ABCF->ab-angle a

Percentage Accurate: 19.3% → 61.7%

Time: 33.2s

Alternatives: 12

Speedup: 6.0×

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: 19.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 61.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\right)\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt (* t_1 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2)))
   (if (<= t_3 (- INFINITY))
     (*
      (sqrt 2.0)
      (-
       (sqrt
        (*
         F
         (/
          (+ A (+ C (hypot B_m (- A C))))
          (fma -4.0 (* A C) (pow B_m 2.0)))))))
     (if (<= t_3 -1e-204)
       (/ (sqrt (* t_1 (+ C (hypot B_m C)))) t_2)
       (if (<= t_3 INFINITY)
         (/ (sqrt (* t_1 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))) t_2)
         (*
          (* (sqrt (+ C (hypot C B_m))) (sqrt F))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = sqrt((t_1 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(2.0) * -sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0)))));
	} else if (t_3 <= -1e-204) {
		tmp = sqrt((t_1 * (C + hypot(B_m, C)))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_2;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	elseif (t_3 <= -1e-204)
		tmp = Float64(sqrt(Float64(t_1 * Float64(C + hypot(B_m, C)))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / t_2);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -1e-204], N[(N[Sqrt[N[(t$95$1 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in F around 0 24.4%

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

      1. mul-1-neg 24.4%

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

      2. *-commutative 24.4%

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

      3. distribute-rgt-neg-in 24.4%

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

      4. associate-/l* 29.1%

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

      5. cancel-sign-sub-inv 29.1%

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

      6. metadata-eval 29.1%

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

      7. +-commutative 29.1%

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

   5. Simplified 57.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-204

   1. Initial program 98.5%

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

   3. Taylor expanded in A around 0 67.7%

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

      1. unpow2 67.7%

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

      2. unpow2 67.7%

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

      3. hypot-define 67.8%

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

   5. Simplified 67.8%

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

   if -1e-204 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 16.4%

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

   3. Taylor expanded in A around -inf 29.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 2.6%

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

      1. mul-1-neg 2.6%

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

      2. *-commutative 2.6%

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

      3. distribute-rgt-neg-in 2.6%

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

      4. unpow2 2.6%

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

      5. unpow2 2.6%

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

      6. hypot-define 18.7%

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

   5. Simplified 18.7%

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

      1. pow1/2 18.8%

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

      2. *-commutative 18.8%

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

      3. unpow-prod-down 29.7%

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

      4. pow1/2 29.7%

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

      5. pow1/2 29.7%

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

   7. Applied egg-rr 29.7%

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

      1. hypot-undefine 2.5%

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

      2. unpow2 2.5%

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

      3. unpow2 2.5%

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

      4. +-commutative 2.5%

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

      5. unpow2 2.5%

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

      6. unpow2 2.5%

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

      7. hypot-define 29.7%

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

   9. Simplified 29.7%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ \mathbf{if}\;B\_m \leq 1.08 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 4000000:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_0) F))))
   (if (<= B_m 1.08e-42)
     (/ (sqrt (* t_2 (* 2.0 C))) t_1)
     (if (<= B_m 4000000.0)
       (/ (sqrt (* t_2 (+ C (hypot B_m C)))) t_1)
       (if (<= B_m 1.7e+40)
         (/ (sqrt (* t_2 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))) t_1)
         (*
          (* (sqrt (+ C (hypot C B_m))) (sqrt F))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double tmp;
	if (B_m <= 1.08e-42) {
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (B_m <= 4000000.0) {
		tmp = sqrt((t_2 * (C + hypot(B_m, C)))) / t_1;
	} else if (B_m <= 1.7e+40) {
		tmp = sqrt((t_2 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double t_2 = 2.0 * ((Math.pow(B_m, 2.0) - t_0) * F);
	double tmp;
	if (B_m <= 1.08e-42) {
		tmp = Math.sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (B_m <= 4000000.0) {
		tmp = Math.sqrt((t_2 * (C + Math.hypot(B_m, C)))) / t_1;
	} else if (B_m <= 1.7e+40) {
		tmp = Math.sqrt((t_2 * ((-0.5 * (Math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1;
	} else {
		tmp = (Math.sqrt((C + Math.hypot(C, B_m))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = t_0 - math.pow(B_m, 2.0)
	t_2 = 2.0 * ((math.pow(B_m, 2.0) - t_0) * F)
	tmp = 0
	if B_m <= 1.08e-42:
		tmp = math.sqrt((t_2 * (2.0 * C))) / t_1
	elif B_m <= 4000000.0:
		tmp = math.sqrt((t_2 * (C + math.hypot(B_m, C)))) / t_1
	elif B_m <= 1.7e+40:
		tmp = math.sqrt((t_2 * ((-0.5 * (math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1
	else:
		tmp = (math.sqrt((C + math.hypot(C, B_m))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	tmp = 0.0
	if (B_m <= 1.08e-42)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * C))) / t_1);
	elseif (B_m <= 4000000.0)
		tmp = Float64(sqrt(Float64(t_2 * Float64(C + hypot(B_m, C)))) / t_1);
	elseif (B_m <= 1.7e+40)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / 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)
	t_0 = (4.0 * A) * C;
	t_1 = t_0 - (B_m ^ 2.0);
	t_2 = 2.0 * (((B_m ^ 2.0) - t_0) * F);
	tmp = 0.0;
	if (B_m <= 1.08e-42)
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	elseif (B_m <= 4000000.0)
		tmp = sqrt((t_2 * (C + hypot(B_m, C)))) / t_1;
	elseif (B_m <= 1.7e+40)
		tmp = sqrt((t_2 * ((-0.5 * ((B_m ^ 2.0) / A)) + (2.0 * C)))) / t_1;
	else
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.08e-42], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 4000000.0], N[(N[Sqrt[N[(t$95$2 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 1.7e+40], N[(N[Sqrt[N[(t$95$2 * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.07999999999999996e-42

