3. Applied egg-rr 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-*r* 43.6%

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

      3. +-commutative 43.6%

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

   5. Simplified 43.6%

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

      1. *-commutative 43.6%

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

      2. sqrt-prod 55.7%

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

      3. *-commutative 55.7%

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

      4. +-commutative 55.7%

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

      5. associate-+l+ 56.1%

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

      6. *-commutative 56.1%

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

   7. Applied egg-rr 56.1%

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

   8. Taylor expanded in B around inf 27.9%

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

   if 1.99999999999999998e-92 < (pow.f64 B 2) < 0.0050000000000000001

   1. Initial program 12.1%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in C around inf 12.7%

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

      1. sqrt-prod 30.2%

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

      2. *-commutative 30.2%

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

      3. associate-+r+ 44.9%

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

      4. +-commutative 44.9%

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

      5. *-commutative 44.9%

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

      6. *-un-lft-identity 44.9%

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

      7. metadata-eval 44.9%

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

      8. distribute-rgt-out 44.9%

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

      9. metadata-eval 44.9%

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

      10. metadata-eval 44.9%

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

      11. associate-*r* 44.9%

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

      12. *-commutative 44.9%

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

      13. *-commutative 44.9%

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

   6. Applied egg-rr 44.9%

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

   if 9.99999999999999953e270 < (pow.f64 B 2)

   1. Initial program 3.1%

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

   2. Taylor expanded in C around 0 3.1%

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

      1. associate-*l* 3.1%

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

      2. +-commutative 3.1%

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

   4. Simplified 3.1%

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

   5. Taylor expanded in A around 0 21.9%

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

      1. mul-1-neg 21.9%

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

   7. Simplified 21.9%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-270}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-92}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;\frac{\sqrt{A \cdot 0 + 2 \cdot C} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 6: 32.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-211} \lor \neg \left({B}^{2} \leq 2 \cdot 10^{-58}\right) \land {B}^{2} \leq 0.005:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot t_0\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0)))))
   (if (or (<= (pow B 2.0) 5e-211)
           (and (not (<= (pow B 2.0) 2e-58)) (<= (pow B 2.0) 0.005)))
     (/ (- (sqrt (* (* (* 2.0 F) t_0) (* 2.0 C)))) t_0)
     (* (sqrt (/ F B)) (- (sqrt 2.0))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if ((pow(B, 2.0) <= 5e-211) || (!(pow(B, 2.0) <= 2e-58) && (pow(B, 2.0) <= 0.005))) {
		tmp = -sqrt((((2.0 * F) * t_0) * (2.0 * C))) / t_0;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (((B ^ 2.0) <= 5e-211) || (!((B ^ 2.0) <= 2e-58) && ((B ^ 2.0) <= 0.005)))
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * F) * t_0) * Float64(2.0 * C)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-211], And[N[Not[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-58]], $MachinePrecision], LessEqual[N[Power[B, 2.0], $MachinePrecision], 0.005]]], N[((-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.0000000000000002e-211 or 2.0000000000000001e-58 < (pow.f64 B 2) < 0.0050000000000000001

   1. Initial program 16.6%

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

   3. Simplified 27.4%

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

   4. Taylor expanded in C around inf 17.6%

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

   5. Taylor expanded in A around 0 25.9%

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

   if 5.0000000000000002e-211 < (pow.f64 B 2) < 2.0000000000000001e-58 or 0.0050000000000000001 < (pow.f64 B 2)

