ABCF->ab-angle a

Percentage Accurate: 19.2% → 52.2%

Time: 51.1s

Alternatives: 20

Speedup: 5.8×

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: 19.2% 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: 52.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \frac{\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot F + F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{if}\;B \leq 1.15 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-249}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{\frac{F}{\frac{C}{\mathsf{fma}\left(0.5, \frac{B}{C} \cdot \frac{B}{C}, -2\right)}}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \cdot \left(-{\left(\frac{B}{\sqrt{2}}\right)}^{-1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* A (* C -4.0)))
        (t_1
         (/
          (*
           (sqrt (* 2.0 (+ (* (* B B) F) (* F t_0))))
           (- (sqrt (+ C (+ A (hypot B (- A C)))))))
          (fma B B t_0))))
   (if (<= B 1.15e-261)
     t_1
     (if (<= B 3e-249)
       (*
        -0.5
        (* (sqrt (/ F (/ C (fma 0.5 (* (/ B C) (/ B C)) -2.0)))) (sqrt 2.0)))
       (if (<= B 8.6e+92)
         t_1
         (*
          (* (sqrt F) (sqrt (+ C (hypot C B))))
          (- (pow (/ B (sqrt 2.0)) -1.0))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = A * (C * -4.0);
	double t_1 = (sqrt((2.0 * (((B * B) * F) + (F * t_0)))) * -sqrt((C + (A + hypot(B, (A - C)))))) / fma(B, B, t_0);
	double tmp;
	if (B <= 1.15e-261) {
		tmp = t_1;
	} else if (B <= 3e-249) {
		tmp = -0.5 * (sqrt((F / (C / fma(0.5, ((B / C) * (B / C)), -2.0)))) * sqrt(2.0));
	} else if (B <= 8.6e+92) {
		tmp = t_1;
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(C, B)))) * -pow((B / sqrt(2.0)), -1.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(A * Float64(C * -4.0))
	t_1 = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(Float64(B * B) * F) + Float64(F * t_0)))) * Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / fma(B, B, t_0))
	tmp = 0.0
	if (B <= 1.15e-261)
		tmp = t_1;
	elseif (B <= 3e-249)
		tmp = Float64(-0.5 * Float64(sqrt(Float64(F / Float64(C / fma(0.5, Float64(Float64(B / C) * Float64(B / C)), -2.0)))) * sqrt(2.0)));
	elseif (B <= 8.6e+92)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(C, B)))) * Float64(-(Float64(B / sqrt(2.0)) ^ -1.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(2.0 * N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] + N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.15e-261], t$95$1, If[LessEqual[B, 3e-249], N[(-0.5 * N[(N[Sqrt[N[(F / N[(C / N[(0.5 * N[(N[(B / C), $MachinePrecision] * N[(B / C), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 8.6e+92], t$95$1, N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Power[N[(B / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.15e-261 or 3.00000000000000004e-249 < B < 8.5999999999999996e92

   1. Initial program 17.3%

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

   3. Simplified 23.0%

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

      1. sqrt-prod 31.7%

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

      2. associate-+r+ 30.8%

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

      3. +-commutative 30.8%

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

      4. associate-+r+ 31.3%

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

   5. Applied egg-rr 31.3%

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

      1. associate-*r* 31.3%

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

      2. *-commutative 31.3%

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

      3. fma-def 31.3%

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

      4. *-commutative 31.3%

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

      5. fma-def 31.3%

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

      6. *-commutative 31.3%

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

      7. associate-*r* 31.3%

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

      8. associate-+r+ 30.8%

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

      9. +-commutative 30.8%

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

      10. associate-+r+ 31.7%

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

   7. Simplified 31.7%

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

      1. fma-udef 31.7%

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

      2. associate-*r* 31.7%

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

      3. *-commutative 31.7%

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

      4. distribute-rgt-in 31.7%

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

      5. *-commutative 31.7%

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

      6. associate-*r* 31.7%

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

   9. Applied egg-rr 31.7%

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

   if 1.15e-261 < B < 3.00000000000000004e-249

   1. Initial program 1.5%

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

   3. Simplified 1.5%

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

   4. Taylor expanded in C around -inf 2.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(A + \left(-1 \cdot C + \left(-0.5 \cdot \frac{A \cdot {B}^{2}}{{C}^{2}} + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 2.8%

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

      2. mul-1-neg 2.8%

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

      3. sub-neg 2.8%

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

      4. distribute-lft-out 2.8%

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

      5. associate-/l* 2.8%

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

      6. unpow2 2.8%

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

      7. unpow2 2.8%

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

      8. unpow2 2.8%

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

   6. Simplified 2.8%

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

   7. Taylor expanded in A around -inf 52.1%

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

      1. associate-/l* 52.1%

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

      2. fma-neg 52.1%

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

      3. unpow2 52.1%

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

      4. unpow2 52.1%

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

      5. times-frac 52.1%

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

      6. metadata-eval 52.1%

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

   9. Simplified 52.1%

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

   if 8.5999999999999996e92 < B

   1. Initial program 6.0%

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

   3. Simplified 6.0%

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

   4. Taylor expanded in A around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. distribute-rgt-neg-in 18.0%

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

      3. +-commutative 18.0%

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

      4. unpow2 18.0%

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

      5. unpow2 18.0%

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

      6. hypot-def 44.5%

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

   6. Simplified 44.5%

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

      1. sqrt-prod 75.7%

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

   8. Applied egg-rr 75.7%

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

      1. clear-num 76.0%

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

      2. inv-pow 76.0%

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

   10. Applied egg-rr 76.0%

       \[\leadsto \color{blue}{{\left(\frac{B}{\sqrt{2}}\right)}^{-1}} \cdot \left(-\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.15 \cdot 10^{-261}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot F + F \cdot \left(A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-249}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{\frac{F}{\frac{C}{\mathsf{fma}\left(0.5, \frac{B}{C} \cdot \frac{B}{C}, -2\right)}}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot F + F \cdot \left(A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \cdot \left(-{\left(\frac{B}{\sqrt{2}}\right)}^{-1}\right)\\ \end{array} \]

Alternative 2: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;{B}^{2} \leq 10^{+186}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot F + F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{2}{B}}}{\sqrt{B}} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* A (* C -4.0))))
   (if (<= (pow B 2.0) 1e+186)
     (/
      (*
       (sqrt (* 2.0 (+ (* (* B B) F) (* F t_0))))
       (- (sqrt (+ C (+ A (hypot B (- A C)))))))
      (fma B B t_0))
     (*
      (/ (sqrt (/ 2.0 B)) (sqrt B))
      (* (sqrt F) (- (sqrt (+ C (hypot C B)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = A * (C * -4.0);
	double tmp;
	if (pow(B, 2.0) <= 1e+186) {
		tmp = (sqrt((2.0 * (((B * B) * F) + (F * t_0)))) * -sqrt((C + (A + hypot(B, (A - C)))))) / fma(B, B, t_0);
	} else {
		tmp = (sqrt((2.0 / B)) / sqrt(B)) * (sqrt(F) * -sqrt((C + hypot(C, B))));
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(A * Float64(C * -4.0))
	tmp = 0.0
	if ((B ^ 2.0) <= 1e+186)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(Float64(B * B) * F) + Float64(F * t_0)))) * Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / fma(B, B, t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 / B)) / sqrt(B)) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(C, B))))));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+186], N[(N[(N[Sqrt[N[(2.0 * N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] + N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 / B), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 9.9999999999999998e185

   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. Simplified 27.3%

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

      1. sqrt-prod 37.0%

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

      2. associate-+r+ 35.9%

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

      3. +-commutative 35.9%

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

      4. associate-+r+ 36.6%

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

   5. Applied egg-rr 36.6%

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

      1. associate-*r* 36.6%

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

      2. *-commutative 36.6%

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

      3. fma-def 36.6%

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

      4. *-commutative 36.6%

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

      5. fma-def 36.6%

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

      6. *-commutative 36.6%

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

      7. associate-*r* 36.6%

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

      8. associate-+r+ 35.9%

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

      9. +-commutative 35.9%

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

      10. associate-+r+ 37.0%

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

   7. Simplified 37.0%

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

      1. fma-udef 37.0%

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

      2. associate-*r* 37.0%

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

      3. *-commutative 37.0%

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

      4. distribute-rgt-in 37.0%

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

      5. *-commutative 37.0%

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

      6. associate-*r* 37.0%

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

   9. Applied egg-rr 37.0%

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

   if 9.9999999999999998e185 < (pow.f64 B 2)