   1. Initial program 19.8%

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

   3. Taylor expanded in A around -inf 14.2%

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

   if 1.07999999999999996e-42 < B < 4e6

   1. Initial program 51.2%

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

   3. Taylor expanded in A around 0 44.7%

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

      1. unpow2 44.7%

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

      2. unpow2 44.7%

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

      3. hypot-define 45.1%

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

   5. Simplified 45.1%

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

   if 4e6 < B < 1.69999999999999994e40

   1. Initial program 14.4%

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

   3. Taylor expanded in A around -inf 26.3%

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

   if 1.69999999999999994e40 < B

   1. Initial program 6.1%

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

   3. Taylor expanded in A around 0 10.0%

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

      1. mul-1-neg 10.0%

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

      2. *-commutative 10.0%

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

      3. distribute-rgt-neg-in 10.0%

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

      4. unpow2 10.0%

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

      5. unpow2 10.0%

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

      6. hypot-define 44.0%

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

   5. Simplified 44.0%

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

      1. pow1/2 44.0%

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

      2. *-commutative 44.0%

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

      3. unpow-prod-down 71.2%

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

      4. pow1/2 71.2%

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

      5. pow1/2 71.2%

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

   7. Applied egg-rr 71.2%

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

      1. hypot-undefine 13.6%

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

      2. unpow2 13.6%

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

      3. unpow2 13.6%

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

      4. +-commutative 13.6%

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

      5. unpow2 13.6%

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

      6. unpow2 13.6%

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

      7. hypot-define 71.2%

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

   9. Simplified 71.2%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.08 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 4000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_3 := \frac{\sqrt{t\_2 \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{if}\;B\_m \leq 1.14 \cdot 10^{-42}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;B\_m \leq 1.14 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 9.6 \cdot 10^{-11}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_3 (/ (sqrt (* t_2 (* 2.0 C))) t_1)))
   (if (<= B_m 1.14e-42)
     t_3
     (if (<= B_m 1.14e-26)
       (/ (sqrt (* t_2 (+ C (hypot B_m C)))) t_1)
       (if (<= B_m 9.6e-11)
         t_3
         (*
          (* (sqrt (+ C (hypot C B_m))) (sqrt F))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_3 = sqrt((t_2 * (2.0 * C))) / t_1;
	double tmp;
	if (B_m <= 1.14e-42) {
		tmp = t_3;
	} else if (B_m <= 1.14e-26) {
		tmp = sqrt((t_2 * (C + hypot(B_m, C)))) / t_1;
	} else if (B_m <= 9.6e-11) {
		tmp = t_3;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double t_2 = 2.0 * ((Math.pow(B_m, 2.0) - t_0) * F);
	double t_3 = Math.sqrt((t_2 * (2.0 * C))) / t_1;
	double tmp;
	if (B_m <= 1.14e-42) {
		tmp = t_3;
	} else if (B_m <= 1.14e-26) {
		tmp = Math.sqrt((t_2 * (C + Math.hypot(B_m, C)))) / t_1;
	} else if (B_m <= 9.6e-11) {
		tmp = t_3;
	} else {
		tmp = (Math.sqrt((C + Math.hypot(C, B_m))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = t_0 - math.pow(B_m, 2.0)
	t_2 = 2.0 * ((math.pow(B_m, 2.0) - t_0) * F)
	t_3 = math.sqrt((t_2 * (2.0 * C))) / t_1
	tmp = 0
	if B_m <= 1.14e-42:
		tmp = t_3
	elif B_m <= 1.14e-26:
		tmp = math.sqrt((t_2 * (C + math.hypot(B_m, C)))) / t_1
	elif B_m <= 9.6e-11:
		tmp = t_3
	else:
		tmp = (math.sqrt((C + math.hypot(C, B_m))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_3 = Float64(sqrt(Float64(t_2 * Float64(2.0 * C))) / t_1)
	tmp = 0.0
	if (B_m <= 1.14e-42)
		tmp = t_3;
	elseif (B_m <= 1.14e-26)
		tmp = Float64(sqrt(Float64(t_2 * Float64(C + hypot(B_m, C)))) / t_1);
	elseif (B_m <= 9.6e-11)
		tmp = t_3;
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / 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)
	t_0 = (4.0 * A) * C;
	t_1 = t_0 - (B_m ^ 2.0);
	t_2 = 2.0 * (((B_m ^ 2.0) - t_0) * F);
	t_3 = sqrt((t_2 * (2.0 * C))) / t_1;
	tmp = 0.0;
	if (B_m <= 1.14e-42)
		tmp = t_3;
	elseif (B_m <= 1.14e-26)
		tmp = sqrt((t_2 * (C + hypot(B_m, C)))) / t_1;
	elseif (B_m <= 9.6e-11)
		tmp = t_3;
	else
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$2 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[B$95$m, 1.14e-42], t$95$3, If[LessEqual[B$95$m, 1.14e-26], N[(N[Sqrt[N[(t$95$2 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 9.6e-11], t$95$3, N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.14e-42 or 1.1399999999999999e-26 < B < 9.6000000000000005e-11