   1. Initial program 14.6%

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

   2. Taylor expanded in C around 0 12.2%

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

      1. associate-*l* 12.2%

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

      2. +-commutative 12.2%

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

   4. Simplified 12.2%

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

   5. Taylor expanded in A around 0 19.8%

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

      1. mul-1-neg 19.8%

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

   7. Simplified 19.8%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-211} \lor \neg \left({B}^{2} \leq 2 \cdot 10^{-58}\right) \land {B}^{2} \leq 0.005:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 7: 32.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot t_0\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-B\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0))))
        (t_1 (/ (- (sqrt (* (* (* 2.0 F) t_0) (* 2.0 C)))) t_0)))
   (if (<= (pow B 2.0) 2e-272)
     t_1
     (if (<= (pow B 2.0) 5e-72)
       (/
        (* (sqrt (* (* 2.0 F) (+ C (+ A (hypot B (- A C)))))) (- B))
        (- (pow B 2.0) (* (* 4.0 A) C)))
       (if (<= (pow B 2.0) 0.005) t_1 (* (sqrt (/ F B)) (- (sqrt 2.0))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double t_1 = -sqrt((((2.0 * F) * t_0) * (2.0 * C))) / t_0;
	double tmp;
	if (pow(B, 2.0) <= 2e-272) {
		tmp = t_1;
	} else if (pow(B, 2.0) <= 5e-72) {
		tmp = (sqrt(((2.0 * F) * (C + (A + hypot(B, (A - C)))))) * -B) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else if (pow(B, 2.0) <= 0.005) {
		tmp = t_1;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * F) * t_0) * Float64(2.0 * C)))) / t_0)
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-272)
		tmp = t_1;
	elseif ((B ^ 2.0) <= 5e-72)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C + Float64(A + hypot(B, Float64(A - C)))))) * Float64(-B)) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	elseif ((B ^ 2.0) <= 0.005)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-272], t$95$1, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-72], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-B)), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 0.005], t$95$1, N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.99999999999999986e-272 or 4.9999999999999996e-72 < (pow.f64 B 2) < 0.0050000000000000001

   1. Initial program 14.6%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in C around inf 16.8%

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

   5. Taylor expanded in A around 0 25.6%

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

   if 1.99999999999999986e-272 < (pow.f64 B 2) < 4.9999999999999996e-72

   1. Initial program 23.5%

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

   3. Applied egg-rr 36.5%

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

      1. *-commutative 36.5%

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

      2. associate-*r* 36.6%

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

      3. +-commutative 36.6%

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

   5. Simplified 36.6%

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

   6. Taylor expanded in B around inf 15.6%

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

   if 0.0050000000000000001 < (pow.f64 B 2)

   1. Initial program 13.6%

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

   2. Taylor expanded in C around 0 13.0%

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

      1. associate-*l* 13.1%

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

      2. +-commutative 13.1%

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

   4. Simplified 13.1%

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

   5. Taylor expanded in A around 0 22.6%

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

      1. mul-1-neg 22.6%

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

   7. Simplified 22.6%

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

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-272}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-B\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 0.005:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 8: 43.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+195}:\\ \;\;\;\;\frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 1e+195)
   (/
    (-
     (*
      (sqrt (+ C (+ A (hypot B (- A C)))))
      (sqrt (* 2.0 (* F (fma B B (* A (* C -4.0))))))))
    (- (pow B 2.0) (* (* 4.0 A) C)))
   (* (sqrt (/ F B)) (- (sqrt 2.0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 1e+195) {
		tmp = -(sqrt((C + (A + hypot(B, (A - C))))) * sqrt((2.0 * (F * fma(B, B, (A * (C * -4.0))))))) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 1e+195)
		tmp = Float64(Float64(-Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * sqrt(Float64(2.0 * Float64(F * fma(B, B, Float64(A * Float64(C * -4.0)))))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+195], N[((-N[(N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 9.99999999999999977e194

   1. Initial program 20.2%

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

   3. Applied egg-rr 36.7%

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

   if 9.99999999999999977e194 < (pow.f64 B 2)

   1. Initial program 5.5%

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

   2. Taylor expanded in C around 0 8.8%

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

      1. associate-*l* 8.8%

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

      2. +-commutative 8.8%

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

   4. Simplified 8.8%

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

   5. Taylor expanded in A around 0 24.4%

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

      1. mul-1-neg 24.4%

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

   7. Simplified 24.4%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+195}:\\ \;\;\;\;\frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 9: 37.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-289}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot t_0\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0)))))
   (if (<= (pow B 2.0) 4e-289)
     (/ (- (sqrt (* (* (* 2.0 F) t_0) (* 2.0 C)))) t_0)
     (if (<= (pow B 2.0) 1e+271)
       (/
        (* B (* (sqrt (+ A (+ C (hypot B (- A C))))) (- (sqrt (* 2.0 F)))))
        (- (pow B 2.0) (* (* 4.0 A) C)))
       (* (sqrt (/ F B)) (- (sqrt 2.0)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if (pow(B, 2.0) <= 4e-289) {
		tmp = -sqrt((((2.0 * F) * t_0) * (2.0 * C))) / t_0;
	} else if (pow(B, 2.0) <= 1e+271) {
		tmp = (B * (sqrt((A + (C + hypot(B, (A - C))))) * -sqrt((2.0 * F)))) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if ((B ^ 2.0) <= 4e-289)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * F) * t_0) * Float64(2.0 * C)))) / t_0);
	elseif ((B ^ 2.0) <= 1e+271)
		tmp = Float64(Float64(B * Float64(sqrt(Float64(A + Float64(C + hypot(B, Float64(A - C))))) * Float64(-sqrt(Float64(2.0 * F))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 4e-289], N[((-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+271], N[(N[(B * N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 4e-289

