ABCF->ab-angle b

Percentage Accurate: 19.0% → 49.4%

Time: 22.2s

Alternatives: 9

Speedup: 3.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 49.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 1.45 \cdot 10^{+144}:\\ \;\;\;\;-\sqrt{\frac{\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)} \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;B\_m \leq 5.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 2e-9)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= B_m 1.45e+144)
       (-
        (sqrt
         (*
          (/ (- (+ A C) (hypot B_m (- A C))) (fma -4.0 (* A C) (pow B_m 2.0)))
          (* 2.0 F))))
       (if (<= B_m 5.5e+254)
         (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m A))))))
         (*
          (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
          (- (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 2e-9) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 1.45e+144) {
		tmp = -sqrt(((((A + C) - hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))) * (2.0 * F)));
	} else if (B_m <= 5.5e+254) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 2e-9)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 1.45e+144)
		tmp = Float64(-sqrt(Float64(Float64(Float64(Float64(A + C) - hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))) * Float64(2.0 * F))));
	elseif (B_m <= 5.5e+254)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 2e-9], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.45e+144], (-N[Sqrt[N[(N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 5.5e+254], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.00000000000000012e-9

   1. Initial program 17.4%

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

   3. Taylor expanded in A around -inf 13.2%

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

   if 2.00000000000000012e-9 < B < 1.44999999999999999e144

   1. Initial program 36.9%

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

   3. Taylor expanded in F around 0 36.8%

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

      1. mul-1-neg 36.8%

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

      2. *-commutative 36.8%

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

      3. associate-/l* 48.0%

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

      4. associate--l+ 48.0%

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

      5. unpow2 48.0%

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

      6. unpow2 48.0%

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

      7. hypot-undefine 57.3%

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

      8. cancel-sign-sub-inv 57.3%

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

   5. Simplified 57.3%

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

      1. *-commutative 57.3%

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

      2. pow1/2 57.3%

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

      3. pow1/2 57.3%

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

      4. pow-prod-down 57.6%

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

      5. *-commutative 57.6%

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

      6. associate-+r- 57.6%

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

      7. *-commutative 57.6%

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

   7. Applied egg-rr 57.6%

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

      1. unpow1/2 57.6%

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

      2. associate-*l* 57.6%

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

      3. hypot-undefine 48.3%

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

      4. unpow2 48.3%

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

      5. unpow2 48.3%

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

      6. +-commutative 48.3%

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

      7. unpow2 48.3%

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

      8. unpow2 48.3%

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

      9. hypot-undefine 57.6%

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

   9. Simplified 57.6%

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

   if 1.44999999999999999e144 < B < 5.50000000000000004e254

   1. Initial program 4.3%

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

   3. Taylor expanded in C around 0 6.7%

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

      1. mul-1-neg 6.7%

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

      2. +-commutative 6.7%

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

      3. unpow2 6.7%

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

      4. unpow2 6.7%

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

      5. hypot-define 53.6%

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

   5. Simplified 53.6%

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

   if 5.50000000000000004e254 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/l* 0.0%

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

      4. associate--l+ 0.0%

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

      5. unpow2 0.0%

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

      6. unpow2 0.0%

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

      7. hypot-undefine 4.5%

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

      8. cancel-sign-sub-inv 4.5%

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

   5. Simplified 4.5%

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

   6. Taylor expanded in B around inf 62.3%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+144}:\\ \;\;\;\;-\sqrt{\frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 1250000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 3.2 \cdot 10^{+254}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 1250000000.0)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= B_m 3.2e+254)
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m A))))))
       (*
        (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
        (- (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 1250000000.0) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 3.2e+254) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 1250000000.0) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (B_m <= 3.2e+254) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (A - Math.hypot(B_m, A))));
	} else {
		tmp = Math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 1250000000.0:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	elif B_m <= 3.2e+254:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (A - math.hypot(B_m, A))))
	else:
		tmp = math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 1250000000.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 3.2e+254)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 1250000000.0)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	elseif (B_m <= 3.2e+254)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	else
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 1250000000.0], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.2e+254], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.25e9

   1. Initial program 19.9%

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

   3. Taylor expanded in A around -inf 13.8%

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

   if 1.25e9 < B < 3.1999999999999998e254

   1. Initial program 15.1%

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

   3. Taylor expanded in C around 0 13.2%

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

      1. mul-1-neg 13.2%

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

      2. +-commutative 13.2%

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

      3. unpow2 13.2%

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

      4. unpow2 13.2%

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

      5. hypot-define 38.9%

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

   5. Simplified 38.9%

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

   if 3.1999999999999998e254 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/l* 0.0%