   1. Initial program 20.1%

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

   3. Taylor expanded in A around -inf 15.2%

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

   if 1.14e-42 < B < 1.1399999999999999e-26

   1. Initial program 51.2%

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

   3. Taylor expanded in A around 0 51.8%

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

      1. unpow2 51.8%

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

      2. unpow2 51.8%

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

      3. hypot-define 52.3%

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

   5. Simplified 52.3%

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

   if 9.6000000000000005e-11 < B

   1. Initial program 9.8%

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

   3. Taylor expanded in A around 0 12.4%

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

      1. mul-1-neg 12.4%

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

      2. *-commutative 12.4%

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

      3. distribute-rgt-neg-in 12.4%

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

      4. unpow2 12.4%

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

      5. unpow2 12.4%

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

      6. hypot-define 40.2%

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

   5. Simplified 40.2%

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

      1. pow1/2 40.2%

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

      2. *-commutative 40.2%

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

      3. unpow-prod-down 62.5%

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

      4. pow1/2 62.5%

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

      5. pow1/2 62.5%

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

   7. Applied egg-rr 62.5%

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

      1. hypot-undefine 15.4%

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

      2. unpow2 15.4%

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

      3. unpow2 15.4%

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

      4. +-commutative 15.4%

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

      5. unpow2 15.4%

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

      6. unpow2 15.4%

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

      7. hypot-define 62.5%

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

   9. Simplified 62.5%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.14 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.14 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 7e-11)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 7e-11) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 7e-11) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt((C + Math.hypot(C, B_m))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 7e-11:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt((C + math.hypot(C, B_m))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 7e-11)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / 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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 7e-11)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 7e-11], 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 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 7.00000000000000038e-11

   1. Initial program 21.1%

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

   3. Taylor expanded in A around -inf 15.3%

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

   if 7.00000000000000038e-11 < B

   1. Initial program 9.8%

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

   3. Taylor expanded in A around 0 12.4%

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

      1. mul-1-neg 12.4%

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

      2. *-commutative 12.4%

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

      3. distribute-rgt-neg-in 12.4%

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

      4. unpow2 12.4%

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

      5. unpow2 12.4%

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

      6. hypot-define 40.2%

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

   5. Simplified 40.2%

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

      1. pow1/2 40.2%

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

      2. *-commutative 40.2%

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

      3. unpow-prod-down 62.5%

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

      4. pow1/2 62.5%

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

      5. pow1/2 62.5%

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

   7. Applied egg-rr 62.5%

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

      1. hypot-undefine 15.4%

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

      2. unpow2 15.4%

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

      3. unpow2 15.4%

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

      4. +-commutative 15.4%

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

      5. unpow2 15.4%

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

      6. unpow2 15.4%

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

      7. hypot-define 62.5%

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

   9. Simplified 62.5%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 6.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{B\_m + C}\right)\\ \end{array} \end{array} \]

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 6.4e+57)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (* (/ (sqrt 2.0) (- B_m)) (* (sqrt F) (sqrt (+ B_m C)))))))

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 <= 6.4e+57) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((B_m + C)));
	}
	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 = (4.0d0 * a) * c
    if (b_m <= 6.4d+57) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m ** 2.0d0))
    else
        tmp = (sqrt(2.0d0) / -b_m) * (sqrt(f) * sqrt((b_m + c)))
    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 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 6.4e+57) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * (Math.sqrt(F) * Math.sqrt((B_m + C)));
	}
	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 <= 6.4e+57:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt(2.0) / -B_m) * (math.sqrt(F) * math.sqrt((B_m + C)))
	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 <= 6.4e+57)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(B_m + C))));
	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 <= 6.4e+57)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((B_m + C)));
	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, 6.4e+57], 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 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 6.40000000000000059e57

   1. Initial program 21.7%

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

   3. Taylor expanded in A around -inf 14.8%

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

   if 6.40000000000000059e57 < B

   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 A around 0 10.2%

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

      1. mul-1-neg 10.2%

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

      2. *-commutative 10.2%

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

      3. distribute-rgt-neg-in 10.2%

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

      4. unpow2 10.2%

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

      5. unpow2 10.2%

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

      6. hypot-define 45.4%

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

   5. Simplified 45.4%

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

      1. pow1/2 45.4%

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

      2. *-commutative 45.4%

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

      3. unpow-prod-down 73.7%

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

      4. pow1/2 73.7%

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

      5. pow1/2 73.7%

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

   7. Applied egg-rr 73.7%

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

      1. hypot-undefine 13.9%

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

      2. unpow2 13.9%

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

      3. unpow2 13.9%

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

      4. +-commutative 13.9%

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

      5. unpow2 13.9%

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

      6. unpow2 13.9%

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

      7. hypot-define 73.7%

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

   9. Simplified 73.7%

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

   10. Taylor expanded in C around 0 71.9%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{\frac{B\_m - C \cdot \frac{4 \cdot A}{B\_m}}{\sqrt{\left(C + \mathsf{hypot}\left(B\_m, C\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{B\_m + C}\right)\\ \end{array} \end{array} \]