   1. Initial program 16.2%

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

   3. Simplified 27.4%

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

   4. Taylor expanded in C around inf 17.7%

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

   5. Taylor expanded in A around 0 24.9%

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

   if 4e-289 < (pow.f64 B 2) < 9.99999999999999953e270

   1. Initial program 22.5%

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

   3. Applied egg-rr 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-*r* 38.6%

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

      3. +-commutative 38.6%

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

   5. Simplified 38.6%

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

      1. *-commutative 38.6%

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

      2. sqrt-prod 49.0%

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

      3. *-commutative 49.0%

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

      4. +-commutative 49.0%

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

      5. associate-+l+ 49.0%

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

      6. *-commutative 49.0%

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

   7. Applied egg-rr 49.0%

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

   8. Taylor expanded in B around inf 23.3%

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

   if 9.99999999999999953e270 < (pow.f64 B 2)

   1. Initial program 3.1%

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

   2. Taylor expanded in C around 0 3.1%

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

      1. associate-*l* 3.1%

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

      2. +-commutative 3.1%

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

   4. Simplified 3.1%

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

   5. Taylor expanded in A around 0 21.9%

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

      1. mul-1-neg 21.9%

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

   7. Simplified 21.9%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-289}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+271}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 10: 37.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+214}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 2e+214)
   (-
    (/
     (sqrt (* 2.0 (* F (+ A (+ C (hypot B (- A C)))))))
     (sqrt (fma B B (* -4.0 (* A C))))))
   (* (sqrt (/ F B)) (- (sqrt 2.0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 2e+214) {
		tmp = -(sqrt((2.0 * (F * (A + (C + hypot(B, (A - C))))))) / sqrt(fma(B, B, (-4.0 * (A * C)))));
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 2e+214)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(F * Float64(A + Float64(C + hypot(B, Float64(A - C))))))) / sqrt(fma(B, B, Float64(-4.0 * Float64(A * C))))));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e+214], (-N[(N[Sqrt[N[(2.0 * N[(F * N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 1.9999999999999999e214

   1. Initial program 19.8%

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

   3. Applied egg-rr 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-*r* 28.6%

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

      3. +-commutative 28.6%

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

   5. Simplified 28.6%

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

   7. Applied egg-rr 28.3%

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

      1. associate-*r/ 28.3%

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

      2. distribute-frac-neg 28.3%

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

      3. *-inverses 28.6%

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

      4. metadata-eval 28.6%

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

      5. neg-mul-1 28.6%

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

      6. distribute-frac-neg 28.6%

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

   9. Simplified 28.6%

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

   if 1.9999999999999999e214 < (pow.f64 B 2)

   1. Initial program 5.6%

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

   2. Taylor expanded in C around 0 7.9%

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

      1. associate-*l* 7.9%

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

      2. +-commutative 7.9%

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

   4. Simplified 7.9%

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

   5. Taylor expanded in A around 0 24.2%

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

      1. mul-1-neg 24.2%

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

   7. Simplified 24.2%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+214}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 11: 37.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 2e+214)
   (/
    (- (sqrt (* (* 2.0 F) (+ C (+ A (hypot B (- A C)))))))
    (sqrt (fma B B (* A (* C -4.0)))))
   (* (sqrt (/ F B)) (- (sqrt 2.0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 2e+214) {
		tmp = -sqrt(((2.0 * F) * (C + (A + hypot(B, (A - C)))))) / sqrt(fma(B, B, (A * (C * -4.0))));
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 2e+214)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / sqrt(fma(B, B, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e+214], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[Sqrt[N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 1.9999999999999999e214

   1. Initial program 19.8%

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

   3. Applied egg-rr 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-*r* 28.6%

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

      3. +-commutative 28.6%

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

   5. Simplified 28.6%

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

      1. *-commutative 28.6%

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

      2. sqrt-prod 35.8%

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

      3. *-commutative 35.8%

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

      4. +-commutative 35.8%

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

      5. associate-+l+ 35.9%

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

      6. *-commutative 35.9%

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

   7. Applied egg-rr 35.9%

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

   9. Applied egg-rr 28.9%

      \[\leadsto \color{blue}{0 - \frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
   10. Step-by-step derivation