   1. Initial program 5.5%

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

   3. Simplified 5.5%

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

   4. Taylor expanded in A around 0 8.6%

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

      1. mul-1-neg 8.6%

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

      2. distribute-rgt-neg-in 8.6%

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

      3. +-commutative 8.6%

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

      4. unpow2 8.6%

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

      5. unpow2 8.6%

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

      6. hypot-def 21.3%

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

   6. Simplified 21.3%

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

      1. sqrt-prod 35.4%

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

   8. Applied egg-rr 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 33.9%

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

      3. times-frac 33.8%

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

      4. sqrt-div 33.9%

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

   10. Applied egg-rr 33.9%

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

       1. associate-*l/ 33.9%

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

       2. *-lft-identity 33.9%

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

   12. Simplified 33.9%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+186}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot F + F \cdot \left(A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{2}{B}}}{\sqrt{B}} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\right)\\ \end{array} \]

Alternative 3: 52.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \frac{\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot F + F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{if}\;B \leq 1.1 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-249}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{\frac{F}{\frac{C}{\mathsf{fma}\left(0.5, \frac{B}{C} \cdot \frac{B}{C}, -2\right)}}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* A (* C -4.0)))
        (t_1
         (/
          (*
           (sqrt (* 2.0 (+ (* (* B B) F) (* F t_0))))
           (- (sqrt (+ C (+ A (hypot B (- A C)))))))
          (fma B B t_0))))
   (if (<= B 1.1e-261)
     t_1
     (if (<= B 4e-249)
       (*
        -0.5
        (* (sqrt (/ F (/ C (fma 0.5 (* (/ B C) (/ B C)) -2.0)))) (sqrt 2.0)))
       (if (<= B 1.55e+93)
         t_1
         (* (* (sqrt F) (sqrt (+ C (hypot C B)))) (/ (- (sqrt 2.0)) B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = A * (C * -4.0);
	double t_1 = (sqrt((2.0 * (((B * B) * F) + (F * t_0)))) * -sqrt((C + (A + hypot(B, (A - C)))))) / fma(B, B, t_0);
	double tmp;
	if (B <= 1.1e-261) {
		tmp = t_1;
	} else if (B <= 4e-249) {
		tmp = -0.5 * (sqrt((F / (C / fma(0.5, ((B / C) * (B / C)), -2.0)))) * sqrt(2.0));
	} else if (B <= 1.55e+93) {
		tmp = t_1;
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(C, B)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(A * Float64(C * -4.0))
	t_1 = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(Float64(B * B) * F) + Float64(F * t_0)))) * Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / fma(B, B, t_0))
	tmp = 0.0
	if (B <= 1.1e-261)
		tmp = t_1;
	elseif (B <= 4e-249)
		tmp = Float64(-0.5 * Float64(sqrt(Float64(F / Float64(C / fma(0.5, Float64(Float64(B / C) * Float64(B / C)), -2.0)))) * sqrt(2.0)));
	elseif (B <= 1.55e+93)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(C, B)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(2.0 * N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] + N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.1e-261], t$95$1, If[LessEqual[B, 4e-249], N[(-0.5 * N[(N[Sqrt[N[(F / N[(C / N[(0.5 * N[(N[(B / C), $MachinePrecision] * N[(B / C), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.55e+93], t$95$1, N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.1000000000000001e-261 or 4.00000000000000022e-249 < B < 1.5500000000000001e93

   1. Initial program 17.3%

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

   3. Simplified 23.0%

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

      1. sqrt-prod 31.7%

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

      2. associate-+r+ 30.8%

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

      3. +-commutative 30.8%

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

      4. associate-+r+ 31.3%

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

   5. Applied egg-rr 31.3%

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

      1. associate-*r* 31.3%

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

      2. *-commutative 31.3%

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

      3. fma-def 31.3%

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

      4. *-commutative 31.3%

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

      5. fma-def 31.3%

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

      6. *-commutative 31.3%

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

      7. associate-*r* 31.3%

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

      8. associate-+r+ 30.8%

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

      9. +-commutative 30.8%

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

      10. associate-+r+ 31.7%

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

   7. Simplified 31.7%

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

      1. fma-udef 31.7%

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

      2. associate-*r* 31.7%

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

      3. *-commutative 31.7%

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

      4. distribute-rgt-in 31.7%

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

      5. *-commutative 31.7%

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

      6. associate-*r* 31.7%

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

   9. Applied egg-rr 31.7%

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

   if 1.1000000000000001e-261 < B < 4.00000000000000022e-249

   1. Initial program 1.5%

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

   3. Simplified 1.5%

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

   4. Taylor expanded in C around -inf 2.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(A + \left(-1 \cdot C + \left(-0.5 \cdot \frac{A \cdot {B}^{2}}{{C}^{2}} + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 2.8%

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

      2. mul-1-neg 2.8%

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

      3. sub-neg 2.8%

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

      4. distribute-lft-out 2.8%

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

      5. associate-/l* 2.8%

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

      6. unpow2 2.8%

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

      7. unpow2 2.8%

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

      8. unpow2 2.8%

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

   6. Simplified 2.8%

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

   7. Taylor expanded in A around -inf 52.1%

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

      1. associate-/l* 52.1%

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

      2. fma-neg 52.1%

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

      3. unpow2 52.1%

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

      4. unpow2 52.1%

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

      5. times-frac 52.1%

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

      6. metadata-eval 52.1%

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

   9. Simplified 52.1%

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

   if 1.5500000000000001e93 < B

   1. Initial program 6.0%

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

   3. Simplified 6.0%

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

   4. Taylor expanded in A around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. distribute-rgt-neg-in 18.0%

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

      3. +-commutative 18.0%

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

      4. unpow2 18.0%

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

      5. unpow2 18.0%

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

      6. hypot-def 44.5%

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

   6. Simplified 44.5%

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

      1. sqrt-prod 75.7%

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

   8. Applied egg-rr 75.7%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-261}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot F + F \cdot \left(A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-249}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{\frac{F}{\frac{C}{\mathsf{fma}\left(0.5, \frac{B}{C} \cdot \frac{B}{C}, -2\right)}}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot F + F \cdot \left(A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4: 52.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= B 3.1e+93)
   (/
    (*
     (sqrt (* 2.0 (* F (+ (* B B) (* -4.0 (* A C))))))
     (- (sqrt (+ A (+ C (hypot B (- A C)))))))
    (- (* B B) (* (* A C) 4.0)))
   (* (* (sqrt F) (sqrt (+ C (hypot C B)))) (/ (- (sqrt 2.0)) B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 3.1e+93) {
		tmp = (sqrt((2.0 * (F * ((B * B) + (-4.0 * (A * C)))))) * -sqrt((A + (C + hypot(B, (A - C)))))) / ((B * B) - ((A * C) * 4.0));
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(C, B)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 3.1e+93) {
		tmp = (Math.sqrt((2.0 * (F * ((B * B) + (-4.0 * (A * C)))))) * -Math.sqrt((A + (C + Math.hypot(B, (A - C)))))) / ((B * B) - ((A * C) * 4.0));
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((C + Math.hypot(C, B)))) * (-Math.sqrt(2.0) / B);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if B <= 3.1e+93:
		tmp = (math.sqrt((2.0 * (F * ((B * B) + (-4.0 * (A * C)))))) * -math.sqrt((A + (C + math.hypot(B, (A - C)))))) / ((B * B) - ((A * C) * 4.0))
	else:
		tmp = (math.sqrt(F) * math.sqrt((C + math.hypot(C, B)))) * (-math.sqrt(2.0) / B)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 3.1e+93)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))))) * Float64(-sqrt(Float64(A + Float64(C + hypot(B, Float64(A - C))))))) / Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0)));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(C, B)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 3.1e+93)
		tmp = (sqrt((2.0 * (F * ((B * B) + (-4.0 * (A * C)))))) * -sqrt((A + (C + hypot(B, (A - C)))))) / ((B * B) - ((A * C) * 4.0));
	else
		tmp = (sqrt(F) * sqrt((C + hypot(C, B)))) * (-sqrt(2.0) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[B, 3.1e+93], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.10000000000000019e93