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 1.7e+85)
   (/
    -1.0
    (/
     (- B_m (* C (/ (* 4.0 A) B_m)))
     (sqrt (* (+ C (hypot B_m C)) (* 2.0 F)))))
   (* (/ (sqrt 2.0) (- B_m)) (* (sqrt F) (sqrt (+ B_m C))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.7e+85) {
		tmp = -1.0 / ((B_m - (C * ((4.0 * A) / B_m))) / sqrt(((C + hypot(B_m, C)) * (2.0 * F))));
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((B_m + C)));
	}
	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 (B_m <= 1.7e+85) {
		tmp = -1.0 / ((B_m - (C * ((4.0 * A) / B_m))) / Math.sqrt(((C + Math.hypot(B_m, C)) * (2.0 * F))));
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * (Math.sqrt(F) * Math.sqrt((B_m + C)));
	}
	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 <= 1.7e+85:
		tmp = -1.0 / ((B_m - (C * ((4.0 * A) / B_m))) / math.sqrt(((C + math.hypot(B_m, C)) * (2.0 * F))))
	else:
		tmp = (math.sqrt(2.0) / -B_m) * (math.sqrt(F) * math.sqrt((B_m + C)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.7e+85)
		tmp = Float64(-1.0 / Float64(Float64(B_m - Float64(C * Float64(Float64(4.0 * A) / B_m))) / sqrt(Float64(Float64(C + hypot(B_m, C)) * Float64(2.0 * F)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(B_m + C))));
	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 <= 1.7e+85)
		tmp = -1.0 / ((B_m - (C * ((4.0 * A) / B_m))) / sqrt(((C + hypot(B_m, C)) * (2.0 * F))));
	else
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((B_m + C)));
	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, 1.7e+85], N[(-1.0 / N[(N[(B$95$m - N[(C * N[(N[(4.0 * A), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.7000000000000002e85