       1. sub0-neg 28.9%

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

       2. distribute-frac-neg 28.9%

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

   11. Simplified 28.9%

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

   if 1.9999999999999999e214 < (pow.f64 B 2)

   1. Initial program 5.6%

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

   2. Taylor expanded in C around 0 7.9%

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

      1. associate-*l* 7.9%

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

      2. +-commutative 7.9%

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

   4. Simplified 7.9%

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

   5. Taylor expanded in A around 0 24.2%

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

      1. mul-1-neg 24.2%

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

   7. Simplified 24.2%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 12: 35.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + \left(2 \cdot C - A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \left(B + 0.5 \cdot \frac{{A}^{2}}{B}\right)\right)} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= F -5e-310)
     (/
      (- (sqrt (* (* -8.0 (* A (* C F))) (+ A (- (* 2.0 C) A)))))
      (fma B B (* C (* A -4.0))))
     (if (<= F 4.2e-37)
       (* (sqrt (* F (+ A (+ B (* 0.5 (/ (pow A 2.0) B)))))) (/ t_0 B))
       (* (sqrt (/ F B)) t_0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (F <= -5e-310) {
		tmp = -sqrt(((-8.0 * (A * (C * F))) * (A + ((2.0 * C) - A)))) / fma(B, B, (C * (A * -4.0)));
	} else if (F <= 4.2e-37) {
		tmp = sqrt((F * (A + (B + (0.5 * (pow(A, 2.0) / B)))))) * (t_0 / B);
	} else {
		tmp = sqrt((F / B)) * t_0;
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(-sqrt(Float64(Float64(-8.0 * Float64(A * Float64(C * F))) * Float64(A + Float64(Float64(2.0 * C) - A))))) / fma(B, B, Float64(C * Float64(A * -4.0))));
	elseif (F <= 4.2e-37)
		tmp = Float64(sqrt(Float64(F * Float64(A + Float64(B + Float64(0.5 * Float64((A ^ 2.0) / B)))))) * Float64(t_0 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_0);
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[F, -5e-310], N[((-N[Sqrt[N[(N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[(N[(2.0 * C), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e-37], N[(N[Sqrt[N[(F * N[(A + N[(B + N[(0.5 * N[(N[Power[A, 2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 18.9%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in C around inf 25.1%

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

   5. Taylor expanded in B around 0 25.2%

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

   if -4.999999999999985e-310 < F < 4.2000000000000002e-37

   1. Initial program 14.5%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in B around inf 7.3%

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

      1. associate-+r+ 7.3%

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

      2. +-commutative 7.3%

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

      3. associate-*r/ 7.3%

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

   6. Simplified 7.3%

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

   7. Taylor expanded in C around 0 15.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \left(B + 0.5 \cdot \frac{{A}^{2}}{B}\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 15.1%

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

   9. Simplified 15.1%

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

   if 4.2000000000000002e-37 < F

   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. Taylor expanded in C around 0 8.5%

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

      1. associate-*l* 8.5%

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

      2. +-commutative 8.5%

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

   4. Simplified 8.5%

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

   5. Taylor expanded in A around 0 15.2%

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

      1. mul-1-neg 15.2%

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

   7. Simplified 15.2%

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

4. Final simplification 16.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + \left(2 \cdot C - A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \left(B + 0.5 \cdot \frac{{A}^{2}}{B}\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 13: 32.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(-16 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 5e-211)
   (/ (sqrt (* A (* -16.0 (* F (* A C))))) (- (fma B B (* -4.0 (* A C)))))
   (* (sqrt (/ F B)) (- (sqrt 2.0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 5e-211) {
		tmp = sqrt((A * (-16.0 * (F * (A * C))))) / -fma(B, B, (-4.0 * (A * C)));
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 5e-211)
		tmp = Float64(sqrt(Float64(A * Float64(-16.0 * Float64(F * Float64(A * C))))) / Float64(-fma(B, B, Float64(-4.0 * Float64(A * C)))));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-211], N[(N[Sqrt[N[(A * N[(-16.0 * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.0000000000000002e-211