   1. Initial program 17.0%

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

   3. Simplified 17.0%

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

      1. sqrt-prod 20.1%

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

      2. *-commutative 20.1%

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

      3. cancel-sign-sub-inv 20.1%

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

      4. metadata-eval 20.1%

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

      5. associate-+l+ 20.3%

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

      6. unpow2 20.3%

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

      7. hypot-udef 30.8%

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

   5. Applied egg-rr 30.8%

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

   if 3.10000000000000019e93 < B

   1. Initial program 6.0%

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

   3. Simplified 6.0%

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

   4. Taylor expanded in A around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. distribute-rgt-neg-in 18.0%

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

      3. +-commutative 18.0%

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

      4. unpow2 18.0%

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

      5. unpow2 18.0%

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

      6. hypot-def 44.5%

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

   6. Simplified 44.5%

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

      1. sqrt-prod 75.7%

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

   8. Applied egg-rr 75.7%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5: 50.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F} \cdot \sqrt{B + C}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= B 9e+94)
   (/
    (*
     (sqrt (* 2.0 (* F (+ (* B B) (* -4.0 (* A C))))))
     (- (sqrt (+ A (+ C (hypot B (- A C)))))))
    (- (* B B) (* (* A C) 4.0)))
   (* (/ (sqrt 2.0) B) (- (* (sqrt F) (sqrt (+ B C)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 9e+94) {
		tmp = (sqrt((2.0 * (F * ((B * B) + (-4.0 * (A * C)))))) * -sqrt((A + (C + hypot(B, (A - C)))))) / ((B * B) - ((A * C) * 4.0));
	} else {
		tmp = (sqrt(2.0) / B) * -(sqrt(F) * sqrt((B + C)));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 9e+94) {
		tmp = (Math.sqrt((2.0 * (F * ((B * B) + (-4.0 * (A * C)))))) * -Math.sqrt((A + (C + Math.hypot(B, (A - C)))))) / ((B * B) - ((A * C) * 4.0));
	} else {
		tmp = (Math.sqrt(2.0) / B) * -(Math.sqrt(F) * Math.sqrt((B + C)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if B <= 9e+94:
		tmp = (math.sqrt((2.0 * (F * ((B * B) + (-4.0 * (A * C)))))) * -math.sqrt((A + (C + math.hypot(B, (A - C)))))) / ((B * B) - ((A * C) * 4.0))
	else:
		tmp = (math.sqrt(2.0) / B) * -(math.sqrt(F) * math.sqrt((B + C)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 9e+94)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))))) * Float64(-sqrt(Float64(A + Float64(C + hypot(B, Float64(A - C))))))) / Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-Float64(sqrt(F) * sqrt(Float64(B + C)))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 9e+94)
		tmp = (sqrt((2.0 * (F * ((B * B) + (-4.0 * (A * C)))))) * -sqrt((A + (C + hypot(B, (A - C)))))) / ((B * B) - ((A * C) * 4.0));
	else
		tmp = (sqrt(2.0) / B) * -(sqrt(F) * sqrt((B + C)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[B, 9e+94], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.99999999999999944e94

   1. Initial program 17.0%

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

   3. Simplified 17.0%

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

      1. sqrt-prod 20.1%

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

      2. *-commutative 20.1%

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

      3. cancel-sign-sub-inv 20.1%

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

      4. metadata-eval 20.1%

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

      5. associate-+l+ 20.3%

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

      6. unpow2 20.3%

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

      7. hypot-udef 30.8%

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

   5. Applied egg-rr 30.8%

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

   if 8.99999999999999944e94 < B

   1. Initial program 6.0%

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

   3. Simplified 6.0%

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

   4. Taylor expanded in A around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. distribute-rgt-neg-in 18.0%

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

      3. +-commutative 18.0%

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

      4. unpow2 18.0%

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

      5. unpow2 18.0%

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

      6. hypot-def 44.5%

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

   6. Simplified 44.5%

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

   7. Taylor expanded in C around 0 44.0%

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

      1. +-commutative 44.0%

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

   9. Simplified 44.0%

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

       1. *-commutative 44.0%

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

       2. sqrt-prod 73.1%

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

       3. +-commutative 73.1%

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

   11. Applied egg-rr 73.1%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F} \cdot \sqrt{B + C}\right)\\ \end{array} \]

Alternative 6: 40.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + A\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F -1e-309)
   (/
    (*
     (sqrt (* 2.0 (* F (fma B B (* A (* C -4.0))))))
     (- (sqrt (+ C (+ A A)))))
    (- (* B B) (* (* A C) 4.0)))
   (if (<= F 4.6e-85)
     (/ (- (sqrt (* F (* 2.0 (+ C (hypot B C)))))) B)
     (* (/ (sqrt 2.0) B) (* (sqrt F) (- (sqrt B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -1e-309) {
		tmp = (sqrt((2.0 * (F * fma(B, B, (A * (C * -4.0)))))) * -sqrt((C + (A + A)))) / ((B * B) - ((A * C) * 4.0));
	} else if (F <= 4.6e-85) {
		tmp = -sqrt((F * (2.0 * (C + hypot(B, C))))) / B;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt(B));
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B, B, Float64(A * Float64(C * -4.0)))))) * Float64(-sqrt(Float64(C + Float64(A + A))))) / Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0)));
	elseif (F <= 4.6e-85)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B, C)))))) / B);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(F) * Float64(-sqrt(B))));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, -1e-309], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.6e-85], N[((-N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.000000000000002e-309

   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. Simplified 25.0%

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

   4. Taylor expanded in A around inf 31.2%

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

      1. sqrt-prod 43.9%

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

      2. cancel-sign-sub-inv 43.9%

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

      3. metadata-eval 43.9%

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

      4. *-commutative 43.9%

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

      5. associate-*r* 43.9%

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

      6. fma-udef 43.9%

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

      7. *-commutative 43.9%

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

      8. +-commutative 43.9%

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

   6. Applied egg-rr 43.9%

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

      1. associate-+r+ 43.9%

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

   8. Simplified 43.9%

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

   if -1.000000000000002e-309 < F < 4.6000000000000001e-85

   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. Simplified 19.8%

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

   4. Taylor expanded in A around 0 16.5%

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

      1. mul-1-neg 16.5%

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

      2. distribute-rgt-neg-in 16.5%

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

      3. +-commutative 16.5%

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

      4. unpow2 16.5%

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

      5. unpow2 16.5%

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

      6. hypot-def 27.5%

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

   6. Simplified 27.5%

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

      1. sqrt-prod 28.5%

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

   8. Applied egg-rr 28.5%

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

      1. sqrt-prod 27.5%

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

      2. distribute-rgt-neg-in 27.5%

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

      3. neg-sub0 27.5%

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

      4. add-sqr-sqrt 26.4%

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

      5. cancel-sign-sub-inv 26.4%

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

      6. associate-*l/ 26.4%

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

      7. sqrt-unprod 26.4%

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

   10. Applied egg-rr 26.4%

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

       1. +-lft-identity 26.4%

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

       2. distribute-lft-neg-out 26.4%

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

       3. rem-square-sqrt 27.7%

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

       4. distribute-neg-frac 27.7%

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

   12. Simplified 27.7%

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

   if 4.6000000000000001e-85 < F

   1. Initial program 10.1%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in A around 0 7.8%

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

      1. mul-1-neg 7.8%

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

      2. distribute-rgt-neg-in 7.8%

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

      3. +-commutative 7.8%

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

      4. unpow2 7.8%

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

      5. unpow2 7.8%

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

      6. hypot-def 10.0%

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

   6. Simplified 10.0%

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

   7. Taylor expanded in C around 0 9.2%

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

      1. sqrt-prod 17.1%

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

   9. Applied egg-rr 17.1%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + A\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 7: 41.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;F \leq 2.6 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= F 2.6e-308)
     (*
      (sqrt (* 2.0 (* (* F t_0) (+ A (+ C (hypot B (- A C)))))))
      (/ -1.0 t_0))
     (if (<= F 3.1e-86)
       (/ (- (sqrt (* F (* 2.0 (+ C (hypot B C)))))) B)
       (* (/ (sqrt 2.0) B) (* (sqrt F) (- (sqrt B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (F <= 2.6e-308) {
		tmp = sqrt((2.0 * ((F * t_0) * (A + (C + hypot(B, (A - C))))))) * (-1.0 / t_0);
	} else if (F <= 3.1e-86) {
		tmp = -sqrt((F * (2.0 * (C + hypot(B, C))))) / B;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt(B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (F <= 2.6e-308) {
		tmp = Math.sqrt((2.0 * ((F * t_0) * (A + (C + Math.hypot(B, (A - C))))))) * (-1.0 / t_0);
	} else if (F <= 3.1e-86) {
		tmp = -Math.sqrt((F * (2.0 * (C + Math.hypot(B, C))))) / B;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt(F) * -Math.sqrt(B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	tmp = 0
	if F <= 2.6e-308:
		tmp = math.sqrt((2.0 * ((F * t_0) * (A + (C + math.hypot(B, (A - C))))))) * (-1.0 / t_0)
	elif F <= 3.1e-86:
		tmp = -math.sqrt((F * (2.0 * (C + math.hypot(B, C))))) / B
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt(F) * -math.sqrt(B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (F <= 2.6e-308)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(A + Float64(C + hypot(B, Float64(A - C))))))) * Float64(-1.0 / t_0));
	elseif (F <= 3.1e-86)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B, C)))))) / B);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(F) * Float64(-sqrt(B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	tmp = 0.0;
	if (F <= 2.6e-308)
		tmp = sqrt((2.0 * ((F * t_0) * (A + (C + hypot(B, (A - C))))))) * (-1.0 / t_0);
	elseif (F <= 3.1e-86)
		tmp = -sqrt((F * (2.0 * (C + hypot(B, C))))) / B;
	else
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt(B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 2.6e-308], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.1e-86], N[((-N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.6e-308