   1. Initial program 21.5%

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

   3. Taylor expanded in A around 0 4.6%

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

      1. associate-*l* 4.6%

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

      2. unpow2 4.6%

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

      3. unpow2 4.6%

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

      4. hypot-define 5.7%

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

   5. Simplified 5.7%

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

      1. clear-num 5.7%

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

      2. inv-pow 5.7%

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

      3. *-commutative 5.7%

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

      4. distribute-rgt-neg-in 5.7%

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

      5. sqrt-unprod 5.7%

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

   7. Applied egg-rr 5.7%

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

      1. unpow-1 5.7%

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

      2. associate-/r* 6.0%

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

      3. associate-*r* 6.0%

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

      4. hypot-undefine 4.6%

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

      5. unpow2 4.6%

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

      6. unpow2 4.6%

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

      7. +-commutative 4.6%

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

      8. unpow2 4.6%

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

      9. unpow2 4.6%

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

      10. hypot-define 6.0%

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

   9. Simplified 6.0%

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

       1. div-inv 6.0%

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

       2. div-sub 6.0%

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

       3. pow1 6.0%

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

       4. pow-div 6.9%

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

       5. metadata-eval 6.9%

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

       6. pow1 6.9%

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

       7. *-commutative 6.9%

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

       8. *-commutative 6.9%

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

   11. Applied egg-rr 6.9%

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

       1. associate-*r/ 6.9%

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

       2. *-rgt-identity 6.9%

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

       3. associate-/l* 6.9%

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

       4. hypot-undefine 5.3%

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

       5. unpow2 5.3%

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

       6. unpow2 5.3%

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

       7. +-commutative 5.3%

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

       8. unpow2 5.3%

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

       9. unpow2 5.3%

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

       10. hypot-undefine 6.9%

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

       11. *-commutative 6.9%

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

   13. Simplified 6.9%

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

   if 1.7000000000000002e85 < B

   1. Initial program 4.5%

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

   3. Taylor expanded in A around 0 10.4%

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

      1. mul-1-neg 10.4%

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

      2. *-commutative 10.4%

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

      3. distribute-rgt-neg-in 10.4%

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

      4. unpow2 10.4%

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

      5. unpow2 10.4%

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

      6. hypot-define 45.2%

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

   5. Simplified 45.2%

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

      1. pow1/2 45.2%

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

      2. *-commutative 45.2%

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

      3. unpow-prod-down 74.6%

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

      4. pow1/2 74.6%

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

      5. pow1/2 74.6%

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

   7. Applied egg-rr 74.6%

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

      1. hypot-undefine 14.3%

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

      2. unpow2 14.3%

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

      3. unpow2 14.3%

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

      4. +-commutative 14.3%

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

      5. unpow2 14.3%

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

      6. unpow2 14.3%

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