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

      4. associate--l+ 0.0%

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

      5. unpow2 0.0%

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

      6. unpow2 0.0%

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

      7. hypot-undefine 4.5%

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

      8. cancel-sign-sub-inv 4.5%

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

   5. Simplified 4.5%

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

   6. Taylor expanded in B around inf 62.3%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1250000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+254}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{elif}\;B\_m \leq 3.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.1e-11)
   (/
    (sqrt (* -8.0 (* (* A C) (* F (+ A A)))))
    (- (fma C (* A -4.0) (pow B_m 2.0))))
   (if (<= B_m 3.5e+254)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m A))))))
     (*
      (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
      (- (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.1e-11) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	} else if (B_m <= 3.5e+254) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.1e-11)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(-fma(C, Float64(A * -4.0), (B_m ^ 2.0))));
	elseif (B_m <= 3.5e+254)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.1e-11], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3.5e+254], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.09999999999999984e-11

   1. Initial program 17.4%

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

   3. Simplified 18.1%

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

   5. Taylor expanded in C around inf 11.5%

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

      1. associate-*r* 12.0%

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

      2. *-commutative 12.0%

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

      3. mul-1-neg 12.0%

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

   7. Simplified 12.0%

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

   if 5.09999999999999984e-11 < B < 3.50000000000000017e254

   1. Initial program 23.7%

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

   3. Taylor expanded in C around 0 20.5%

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

      1. mul-1-neg 20.5%

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

      2. +-commutative 20.5%

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

      3. unpow2 20.5%

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

      4. unpow2 20.5%

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

      5. hypot-define 43.0%

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

   5. Simplified 43.0%

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

   if 3.50000000000000017e254 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/l* 0.0%

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

      4. associate--l+ 0.0%

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

      5. unpow2 0.0%

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

      6. unpow2 0.0%

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

      7. hypot-undefine 4.5%

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

      8. cancel-sign-sub-inv 4.5%

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

   5. Simplified 4.5%

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

   6. Taylor expanded in B around inf 62.3%

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

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 9 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 9e-49)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m A))))))
   (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 9e-49) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	} else {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 9e-49) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (A - Math.hypot(B_m, A))));
	} else {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 9e-49:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (A - math.hypot(B_m, A))))
	else:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 9e-49)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 9e-49)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	else
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 9e-49], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 9.0000000000000004e-49

   1. Initial program 23.2%

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

   3. Taylor expanded in C around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. unpow2 9.8%

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

      4. unpow2 9.8%

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

      5. hypot-define 16.5%

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

   5. Simplified 16.5%

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

   if 9.0000000000000004e-49 < C

   1. Initial program 4.4%

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

   3. Taylor expanded in F around 0 7.2%

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

      1. mul-1-neg 7.2%

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

      2. *-commutative 7.2%

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

      3. associate-/l* 7.1%

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

      4. associate--l+ 8.6%

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

      5. unpow2 8.6%

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

      6. unpow2 8.6%

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

      7. hypot-undefine 13.8%

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

      8. cancel-sign-sub-inv 13.8%

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

   5. Simplified 13.8%

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

   6. Taylor expanded in A around -inf 31.6%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 9 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 7.5 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 7.5e+83)
     (* (sqrt (* F (/ -0.5 C))) t_0)
     (* (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m))) t_0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 7.5e+83) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (b_m <= 7.5d+83) then
        tmp = sqrt((f * ((-0.5d0) / c))) * t_0
    else
        tmp = sqrt((f * ((((a / b_m) + (c / b_m)) + (-1.0d0)) / b_m))) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 7.5e+83) {
		tmp = Math.sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = Math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 7.5e+83:
		tmp = math.sqrt((F * (-0.5 / C))) * t_0
	else:
		tmp = math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 7.5e+83)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 7.5e+83)
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	else
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 7.5e+83], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 7.49999999999999989e83

   1. Initial program 21.0%

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

   3. Taylor expanded in F around 0 19.4%

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

      1. mul-1-neg 19.4%

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

      2. *-commutative 19.4%

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

      3. associate-/l* 22.3%

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

      4. associate--l+ 22.8%

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

      5. unpow2 22.8%

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

      6. unpow2 22.8%

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

      7. hypot-undefine 30.3%

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

      8. cancel-sign-sub-inv 30.3%

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

   5. Simplified 30.3%

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

   6. Taylor expanded in A around -inf 12.6%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 7.49999999999999989e83 < B

   1. Initial program 4.7%

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

   3. Taylor expanded in F around 0 2.8%

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

      1. mul-1-neg 2.8%

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

      2. *-commutative 2.8%

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

      3. associate-/l* 8.5%

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

      4. associate--l+ 8.5%

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

      5. unpow2 8.5%

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

      6. unpow2 8.5%

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

      7. hypot-undefine 15.6%

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

      8. cancel-sign-sub-inv 15.6%

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

   5. Simplified 15.6%

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

   6. Taylor expanded in B around inf 48.1%

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

4. Final simplification 19.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.5 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.2 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{-B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8.2e+83)
   (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))
   (* (sqrt 2.0) (- (sqrt (/ F (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.2e+83) {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / -B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 8.2d+83) then
        tmp = sqrt((f * ((-0.5d0) / c))) * -sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * -sqrt((f / -b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.2e+83) {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / -B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 8.2e+83:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / -B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 8.2e+83)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / Float64(-B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 8.2e+83)
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	else
		tmp = sqrt(2.0) * -sqrt((F / -B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 8.2e+83], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / (-B$95$m)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.2000000000000002e83