   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. Simplified 25.0%

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

      1. div-inv 25.1%

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

   5. Applied egg-rr 41.6%

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

   if 2.6e-308 < F < 3.09999999999999989e-86

   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. Simplified 19.8%

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

   4. Taylor expanded in A around 0 16.5%

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

      1. mul-1-neg 16.5%

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

      2. distribute-rgt-neg-in 16.5%

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

      3. +-commutative 16.5%

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

      4. unpow2 16.5%

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

      5. unpow2 16.5%

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

      6. hypot-def 27.5%

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

   6. Simplified 27.5%

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

      1. sqrt-prod 28.5%

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

   8. Applied egg-rr 28.5%

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

      1. sqrt-prod 27.5%

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

      2. distribute-rgt-neg-in 27.5%

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

      3. neg-sub0 27.5%

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

      4. add-sqr-sqrt 26.4%

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

      5. cancel-sign-sub-inv 26.4%

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

      6. associate-*l/ 26.4%

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

      7. sqrt-unprod 26.4%

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

   10. Applied egg-rr 26.4%

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

       1. +-lft-identity 26.4%

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

       2. distribute-lft-neg-out 26.4%

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

       3. rem-square-sqrt 27.7%

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

       4. distribute-neg-frac 27.7%

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

   12. Simplified 27.7%

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

   if 3.09999999999999989e-86 < F

   1. Initial program 10.1%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in A around 0 7.8%

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

      1. mul-1-neg 7.8%

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

      2. distribute-rgt-neg-in 7.8%

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

      3. +-commutative 7.8%

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

      4. unpow2 7.8%

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

      5. unpow2 7.8%

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

      6. hypot-def 10.0%

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

   6. Simplified 10.0%

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

   7. Taylor expanded in C around 0 9.2%

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

      1. sqrt-prod 17.1%

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

   9. Applied egg-rr 17.1%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.6 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 8: 41.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;F \leq 1.8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= F 1.8e-308)
     (*
      (sqrt (* 2.0 (* (* F t_0) (+ A (+ C (hypot B (- A C)))))))
      (/ -1.0 t_0))
     (if (<= F 3.5e-30)
       (/ (- (sqrt (* F (* 2.0 (+ C (hypot B C)))))) B)
       (* (sqrt 2.0) (- (sqrt (/ F B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (F <= 1.8e-308) {
		tmp = sqrt((2.0 * ((F * t_0) * (A + (C + hypot(B, (A - C))))))) * (-1.0 / t_0);
	} else if (F <= 3.5e-30) {
		tmp = -sqrt((F * (2.0 * (C + hypot(B, C))))) / B;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (F <= 1.8e-308) {
		tmp = Math.sqrt((2.0 * ((F * t_0) * (A + (C + Math.hypot(B, (A - C))))))) * (-1.0 / t_0);
	} else if (F <= 3.5e-30) {
		tmp = -Math.sqrt((F * (2.0 * (C + Math.hypot(B, C))))) / B;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	tmp = 0
	if F <= 1.8e-308:
		tmp = math.sqrt((2.0 * ((F * t_0) * (A + (C + math.hypot(B, (A - C))))))) * (-1.0 / t_0)
	elif F <= 3.5e-30:
		tmp = -math.sqrt((F * (2.0 * (C + math.hypot(B, C))))) / B
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (F <= 1.8e-308)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(A + Float64(C + hypot(B, Float64(A - C))))))) * Float64(-1.0 / t_0));
	elseif (F <= 3.5e-30)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B, C)))))) / B);
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	tmp = 0.0;
	if (F <= 1.8e-308)
		tmp = sqrt((2.0 * ((F * t_0) * (A + (C + hypot(B, (A - C))))))) * (-1.0 / t_0);
	elseif (F <= 3.5e-30)
		tmp = -sqrt((F * (2.0 * (C + hypot(B, C))))) / B;
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 1.8e-308], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-30], N[((-N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.7999999999999999e-308

   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. Simplified 25.0%

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

      1. div-inv 25.1%

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

   5. Applied egg-rr 41.6%

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

   if 1.7999999999999999e-308 < F < 3.5000000000000003e-30

   1. Initial program 19.1%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in A around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. distribute-rgt-neg-in 15.7%

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

      3. +-commutative 15.7%

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

      4. unpow2 15.7%

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

      5. unpow2 15.7%

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

      6. hypot-def 25.7%

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

   6. Simplified 25.7%

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

      1. sqrt-prod 26.5%

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

   8. Applied egg-rr 26.5%

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

      1. sqrt-prod 25.7%

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

      2. distribute-rgt-neg-in 25.7%

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

      3. neg-sub0 25.7%

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

      4. add-sqr-sqrt 24.7%

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

      5. cancel-sign-sub-inv 24.7%

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

      6. associate-*l/ 24.7%

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

      7. sqrt-unprod 24.7%

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

   10. Applied egg-rr 24.7%

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

       1. +-lft-identity 24.7%

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

       2. distribute-lft-neg-out 24.7%

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

       3. rem-square-sqrt 25.9%

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

       4. distribute-neg-frac 25.9%

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

   12. Simplified 25.9%

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

   if 3.5000000000000003e-30 < F

   1. Initial program 9.0%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. distribute-rgt-neg-in 7.0%

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

      3. +-commutative 7.0%

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

      4. unpow2 7.0%

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

      5. unpow2 7.0%

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

      6. hypot-def 8.6%

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

   6. Simplified 8.6%

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

   7. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   9. Simplified 16.1%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 9: 41.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= F -1e-309)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (+ A (+ C (hypot B (- A C)))))))) t_0)
     (if (<= F 3.5e-30)
       (/ (- (sqrt (* F (* 2.0 (+ C (hypot B C)))))) B)
       (* (sqrt 2.0) (- (sqrt (/ F B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (F <= -1e-309) {
		tmp = -sqrt((2.0 * ((F * t_0) * (A + (C + hypot(B, (A - C))))))) / t_0;
	} else if (F <= 3.5e-30) {
		tmp = -sqrt((F * (2.0 * (C + hypot(B, C))))) / B;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (F <= -1e-309) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (A + (C + Math.hypot(B, (A - C))))))) / t_0;
	} else if (F <= 3.5e-30) {
		tmp = -Math.sqrt((F * (2.0 * (C + Math.hypot(B, C))))) / B;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	tmp = 0
	if F <= -1e-309:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (A + (C + math.hypot(B, (A - C))))))) / t_0
	elif F <= 3.5e-30:
		tmp = -math.sqrt((F * (2.0 * (C + math.hypot(B, C))))) / B
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (F <= -1e-309)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(A + Float64(C + hypot(B, Float64(A - C)))))))) / t_0);
	elseif (F <= 3.5e-30)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B, C)))))) / B);
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	tmp = 0.0;
	if (F <= -1e-309)
		tmp = -sqrt((2.0 * ((F * t_0) * (A + (C + hypot(B, (A - C))))))) / t_0;
	elseif (F <= 3.5e-30)
		tmp = -sqrt((F * (2.0 * (C + hypot(B, C))))) / B;
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-309], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[F, 3.5e-30], N[((-N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.000000000000002e-309