      7. hypot-define 74.6%

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

   9. Simplified 74.6%

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

   10. Taylor expanded in C around 0 74.3%

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

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{\frac{B - C \cdot \frac{4 \cdot A}{B}}{\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9.2 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1}{\frac{B\_m - C \cdot \frac{4 \cdot A}{B\_m}}{\sqrt{\left(C + \mathsf{hypot}\left(B\_m, C\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 9.2e+174)
   (/
    -1.0
    (/
     (- B_m (* C (/ (* 4.0 A) B_m)))
     (sqrt (* (+ C (hypot B_m C)) (* 2.0 F)))))
   (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.2e+174) {
		tmp = -1.0 / ((B_m - (C * ((4.0 * A) / B_m))) / sqrt(((C + hypot(B_m, C)) * (2.0 * F))));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.2e+174) {
		tmp = -1.0 / ((B_m - (C * ((4.0 * A) / B_m))) / Math.sqrt(((C + Math.hypot(B_m, C)) * (2.0 * F))));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9.2e+174:
		tmp = -1.0 / ((B_m - (C * ((4.0 * A) / B_m))) / math.sqrt(((C + math.hypot(B_m, C)) * (2.0 * F))))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 9.2e+174)
		tmp = Float64(-1.0 / Float64(Float64(B_m - Float64(C * Float64(Float64(4.0 * A) / B_m))) / sqrt(Float64(Float64(C + hypot(B_m, C)) * Float64(2.0 * F)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 9.2e+174)
		tmp = -1.0 / ((B_m - (C * ((4.0 * A) / B_m))) / sqrt(((C + hypot(B_m, C)) * (2.0 * F))));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9.2e+174], N[(-1.0 / N[(N[(B$95$m - N[(C * N[(N[(4.0 * A), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9.1999999999999991e174

   1. Initial program 21.1%

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

   3. Taylor expanded in A around 0 6.2%

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

      1. associate-*l* 6.2%

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

      2. unpow2 6.2%

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

      3. unpow2 6.2%

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

      4. hypot-define 7.3%

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

   5. Simplified 7.3%

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

      1. clear-num 7.3%

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

      2. inv-pow 7.3%

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

      3. *-commutative 7.3%

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

      4. distribute-rgt-neg-in 7.3%

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

      5. sqrt-unprod 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. unpow-1 7.3%

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

      2. associate-/r* 7.5%

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

      3. associate-*r* 7.5%

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

      4. hypot-undefine 6.2%

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

      5. unpow2 6.2%

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

      6. unpow2 6.2%

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

      7. +-commutative 6.2%

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

      8. unpow2 6.2%

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

      9. unpow2 6.2%

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

      10. hypot-define 7.5%

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

   9. Simplified 7.5%

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

       1. div-inv 7.5%

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

       2. div-sub 7.6%

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

       3. pow1 7.6%

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

       4. pow-div 9.0%

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

       5. metadata-eval 9.0%

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

       6. pow1 9.0%

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

       7. *-commutative 9.0%

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

       8. *-commutative 9.0%

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

   11. Applied egg-rr 9.0%

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

       1. associate-*r/ 9.0%

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

       2. *-rgt-identity 9.0%

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

       3. associate-/l* 9.0%

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

       4. hypot-undefine 6.9%

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

       5. unpow2 6.9%

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

       6. unpow2 6.9%

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

       7. +-commutative 6.9%

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

       8. unpow2 6.9%

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

       9. unpow2 6.9%

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

       10. hypot-undefine 9.0%

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

       11. *-commutative 9.0%

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

   13. Simplified 9.0%

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

   if 9.1999999999999991e174 < 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 B around inf 57.5%