   1. Initial program 17.5%

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

   2. Taylor expanded in A around inf 19.7%

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

      1. frac-2neg 19.7%

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

      2. remove-double-neg 19.7%

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

      3. div-inv 19.7%

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

   4. Applied egg-rr 19.7%

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

      1. associate-*r/ 19.7%

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

      2. *-rgt-identity 19.7%

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

      3. associate-*l* 19.7%

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

      4. associate-*r* 19.7%

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

      5. associate-*r* 19.7%

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

   6. Simplified 19.7%

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

   7. Taylor expanded in B around 0 17.6%

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

      1. associate-*r* 19.6%

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

   9. Simplified 19.6%

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

   if 5.0000000000000002e-211 < (pow.f64 B 2)

   1. Initial program 14.3%

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

   2. Taylor expanded in C around 0 12.0%

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

      1. associate-*l* 12.0%

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

      2. +-commutative 12.0%

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

   4. Simplified 12.0%

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

   5. Taylor expanded in A around 0 18.4%

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

      1. mul-1-neg 18.4%

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

   7. Simplified 18.4%

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

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(-16 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 14: 31.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(A \cdot \left(C \cdot F\right)\right) \cdot -16\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= B 1.75e-105)
   (/ (sqrt (* A (* (* A (* C F)) -16.0))) (- (fma B B (* -4.0 (* A C)))))
   (* (sqrt (/ F B)) (- (sqrt 2.0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 1.75e-105) {
		tmp = sqrt((A * ((A * (C * F)) * -16.0))) / -fma(B, B, (-4.0 * (A * C)));
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 1.75e-105)
		tmp = Float64(sqrt(Float64(A * Float64(Float64(A * Float64(C * F)) * -16.0))) / Float64(-fma(B, B, Float64(-4.0 * Float64(A * C)))));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[B, 1.75e-105], N[(N[Sqrt[N[(A * N[(N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.75e-105

   1. Initial program 13.9%

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

   2. Taylor expanded in A around inf 12.1%

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

      1. frac-2neg 12.1%

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

      2. remove-double-neg 12.1%

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

      3. div-inv 12.0%

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

   4. Applied egg-rr 12.0%

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

      1. associate-*r/ 12.1%

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

      2. *-rgt-identity 12.1%

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

      3. associate-*l* 12.1%

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

      4. associate-*r* 12.1%

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

      5. associate-*r* 12.1%

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

   6. Simplified 12.1%

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

   7. Taylor expanded in B around 0 10.5%

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

   if 1.75e-105 < B

   1. Initial program 18.8%

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

   2. Taylor expanded in C around 0 24.4%

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

      1. associate-*l* 24.5%

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

      2. +-commutative 24.5%

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

   4. Simplified 24.5%

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

   5. Taylor expanded in A around 0 37.5%

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

      1. mul-1-neg 37.5%

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

   7. Simplified 37.5%

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

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(A \cdot \left(C \cdot F\right)\right) \cdot -16\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 15: 27.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 1.95 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A}}{\frac{B}{\sqrt{F}}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= A 1.95e+95)
   (* (sqrt (/ F B)) (- (sqrt 2.0)))
   (* -2.0 (/ (sqrt A) (/ B (sqrt F))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= 1.95e+95) {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	} else {
		tmp = -2.0 * (sqrt(A) / (B / sqrt(F)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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) :: tmp
    if (a <= 1.95d+95) then
        tmp = sqrt((f / b)) * -sqrt(2.0d0)
    else
        tmp = (-2.0d0) * (sqrt(a) / (b / sqrt(f)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= 1.95e+95) {
		tmp = Math.sqrt((F / B)) * -Math.sqrt(2.0);
	} else {
		tmp = -2.0 * (Math.sqrt(A) / (B / Math.sqrt(F)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if A <= 1.95e+95:
		tmp = math.sqrt((F / B)) * -math.sqrt(2.0)
	else:
		tmp = -2.0 * (math.sqrt(A) / (B / math.sqrt(F)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (A <= 1.95e+95)
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(A) / Float64(B / sqrt(F))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= 1.95e+95)
		tmp = sqrt((F / B)) * -sqrt(2.0);
	else
		tmp = -2.0 * (sqrt(A) / (B / sqrt(F)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[A, 1.95e+95], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[A], $MachinePrecision] / N[(B / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 1.9499999999999999e95