   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. Simplified 25.0%

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

      1. distribute-frac-neg 25.0%

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

   5. Applied egg-rr 41.6%

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

   if -1.000000000000002e-309 < F < 3.5000000000000003e-30

   1. Initial program 19.1%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in A around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. distribute-rgt-neg-in 15.7%

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

      3. +-commutative 15.7%

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

      4. unpow2 15.7%

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

      5. unpow2 15.7%

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

      6. hypot-def 25.7%

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

   6. Simplified 25.7%

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

      1. sqrt-prod 26.5%

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

   8. Applied egg-rr 26.5%

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

      1. sqrt-prod 25.7%

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

      2. distribute-rgt-neg-in 25.7%

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

      3. neg-sub0 25.7%

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

      4. add-sqr-sqrt 24.7%

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

      5. cancel-sign-sub-inv 24.7%

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

      6. associate-*l/ 24.7%

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

      7. sqrt-unprod 24.7%

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

   10. Applied egg-rr 24.7%

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

       1. +-lft-identity 24.7%

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

       2. distribute-lft-neg-out 24.7%

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

       3. rem-square-sqrt 25.9%

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

       4. distribute-neg-frac 25.9%

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

   12. Simplified 25.9%

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

   if 3.5000000000000003e-30 < F

   1. Initial program 9.0%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. distribute-rgt-neg-in 7.0%

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

      3. +-commutative 7.0%

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

      4. unpow2 7.0%

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

      5. unpow2 7.0%

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

      6. hypot-def 8.6%

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

   6. Simplified 8.6%

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

   7. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   9. Simplified 16.1%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 10: 39.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0))))
   (if (<= F -1e-309)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (+ A C))))) t_0)
     (if (<= F 3.5e-30)
       (/ (- (sqrt (* F (* 2.0 (+ C (hypot B C)))))) B)
       (* (sqrt 2.0) (- (sqrt (/ F B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 3.5e-30) {
		tmp = -sqrt((F * (2.0 * (C + hypot(B, C))))) / B;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 3.5e-30) {
		tmp = -Math.sqrt((F * (2.0 * (C + Math.hypot(B, C))))) / B;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - ((A * C) * 4.0)
	tmp = 0
	if F <= -1e-309:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0
	elif F <= 3.5e-30:
		tmp = -math.sqrt((F * (2.0 * (C + math.hypot(B, C))))) / B
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (F <= -1e-309)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + Float64(A + C))))) / t_0);
	elseif (F <= 3.5e-30)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B, C)))))) / B);
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	tmp = 0.0;
	if (F <= -1e-309)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	elseif (F <= 3.5e-30)
		tmp = -sqrt((F * (2.0 * (C + hypot(B, C))))) / B;
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-309], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[F, 3.5e-30], N[((-N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.000000000000002e-309

   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. Simplified 25.0%

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

   4. Taylor expanded in A around inf 31.2%

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

   if -1.000000000000002e-309 < F < 3.5000000000000003e-30

   1. Initial program 19.1%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in A around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. distribute-rgt-neg-in 15.7%

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

      3. +-commutative 15.7%

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

      4. unpow2 15.7%

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

      5. unpow2 15.7%

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

      6. hypot-def 25.7%

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

   6. Simplified 25.7%

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

      1. sqrt-prod 26.5%

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

   8. Applied egg-rr 26.5%

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

      1. sqrt-prod 25.7%

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

      2. distribute-rgt-neg-in 25.7%

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

      3. neg-sub0 25.7%

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

      4. add-sqr-sqrt 24.7%

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

      5. cancel-sign-sub-inv 24.7%

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

      6. associate-*l/ 24.7%

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

      7. sqrt-unprod 24.7%

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

   10. Applied egg-rr 24.7%

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

       1. +-lft-identity 24.7%

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

       2. distribute-lft-neg-out 24.7%

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

       3. rem-square-sqrt 25.9%

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

       4. distribute-neg-frac 25.9%

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

   12. Simplified 25.9%

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

   if 3.5000000000000003e-30 < F

   1. Initial program 9.0%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. distribute-rgt-neg-in 7.0%

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

      3. +-commutative 7.0%

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

      4. unpow2 7.0%

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

      5. unpow2 7.0%

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

      6. hypot-def 8.6%

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

   6. Simplified 8.6%

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

   7. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   9. Simplified 16.1%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 11: 37.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 7.1 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{B \cdot F} \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0))))
   (if (<= F -1e-309)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (+ A C))))) t_0)
     (if (<= F 7.1e-31)
       (* (sqrt (* B F)) (* (sqrt 2.0) (/ -1.0 B)))
       (* (sqrt 2.0) (- (sqrt (/ F B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 7.1e-31) {
		tmp = sqrt((B * F)) * (sqrt(2.0) * (-1.0 / B));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - ((a * c) * 4.0d0)
    if (f <= (-1d-309)) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (a + (a + c)))) / t_0
    else if (f <= 7.1d-31) then
        tmp = sqrt((b * f)) * (sqrt(2.0d0) * ((-1.0d0) / b))
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 7.1e-31) {
		tmp = Math.sqrt((B * F)) * (Math.sqrt(2.0) * (-1.0 / B));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - ((A * C) * 4.0)
	tmp = 0
	if F <= -1e-309:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0
	elif F <= 7.1e-31:
		tmp = math.sqrt((B * F)) * (math.sqrt(2.0) * (-1.0 / B))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (F <= -1e-309)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + Float64(A + C))))) / t_0);
	elseif (F <= 7.1e-31)
		tmp = Float64(sqrt(Float64(B * F)) * Float64(sqrt(2.0) * Float64(-1.0 / B)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	tmp = 0.0;
	if (F <= -1e-309)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	elseif (F <= 7.1e-31)
		tmp = sqrt((B * F)) * (sqrt(2.0) * (-1.0 / B));
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-309], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[F, 7.1e-31], N[(N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.000000000000002e-309

   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. Simplified 25.0%

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

   4. Taylor expanded in A around inf 31.2%

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

   if -1.000000000000002e-309 < F < 7.0999999999999999e-31

   1. Initial program 19.1%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in A around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. distribute-rgt-neg-in 15.7%

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

      3. +-commutative 15.7%

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

      4. unpow2 15.7%

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

      5. unpow2 15.7%

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

      6. hypot-def 25.7%

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

   6. Simplified 25.7%

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

   7. Taylor expanded in C around 0 21.2%

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

      1. div-inv 21.4%

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

   9. Applied egg-rr 21.4%

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

   if 7.0999999999999999e-31 < F

   1. Initial program 9.0%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. distribute-rgt-neg-in 7.0%

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

      3. +-commutative 7.0%

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

      4. unpow2 7.0%

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

      5. unpow2 7.0%

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

      6. hypot-def 8.6%

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

   6. Simplified 8.6%

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

   7. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   9. Simplified 16.1%

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

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;F \leq 7.1 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{B \cdot F} \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 12: 37.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{B \cdot F}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0))))
   (if (<= F -1e-309)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (+ A C))))) t_0)
     (if (<= F 3.2e-30)
       (/ (* (sqrt 2.0) (- (sqrt (* B F)))) B)
       (* (sqrt 2.0) (- (sqrt (/ F B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 3.2e-30) {
		tmp = (sqrt(2.0) * -sqrt((B * F))) / B;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - ((a * c) * 4.0d0)
    if (f <= (-1d-309)) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (a + (a + c)))) / t_0
    else if (f <= 3.2d-30) then
        tmp = (sqrt(2.0d0) * -sqrt((b * f))) / b
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 3.2e-30) {
		tmp = (Math.sqrt(2.0) * -Math.sqrt((B * F))) / B;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - ((A * C) * 4.0)
	tmp = 0
	if F <= -1e-309:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0
	elif F <= 3.2e-30:
		tmp = (math.sqrt(2.0) * -math.sqrt((B * F))) / B
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (F <= -1e-309)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + Float64(A + C))))) / t_0);
	elseif (F <= 3.2e-30)
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(B * F)))) / B);
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	tmp = 0.0;
	if (F <= -1e-309)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	elseif (F <= 3.2e-30)
		tmp = (sqrt(2.0) * -sqrt((B * F))) / B;
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-309], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[F, 3.2e-30], N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / B), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.000000000000002e-309

   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. Simplified 25.0%

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

   4. Taylor expanded in A around inf 31.2%

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

   if -1.000000000000002e-309 < F < 3.2e-30

   1. Initial program 19.1%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in A around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. distribute-rgt-neg-in 15.7%