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

      1. mul-1-neg 57.5%

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

      2. distribute-rgt-neg-in 57.5%

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

   5. Simplified 57.5%

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

      1. pow1 57.5%

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

      2. distribute-rgt-neg-out 57.5%

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

      3. pow1/2 57.5%

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

      4. pow1/2 57.5%

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

      5. pow-prod-down 57.5%

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

   7. Applied egg-rr 57.5%

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

      1. unpow1 57.5%

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

      2. unpow1/2 57.5%

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

      3. associate-*l/ 57.7%

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

   9. Simplified 57.7%

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

       1. sqrt-div 82.6%

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

   11. Applied egg-rr 82.6%

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

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.2 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1}{\frac{B - C \cdot \frac{4 \cdot A}{B}}{\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.8% accurate, 3.1× speedup

Math

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

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 F) (- (sqrt (/ 2.0 B_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.2%

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

3. Taylor expanded in B around inf 14.2%

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

   1. mul-1-neg 14.2%

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

   2. distribute-rgt-neg-in 14.2%

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

5. Simplified 14.2%

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

   1. pow1 14.2%

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

   2. distribute-rgt-neg-out 14.2%

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

   3. pow1/2 14.4%

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

   4. pow1/2 14.4%

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

   5. pow-prod-down 14.4%

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

7. Applied egg-rr 14.4%

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

   1. unpow1 14.4%

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

   2. unpow1/2 14.2%

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

   3. associate-*l/ 14.3%

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

9. Simplified 14.3%

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

    1. pow1/2 14.4%

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

    2. associate-/l* 14.4%

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

    3. unpow-prod-down 18.1%

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

    4. pow1/2 18.1%

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

11. Applied egg-rr 18.1%

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

    1. unpow1/2 18.1%

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

13. Simplified 18.1%

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

14. Final simplification 18.1%

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

Alternative 9: 34.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.2%

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

3. Taylor expanded in B around inf 14.2%

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

   1. mul-1-neg 14.2%

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

   2. distribute-rgt-neg-in 14.2%

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

5. Simplified 14.2%

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

   1. pow1 14.2%

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

   2. distribute-rgt-neg-out 14.2%

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

   3. pow1/2 14.4%

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

   4. pow1/2 14.4%

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

   5. pow-prod-down 14.4%

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

7. Applied egg-rr 14.4%

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

   1. unpow1 14.4%

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

   2. unpow1/2 14.2%

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

   3. associate-*l/ 14.3%

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

9. Simplified 14.3%

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

    1. sqrt-div 18.1%

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

11. Applied egg-rr 18.1%

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

12. Final simplification 18.1%

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

Alternative 10: 33.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\frac{B\_m + -4 \cdot \left(A \cdot \frac{C}{B\_m}\right)}{\sqrt{\left(2 \cdot F\right) \cdot \left(B\_m + C \cdot \left(1 + \frac{C}{B\_m} \cdot 0.5\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{B\_m}{2 \cdot F}}}\\ \end{array} \end{array} \]