   1. Initial program 21.0%

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

   3. Taylor expanded in F around 0 19.4%

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

      1. mul-1-neg 19.4%

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

      2. *-commutative 19.4%

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

      3. associate-/l* 22.3%

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

      4. associate--l+ 22.8%

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

      5. unpow2 22.8%

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

      6. unpow2 22.8%

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

      7. hypot-undefine 30.3%

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

      8. cancel-sign-sub-inv 30.3%

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

   5. Simplified 30.3%

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

   6. Taylor expanded in A around -inf 12.6%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 8.2000000000000002e83 < B

   1. Initial program 4.7%

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

   3. Taylor expanded in F around 0 2.8%

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

      1. mul-1-neg 2.8%

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

      2. *-commutative 2.8%

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

      3. associate--l+ 2.8%

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

      4. unpow2 2.8%

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

      5. unpow2 2.8%

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

      6. hypot-undefine 8.9%

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

      7. cancel-sign-sub-inv 8.9%

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

      8. metadata-eval 8.9%

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

   5. Simplified 8.9%

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

   6. Taylor expanded in B around inf 48.5%

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

      1. mul-1-neg 48.5%

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

   8. Simplified 48.5%

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

4. Final simplification 19.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.2 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{-B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.1 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{-B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.1e+83)
   (* (sqrt (* -0.5 (/ F C))) (- (sqrt 2.0)))
   (* (sqrt 2.0) (- (sqrt (/ F (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.1e+83) {
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / -B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 7.1d+83) then
        tmp = sqrt(((-0.5d0) * (f / c))) * -sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * -sqrt((f / -b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.1e+83) {
		tmp = Math.sqrt((-0.5 * (F / C))) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / -B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7.1e+83:
		tmp = math.sqrt((-0.5 * (F / C))) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / -B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.1e+83)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / Float64(-B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 7.1e+83)
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	else
		tmp = sqrt(2.0) * -sqrt((F / -B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7.1e+83], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / (-B$95$m)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.10000000000000004e83

   1. Initial program 21.0%

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

   3. Taylor expanded in F around 0 19.4%

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

      1. mul-1-neg 19.4%

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

      2. *-commutative 19.4%

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

      3. associate--l+ 19.8%

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

      4. unpow2 19.8%

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

      5. unpow2 19.8%

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

      6. hypot-undefine 26.2%

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

      7. cancel-sign-sub-inv 26.2%

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

      8. metadata-eval 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in A around -inf 12.6%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \]

   if 7.10000000000000004e83 < B

   1. Initial program 4.7%

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

   3. Taylor expanded in F around 0 2.8%

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

      1. mul-1-neg 2.8%

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

      2. *-commutative 2.8%

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

      3. associate--l+ 2.8%

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

      4. unpow2 2.8%

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

      5. unpow2 2.8%

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

      6. hypot-undefine 8.9%

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

      7. cancel-sign-sub-inv 8.9%

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

      8. metadata-eval 8.9%

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

   5. Simplified 8.9%

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

   6. Taylor expanded in B around inf 48.5%

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

      1. mul-1-neg 48.5%

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

   8. Simplified 48.5%

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

4. Final simplification 19.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.1 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{-B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

3. Taylor expanded in F around 0 16.1%

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

   1. mul-1-neg 16.1%

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

   2. *-commutative 16.1%

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

   3. associate--l+ 16.5%

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

   4. unpow2 16.5%

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

   5. unpow2 16.5%

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

   6. hypot-undefine 22.8%

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

   7. cancel-sign-sub-inv 22.8%

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

   8. metadata-eval 22.8%

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

5. Simplified 22.8%

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

6. Taylor expanded in B around inf 15.0%

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

   1. mul-1-neg 15.0%

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

8. Simplified 15.0%

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

9. Final simplification 15.0%

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

Alternative 9: 0.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

3. Taylor expanded in F around 0 16.1%

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

   1. mul-1-neg 16.1%

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

   2. *-commutative 16.1%

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

   3. associate--l+ 16.5%

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

   4. unpow2 16.5%

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

   5. unpow2 16.5%

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

   6. hypot-undefine 22.8%

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

   7. cancel-sign-sub-inv 22.8%

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

   8. metadata-eval 22.8%

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

5. Simplified 22.8%

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

6. Taylor expanded in B around -inf 13.3%

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

7. Final simplification 13.3%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))