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

      3. +-commutative 15.7%

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

      4. unpow2 15.7%

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

      5. unpow2 15.7%

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

      6. hypot-def 25.7%

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

   6. Simplified 25.7%

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

   7. Taylor expanded in C around 0 21.2%

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

      1. associate-*l/ 21.3%

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

      2. *-commutative 21.3%

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

   9. Applied egg-rr 21.3%

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

   if 3.2e-30 < F

   1. Initial program 9.0%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. distribute-rgt-neg-in 7.0%

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

      3. +-commutative 7.0%

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

      4. unpow2 7.0%

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

      5. unpow2 7.0%

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

      6. hypot-def 8.6%

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

   6. Simplified 8.6%

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

   7. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   9. Simplified 16.1%

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

4. Final simplification 20.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{B \cdot F}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 13: 37.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(B + C\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0))))
   (if (<= F -1e-309)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (+ A C))))) t_0)
     (if (<= F 3.4e-30)
       (/ (- (sqrt (* F (* 2.0 (+ B C))))) B)
       (* (sqrt 2.0) (- (sqrt (/ F B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 3.4e-30) {
		tmp = -sqrt((F * (2.0 * (B + C)))) / B;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - ((a * c) * 4.0d0)
    if (f <= (-1d-309)) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (a + (a + c)))) / t_0
    else if (f <= 3.4d-30) then
        tmp = -sqrt((f * (2.0d0 * (b + c)))) / b
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 3.4e-30) {
		tmp = -Math.sqrt((F * (2.0 * (B + C)))) / B;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - ((A * C) * 4.0)
	tmp = 0
	if F <= -1e-309:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0
	elif F <= 3.4e-30:
		tmp = -math.sqrt((F * (2.0 * (B + C)))) / B
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (F <= -1e-309)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + Float64(A + C))))) / t_0);
	elseif (F <= 3.4e-30)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(B + C))))) / B);
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	tmp = 0.0;
	if (F <= -1e-309)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	elseif (F <= 3.4e-30)
		tmp = -sqrt((F * (2.0 * (B + C)))) / B;
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-309], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[F, 3.4e-30], N[((-N[Sqrt[N[(F * N[(2.0 * N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.000000000000002e-309

   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. Simplified 25.0%

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

   4. Taylor expanded in A around inf 31.2%

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

   if -1.000000000000002e-309 < F < 3.4000000000000003e-30

   1. Initial program 19.1%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in A around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. distribute-rgt-neg-in 15.7%

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

      3. +-commutative 15.7%

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

      4. unpow2 15.7%

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

      5. unpow2 15.7%

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

      6. hypot-def 25.7%

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

   6. Simplified 25.7%

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

   7. Taylor expanded in C around 0 21.2%

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

      1. +-commutative 21.2%

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

   9. Simplified 21.2%

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

       1. *-commutative 21.2%

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

       2. sqrt-prod 21.9%

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

       3. +-commutative 21.9%

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

   11. Applied egg-rr 21.9%

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

       1. expm1-log1p-u 21.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B + C} \cdot \sqrt{F}\right)\right)\right)} \]

       2. expm1-udef 2.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B + C} \cdot \sqrt{F}\right)\right)} - 1} \]

       3. distribute-rgt-neg-out 2.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \sqrt{F}\right)}\right)} - 1 \]

       4. *-commutative 2.0%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{B + C}\right)}\right)} - 1 \]

       5. sqrt-prod 2.2%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(B + C\right)}}\right)} - 1 \]

       6. associate-*l/ 2.2%

          \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(B + C\right)}}{B}}\right)} - 1 \]

       7. pow1/2 2.2%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(B + C\right)}}{B}\right)} - 1 \]

       8. pow1/2 2.2%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(B + C\right)\right)}^{0.5}}}{B}\right)} - 1 \]

       9. pow-prod-down 2.2%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(B + C\right)\right)\right)}^{0.5}}}{B}\right)} - 1 \]

   13. Applied egg-rr 2.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(B + C\right)\right)\right)}^{0.5}}{B}\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 20.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(B + C\right)\right)\right)}^{0.5}}{B}\right)\right)} \]

       2. expm1-log1p 21.3%

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

       3. distribute-neg-frac 21.3%

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

       4. unpow1/2 21.3%

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

       5. *-commutative 21.3%

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

       6. associate-*r* 21.3%

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

   15. Simplified 21.3%

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

   if 3.4000000000000003e-30 < F

   1. Initial program 9.0%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. distribute-rgt-neg-in 7.0%

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

      3. +-commutative 7.0%

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

      4. unpow2 7.0%

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

      5. unpow2 7.0%

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

      6. hypot-def 8.6%

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

   6. Simplified 8.6%

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

   7. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   9. Simplified 16.1%

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

4. Final simplification 20.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(B + C\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 14: 29.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(B + C\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0))))
   (if (<= F -1e-309)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (+ A C))))) t_0)
     (/ (- (sqrt (* F (* 2.0 (+ B C))))) B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else {
		tmp = -sqrt((F * (2.0 * (B + C)))) / B;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - ((a * c) * 4.0d0)
    if (f <= (-1d-309)) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (a + (a + c)))) / t_0
    else
        tmp = -sqrt((f * (2.0d0 * (b + c)))) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (F <= -1e-309) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else {
		tmp = -Math.sqrt((F * (2.0 * (B + C)))) / B;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - ((A * C) * 4.0)
	tmp = 0
	if F <= -1e-309:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0
	else:
		tmp = -math.sqrt((F * (2.0 * (B + C)))) / B
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (F <= -1e-309)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + Float64(A + C))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(B + C))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	tmp = 0.0;
	if (F <= -1e-309)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	else
		tmp = -sqrt((F * (2.0 * (B + C)))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-309], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(F * N[(2.0 * N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < -1.000000000000002e-309

   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. Simplified 25.0%

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

   4. Taylor expanded in A around inf 31.2%

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

   if -1.000000000000002e-309 < F

   1. Initial program 13.9%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in A around 0 11.2%

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

      1. mul-1-neg 11.2%

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

      2. distribute-rgt-neg-in 11.2%

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

      3. +-commutative 11.2%

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

      4. unpow2 11.2%

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

      5. unpow2 11.2%

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

      6. hypot-def 16.9%

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

   6. Simplified 16.9%

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

   7. Taylor expanded in C around 0 14.0%

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

      1. +-commutative 14.0%

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

   9. Simplified 14.0%

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

       1. *-commutative 14.0%

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

       2. sqrt-prod 19.3%

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

       3. +-commutative 19.3%

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

   11. Applied egg-rr 19.3%

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

       1. expm1-log1p-u 14.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B + C} \cdot \sqrt{F}\right)\right)\right)} \]

       2. expm1-udef 2.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B + C} \cdot \sqrt{F}\right)\right)} - 1} \]

       3. distribute-rgt-neg-out 2.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \sqrt{F}\right)}\right)} - 1 \]

       4. *-commutative 2.0%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{B + C}\right)}\right)} - 1 \]

       5. sqrt-prod 1.2%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(B + C\right)}}\right)} - 1 \]

       6. associate-*l/ 1.2%

          \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(B + C\right)}}{B}}\right)} - 1 \]

       7. pow1/2 1.2%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(B + C\right)}}{B}\right)} - 1 \]

       8. pow1/2 1.3%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(B + C\right)\right)}^{0.5}}}{B}\right)} - 1 \]

       9. pow-prod-down 1.3%

          \[\leadsto e^{\mathsf{log1p}\left(-\frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(B + C\right)\right)\right)}^{0.5}}}{B}\right)} - 1 \]

   13. Applied egg-rr 1.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(B + C\right)\right)\right)}^{0.5}}{B}\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 10.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(B + C\right)\right)\right)}^{0.5}}{B}\right)\right)} \]

       2. expm1-log1p 14.2%

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

       3. distribute-neg-frac 14.2%

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

       4. unpow1/2 14.1%

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

       5. *-commutative 14.1%

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

       6. associate-*r* 14.1%

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

   15. Simplified 14.1%

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

4. Final simplification 16.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(B + C\right)\right)}}{B}\\ \end{array} \]