   1. Initial program 16.5%

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

   2. Taylor expanded in C around 0 9.7%

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

      1. associate-*l* 9.7%

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

      2. +-commutative 9.7%

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

   4. Simplified 9.7%

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

   5. Taylor expanded in A around 0 14.2%

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

      1. mul-1-neg 14.2%

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

   7. Simplified 14.2%

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

   if 1.9499999999999999e95 < A

   1. Initial program 7.9%

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

   2. Taylor expanded in A around inf 14.2%

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

   3. Taylor expanded in B around inf 5.3%

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

      1. un-div-inv 5.3%

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

      2. sqrt-prod 13.8%

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

      3. associate-/l* 13.7%

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

   5. Applied egg-rr 13.7%

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

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 1.95 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A}}{\frac{B}{\sqrt{F}}}\\ \end{array} \]

Alternative 16: 26.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right) \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F) :precision binary64 (* (sqrt (/ F B)) (- (sqrt 2.0))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	return sqrt((F / B)) * -sqrt(2.0);
}

Fortran

NOTE: B should be positive before calling this function
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
    code = sqrt((f / b)) * -sqrt(2.0d0)
end function

Java

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

Python

B = abs(B)
def code(A, B, C, F):
	return math.sqrt((F / B)) * -math.sqrt(2.0)

Julia

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

MATLAB

B = abs(B)
function tmp = code(A, B, C, F)
	tmp = sqrt((F / B)) * -sqrt(2.0);
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 15.4%

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

2. Taylor expanded in C around 0 9.0%

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

   1. associate-*l* 9.0%

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

   2. +-commutative 9.0%

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

4. Simplified 9.0%

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

5. Taylor expanded in A around 0 12.9%

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

   1. mul-1-neg 12.9%

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

7. Simplified 12.9%

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

8. Final simplification 12.9%

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

Alternative 17: 5.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ -2 \cdot \left({\left(A \cdot F\right)}^{0.5} \cdot \frac{1}{B}\right) \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F) :precision binary64 (* -2.0 (* (pow (* A F) 0.5) (/ 1.0 B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	return -2.0 * (pow((A * F), 0.5) * (1.0 / B));
}

Fortran

NOTE: B should be positive before calling this function
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
    code = (-2.0d0) * (((a * f) ** 0.5d0) * (1.0d0 / b))
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.pow((A * F), 0.5) * (1.0 / B));
}

Python

B = abs(B)
def code(A, B, C, F):
	return -2.0 * (math.pow((A * F), 0.5) * (1.0 / B))

Julia

B = abs(B)
function code(A, B, C, F)
	return Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) * Float64(1.0 / B)))
end

MATLAB

B = abs(B)
function tmp = code(A, B, C, F)
	tmp = -2.0 * (((A * F) ^ 0.5) * (1.0 / B));
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.4%

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

2. Taylor expanded in A around inf 9.6%

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

3. Taylor expanded in B around inf 2.5%

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

   1. pow1/2 2.7%

      \[\leadsto -2 \cdot \left(\color{blue}{{\left(A \cdot F\right)}^{0.5}} \cdot \frac{1}{B}\right) \]

5. Applied egg-rr 2.7%

   \[\leadsto -2 \cdot \left(\color{blue}{{\left(A \cdot F\right)}^{0.5}} \cdot \frac{1}{B}\right) \]

6. Final simplification 2.7%

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

Alternative 18: 4.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ -2 \cdot \frac{\sqrt{A \cdot F}}{B} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (sqrt (* A F)) B)))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	return -2.0 * (sqrt((A * F)) / B);
}

Fortran

NOTE: B should be positive before calling this function
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
    code = (-2.0d0) * (sqrt((a * f)) / b)
end function

Java

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

Python

B = abs(B)
def code(A, B, C, F):
	return -2.0 * (math.sqrt((A * F)) / B)

Julia

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

MATLAB

B = abs(B)
function tmp = code(A, B, C, F)
	tmp = -2.0 * (sqrt((A * F)) / B);
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.4%

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

2. Taylor expanded in A around inf 9.6%

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

   1. frac-2neg 9.6%

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

   2. remove-double-neg 9.6%

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

   3. div-inv 9.6%

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

4. Applied egg-rr 9.6%

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

   1. associate-*r/ 9.6%

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

   2. *-rgt-identity 9.6%

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

   3. associate-*l* 9.6%

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

   4. associate-*r* 9.6%

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

   5. associate-*r* 9.6%

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

6. Simplified 9.6%

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

7. Taylor expanded in B around inf 2.5%

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

   1. associate-*r/ 2.5%

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

   2. *-rgt-identity 2.5%

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

9. Simplified 2.5%

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

10. Final simplification 2.5%

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

Reproduce

herbie shell --seed 2023301 
(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))))