Alternative 15: 8.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.35 \cdot 10^{-229}:\\ \;\;\;\;\frac{2}{B} \cdot \left(-{\left(F \cdot A\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(F \cdot C\right)}^{0.5} \cdot \frac{-2}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= C 1.35e-229)
   (* (/ 2.0 B) (- (pow (* F A) 0.5)))
   (* (pow (* F C) 0.5) (/ -2.0 B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 1.35e-229) {
		tmp = (2.0 / B) * -pow((F * A), 0.5);
	} else {
		tmp = pow((F * C), 0.5) * (-2.0 / B);
	}
	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 (c <= 1.35d-229) then
        tmp = (2.0d0 / b) * -((f * a) ** 0.5d0)
    else
        tmp = ((f * c) ** 0.5d0) * ((-2.0d0) / b)
    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 (C <= 1.35e-229) {
		tmp = (2.0 / B) * -Math.pow((F * A), 0.5);
	} else {
		tmp = Math.pow((F * C), 0.5) * (-2.0 / B);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if C <= 1.35e-229:
		tmp = (2.0 / B) * -math.pow((F * A), 0.5)
	else:
		tmp = math.pow((F * C), 0.5) * (-2.0 / B)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (C <= 1.35e-229)
		tmp = Float64(Float64(2.0 / B) * Float64(-(Float64(F * A) ^ 0.5)));
	else
		tmp = Float64((Float64(F * C) ^ 0.5) * Float64(-2.0 / B));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= 1.35e-229)
		tmp = (2.0 / B) * -((F * A) ^ 0.5);
	else
		tmp = ((F * C) ^ 0.5) * (-2.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[C, 1.35e-229], N[(N[(2.0 / B), $MachinePrecision] * (-N[Power[N[(F * A), $MachinePrecision], 0.5], $MachinePrecision])), $MachinePrecision], N[(N[Power[N[(F * C), $MachinePrecision], 0.5], $MachinePrecision] * N[(-2.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.3499999999999999e-229

   1. Initial program 10.4%

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

   3. Simplified 10.4%

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

   4. Taylor expanded in A around inf 3.2%

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

   5. Taylor expanded in C around 0 2.7%

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

      1. mul-1-neg 2.7%

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

      2. unpow2 2.7%

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

      3. rem-square-sqrt 2.7%

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

   7. Simplified 2.7%

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

      1. pow1/2 2.8%

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

      2. *-commutative 2.8%

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

   9. Applied egg-rr 2.8%

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

   if 1.3499999999999999e-229 < C

   1. Initial program 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around 0 14.2%

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

      1. mul-1-neg 14.2%

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

      2. distribute-rgt-neg-in 14.2%

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

      3. +-commutative 14.2%

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

      4. unpow2 14.2%

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

      5. unpow2 14.2%

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

      6. hypot-def 17.0%

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

   6. Simplified 17.0%

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

      1. sqrt-prod 22.7%

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

   8. Applied egg-rr 22.7%

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

   9. Taylor expanded in B around 0 5.5%

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

       1. mul-1-neg 5.5%

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

       2. *-commutative 5.5%

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

       3. distribute-rgt-neg-in 5.5%

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

       4. mul-1-neg 5.5%

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

       5. unpow2 5.5%

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

       6. rem-square-sqrt 5.5%

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

       7. associate-*r/ 5.5%

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

       8. metadata-eval 5.5%

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

   11. Simplified 5.5%

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

       1. pow1/2 5.6%

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

       2. *-commutative 5.6%

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

   13. Applied egg-rr 5.6%

       \[\leadsto \color{blue}{{\left(F \cdot C\right)}^{0.5}} \cdot \frac{-2}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.35 \cdot 10^{-229}:\\ \;\;\;\;\frac{2}{B} \cdot \left(-{\left(F \cdot A\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(F \cdot C\right)}^{0.5} \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 16: 8.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{2}{B} \cdot \left(-\sqrt{F \cdot A}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B} \cdot \sqrt{F \cdot C}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= C 1.3e-230)
   (* (/ 2.0 B) (- (sqrt (* F A))))
   (* (/ -2.0 B) (sqrt (* F C)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 1.3e-230) {
		tmp = (2.0 / B) * -sqrt((F * A));
	} else {
		tmp = (-2.0 / B) * sqrt((F * C));
	}
	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 (c <= 1.3d-230) then
        tmp = (2.0d0 / b) * -sqrt((f * a))
    else
        tmp = ((-2.0d0) / b) * sqrt((f * c))
    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 (C <= 1.3e-230) {
		tmp = (2.0 / B) * -Math.sqrt((F * A));
	} else {
		tmp = (-2.0 / B) * Math.sqrt((F * C));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if C <= 1.3e-230:
		tmp = (2.0 / B) * -math.sqrt((F * A))
	else:
		tmp = (-2.0 / B) * math.sqrt((F * C))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (C <= 1.3e-230)
		tmp = Float64(Float64(2.0 / B) * Float64(-sqrt(Float64(F * A))));
	else
		tmp = Float64(Float64(-2.0 / B) * sqrt(Float64(F * C)));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= 1.3e-230)
		tmp = (2.0 / B) * -sqrt((F * A));
	else
		tmp = (-2.0 / B) * sqrt((F * C));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[C, 1.3e-230], N[(N[(2.0 / B), $MachinePrecision] * (-N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(-2.0 / B), $MachinePrecision] * N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.3000000000000001e-230

   1. Initial program 10.4%

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

   3. Simplified 10.4%

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

   4. Taylor expanded in A around inf 3.2%

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

   5. Taylor expanded in C around 0 2.7%

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

      1. mul-1-neg 2.7%

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

      2. unpow2 2.7%

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

      3. rem-square-sqrt 2.7%

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

   7. Simplified 2.7%

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

   if 1.3000000000000001e-230 < C

   1. Initial program 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around 0 14.2%

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

      1. mul-1-neg 14.2%

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

      2. distribute-rgt-neg-in 14.2%

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

      3. +-commutative 14.2%

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

      4. unpow2 14.2%

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

      5. unpow2 14.2%

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

      6. hypot-def 17.0%

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

   6. Simplified 17.0%

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

      1. sqrt-prod 22.7%

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

   8. Applied egg-rr 22.7%

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

   9. Taylor expanded in B around 0 5.5%

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

       1. mul-1-neg 5.5%

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

       2. *-commutative 5.5%

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

       3. distribute-rgt-neg-in 5.5%

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

       4. mul-1-neg 5.5%

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

       5. unpow2 5.5%

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

       6. rem-square-sqrt 5.5%

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

       7. associate-*r/ 5.5%

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

       8. metadata-eval 5.5%

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

   11. Simplified 5.5%

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

4. Final simplification 4.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{2}{B} \cdot \left(-\sqrt{F \cdot A}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B} \cdot \sqrt{F \cdot C}\\ \end{array} \]

Alternative 17: 8.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 3.3 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt{F \cdot A} \cdot \left(-2\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B} \cdot \sqrt{F \cdot C}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= C 3.3e-233)
   (/ (* (sqrt (* F A)) (- 2.0)) B)
   (* (/ -2.0 B) (sqrt (* F C)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 3.3e-233) {
		tmp = (sqrt((F * A)) * -2.0) / B;
	} else {
		tmp = (-2.0 / B) * sqrt((F * C));
	}
	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 (c <= 3.3d-233) then
        tmp = (sqrt((f * a)) * -2.0d0) / b
    else
        tmp = ((-2.0d0) / b) * sqrt((f * c))
    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 (C <= 3.3e-233) {
		tmp = (Math.sqrt((F * A)) * -2.0) / B;
	} else {
		tmp = (-2.0 / B) * Math.sqrt((F * C));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if C <= 3.3e-233:
		tmp = (math.sqrt((F * A)) * -2.0) / B
	else:
		tmp = (-2.0 / B) * math.sqrt((F * C))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (C <= 3.3e-233)
		tmp = Float64(Float64(sqrt(Float64(F * A)) * Float64(-2.0)) / B);
	else
		tmp = Float64(Float64(-2.0 / B) * sqrt(Float64(F * C)));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= 3.3e-233)
		tmp = (sqrt((F * A)) * -2.0) / B;
	else
		tmp = (-2.0 / B) * sqrt((F * C));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[C, 3.3e-233], N[(N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] * (-2.0)), $MachinePrecision] / B), $MachinePrecision], N[(N[(-2.0 / B), $MachinePrecision] * N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 3.3e-233

   1. Initial program 10.4%

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

   3. Simplified 10.4%

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

   4. Taylor expanded in A around inf 3.2%

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

   5. Taylor expanded in C around 0 2.7%

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

      1. mul-1-neg 2.7%

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

      2. unpow2 2.7%

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

      3. rem-square-sqrt 2.7%

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

   7. Simplified 2.7%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

   9. Applied egg-rr 2.7%

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

   if 3.3e-233 < C

   1. Initial program 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around 0 14.2%

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

      1. mul-1-neg 14.2%

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

      2. distribute-rgt-neg-in 14.2%

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

      3. +-commutative 14.2%

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

      4. unpow2 14.2%

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

      5. unpow2 14.2%

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

      6. hypot-def 17.0%

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

   6. Simplified 17.0%

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

      1. sqrt-prod 22.7%

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

   8. Applied egg-rr 22.7%

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

   9. Taylor expanded in B around 0 5.5%

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

       1. mul-1-neg 5.5%

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

       2. *-commutative 5.5%

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

       3. distribute-rgt-neg-in 5.5%

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

       4. mul-1-neg 5.5%

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

       5. unpow2 5.5%

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

       6. rem-square-sqrt 5.5%

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

       7. associate-*r/ 5.5%

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

       8. metadata-eval 5.5%

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

   11. Simplified 5.5%

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

4. Final simplification 4.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3.3 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt{F \cdot A} \cdot \left(-2\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B} \cdot \sqrt{F \cdot C}\\ \end{array} \]

Alternative 18: 8.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 8.4 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{F \cdot A} \cdot \left(-2\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;{\left(F \cdot C\right)}^{0.5} \cdot \frac{-2}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= C 8.4e-227)
   (/ (* (sqrt (* F A)) (- 2.0)) B)
   (* (pow (* F C) 0.5) (/ -2.0 B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 8.4e-227) {
		tmp = (sqrt((F * A)) * -2.0) / B;
	} else {
		tmp = pow((F * C), 0.5) * (-2.0 / B);
	}
	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 (c <= 8.4d-227) then
        tmp = (sqrt((f * a)) * -2.0d0) / b
    else
        tmp = ((f * c) ** 0.5d0) * ((-2.0d0) / b)
    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 (C <= 8.4e-227) {
		tmp = (Math.sqrt((F * A)) * -2.0) / B;
	} else {
		tmp = Math.pow((F * C), 0.5) * (-2.0 / B);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if C <= 8.4e-227:
		tmp = (math.sqrt((F * A)) * -2.0) / B
	else:
		tmp = math.pow((F * C), 0.5) * (-2.0 / B)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (C <= 8.4e-227)
		tmp = Float64(Float64(sqrt(Float64(F * A)) * Float64(-2.0)) / B);
	else
		tmp = Float64((Float64(F * C) ^ 0.5) * Float64(-2.0 / B));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= 8.4e-227)
		tmp = (sqrt((F * A)) * -2.0) / B;
	else
		tmp = ((F * C) ^ 0.5) * (-2.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[C, 8.4e-227], N[(N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] * (-2.0)), $MachinePrecision] / B), $MachinePrecision], N[(N[Power[N[(F * C), $MachinePrecision], 0.5], $MachinePrecision] * N[(-2.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 8.3999999999999999e-227

   1. Initial program 10.4%

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

   3. Simplified 10.4%

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

   4. Taylor expanded in A around inf 3.2%

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

   5. Taylor expanded in C around 0 2.7%

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

      1. mul-1-neg 2.7%

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

      2. unpow2 2.7%

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

      3. rem-square-sqrt 2.7%

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

   7. Simplified 2.7%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

   9. Applied egg-rr 2.7%

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

   if 8.3999999999999999e-227 < C

   1. Initial program 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around 0 14.2%

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

      1. mul-1-neg 14.2%

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

      2. distribute-rgt-neg-in 14.2%

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

      3. +-commutative 14.2%

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

      4. unpow2 14.2%

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

      5. unpow2 14.2%

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

      6. hypot-def 17.0%

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

   6. Simplified 17.0%

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

      1. sqrt-prod 22.7%

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

   8. Applied egg-rr 22.7%

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

   9. Taylor expanded in B around 0 5.5%

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

       1. mul-1-neg 5.5%

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

       2. *-commutative 5.5%

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

       3. distribute-rgt-neg-in 5.5%

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

       4. mul-1-neg 5.5%

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

       5. unpow2 5.5%

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

       6. rem-square-sqrt 5.5%

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

       7. associate-*r/ 5.5%

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

       8. metadata-eval 5.5%

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

   11. Simplified 5.5%

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

       1. pow1/2 5.6%

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

       2. *-commutative 5.6%

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

   13. Applied egg-rr 5.6%

       \[\leadsto \color{blue}{{\left(F \cdot C\right)}^{0.5}} \cdot \frac{-2}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 8.4 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{F \cdot A} \cdot \left(-2\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;{\left(F \cdot C\right)}^{0.5} \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 19: 25.7% accurate, 5.8× speedup

Math

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

FPCore

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

C

B = abs(B);
double code(double A, double B, double C, double F) {
	return -sqrt((F * (2.0 * (B + C)))) / 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 = -sqrt((f * (2.0d0 * (b + c)))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.4%

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

3. Simplified 15.4%

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

4. Taylor expanded in A around 0 9.7%

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

   1. mul-1-neg 9.7%

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

   2. distribute-rgt-neg-in 9.7%

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

   3. +-commutative 9.7%

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

   4. unpow2 9.7%

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

   5. unpow2 9.7%

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

   6. hypot-def 14.6%

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

6. Simplified 14.6%

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

7. Taylor expanded in C around 0 12.2%

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

   1. +-commutative 12.2%

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

9. Simplified 12.2%

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

    1. *-commutative 12.2%

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

    2. sqrt-prod 16.7%

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

    3. +-commutative 16.7%

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

11. Applied egg-rr 16.7%

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

    1. expm1-log1p-u 12.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B + C} \cdot \sqrt{F}\right)\right)\right)} \]

    2. expm1-udef 1.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B + C} \cdot \sqrt{F}\right)\right)} - 1} \]

    3. distribute-rgt-neg-out 1.7%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \sqrt{F}\right)}\right)} - 1 \]

    4. *-commutative 1.7%

       \[\leadsto e^{\mathsf{log1p}\left(-\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{B + C}\right)}\right)} - 1 \]

    5. sqrt-prod 1.1%

       \[\leadsto e^{\mathsf{log1p}\left(-\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(B + C\right)}}\right)} - 1 \]

    6. associate-*l/ 1.1%

       \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(B + C\right)}}{B}}\right)} - 1 \]

    7. pow1/2 1.1%

       \[\leadsto e^{\mathsf{log1p}\left(-\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(B + C\right)}}{B}\right)} - 1 \]

    8. pow1/2 1.2%

       \[\leadsto e^{\mathsf{log1p}\left(-\frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(B + C\right)\right)}^{0.5}}}{B}\right)} - 1 \]

    9. pow-prod-down 1.2%

       \[\leadsto e^{\mathsf{log1p}\left(-\frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(B + C\right)\right)\right)}^{0.5}}}{B}\right)} - 1 \]

13. Applied egg-rr 1.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(B + C\right)\right)\right)}^{0.5}}{B}\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 9.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(B + C\right)\right)\right)}^{0.5}}{B}\right)\right)} \]

    2. expm1-log1p 12.4%

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

    3. distribute-neg-frac 12.4%

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

    4. unpow1/2 12.2%

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

    5. *-commutative 12.2%

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

    6. associate-*r* 12.2%

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

15. Simplified 12.2%

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

16. Final simplification 12.2%

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

Alternative 20: 5.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

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) / b) * sqrt((f * c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.4%

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

3. Simplified 15.4%

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

4. Taylor expanded in A around 0 9.7%

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

   1. mul-1-neg 9.7%

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

   2. distribute-rgt-neg-in 9.7%

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

   3. +-commutative 9.7%

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

   4. unpow2 9.7%

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

   5. unpow2 9.7%

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

   6. hypot-def 14.6%

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

6. Simplified 14.6%

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

   1. sqrt-prod 20.2%

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

8. Applied egg-rr 20.2%

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

9. Taylor expanded in B around 0 2.9%

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

    1. mul-1-neg 2.9%

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

    2. *-commutative 2.9%

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

    3. distribute-rgt-neg-in 2.9%

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

    4. mul-1-neg 2.9%

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

    5. unpow2 2.9%

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

    6. rem-square-sqrt 2.9%

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

    7. associate-*r/ 2.9%

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

    8. metadata-eval 2.9%

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

11. Simplified 2.9%

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

12. Final simplification 2.9%

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

Reproduce

herbie shell --seed 2023275 
(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))))