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 (<= F 1.3e-23)
   (/
    -1.0
    (/
     (+ B_m (* -4.0 (* A (/ C B_m))))
     (sqrt (* (* 2.0 F) (+ B_m (* C (+ 1.0 (* (/ C B_m) 0.5))))))))
   (/ -1.0 (sqrt (/ B_m (* 2.0 F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.3e-23) {
		tmp = -1.0 / ((B_m + (-4.0 * (A * (C / B_m)))) / sqrt(((2.0 * F) * (B_m + (C * (1.0 + ((C / B_m) * 0.5)))))));
	} else {
		tmp = -1.0 / sqrt((B_m / (2.0 * F)));
	}
	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 (f <= 1.3d-23) then
        tmp = (-1.0d0) / ((b_m + ((-4.0d0) * (a * (c / b_m)))) / sqrt(((2.0d0 * f) * (b_m + (c * (1.0d0 + ((c / b_m) * 0.5d0)))))))
    else
        tmp = (-1.0d0) / sqrt((b_m / (2.0d0 * f)))
    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 (F <= 1.3e-23) {
		tmp = -1.0 / ((B_m + (-4.0 * (A * (C / B_m)))) / Math.sqrt(((2.0 * F) * (B_m + (C * (1.0 + ((C / B_m) * 0.5)))))));
	} else {
		tmp = -1.0 / Math.sqrt((B_m / (2.0 * F)));
	}
	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 F <= 1.3e-23:
		tmp = -1.0 / ((B_m + (-4.0 * (A * (C / B_m)))) / math.sqrt(((2.0 * F) * (B_m + (C * (1.0 + ((C / B_m) * 0.5)))))))
	else:
		tmp = -1.0 / math.sqrt((B_m / (2.0 * F)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.3e-23)
		tmp = Float64(-1.0 / Float64(Float64(B_m + Float64(-4.0 * Float64(A * Float64(C / B_m)))) / sqrt(Float64(Float64(2.0 * F) * Float64(B_m + Float64(C * Float64(1.0 + Float64(Float64(C / B_m) * 0.5))))))));
	else
		tmp = Float64(-1.0 / sqrt(Float64(B_m / Float64(2.0 * F))));
	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 (F <= 1.3e-23)
		tmp = -1.0 / ((B_m + (-4.0 * (A * (C / B_m)))) / sqrt(((2.0 * F) * (B_m + (C * (1.0 + ((C / B_m) * 0.5)))))));
	else
		tmp = -1.0 / sqrt((B_m / (2.0 * F)));
	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[F, 1.3e-23], N[(-1.0 / N[(N[(B$95$m + N[(-4.0 * N[(A * N[(C / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(B$95$m + N[(C * N[(1.0 + N[(N[(C / B$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sqrt[N[(B$95$m / N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.3e-23

   1. Initial program 21.8%

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

   3. Taylor expanded in A around 0 3.3%

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

      1. associate-*l* 3.3%

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

      2. unpow2 3.3%

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

      3. unpow2 3.3%

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

      4. hypot-define 4.4%

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

   5. Simplified 4.4%

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

      1. clear-num 4.4%

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

      2. inv-pow 4.4%

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

      3. *-commutative 4.4%

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

      4. distribute-rgt-neg-in 4.4%

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

      5. sqrt-unprod 4.4%

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

   7. Applied egg-rr 4.4%

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

      1. unpow-1 4.4%

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

      2. associate-/r* 5.0%

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

      3. associate-*r* 5.0%

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

      4. hypot-undefine 3.2%

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

      5. unpow2 3.2%

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

      6. unpow2 3.2%

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

      7. +-commutative 3.2%

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

      8. unpow2 3.2%

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

      9. unpow2 3.2%

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

      10. hypot-define 5.0%

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

   9. Simplified 5.0%

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

   10. Taylor expanded in C around 0 3.4%

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

   11. Taylor expanded in C around 0 11.6%

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

       1. associate-/l* 13.1%

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

   13. Simplified 13.1%

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

   if 1.3e-23 < F

   1. Initial program 14.8%

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

   3. Taylor expanded in B around inf 22.8%

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

      1. mul-1-neg 22.8%

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

      2. distribute-rgt-neg-in 22.8%

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

   5. Simplified 22.8%

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

      1. pow1 22.8%

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

      2. distribute-rgt-neg-out 22.8%

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

      3. pow1/2 23.1%

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

      4. pow1/2 23.1%

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

      5. pow-prod-down 23.1%

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

   7. Applied egg-rr 23.1%

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

      1. unpow1 23.1%

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

      2. unpow1/2 22.8%

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

      3. associate-*l/ 22.8%

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

   9. Simplified 22.8%

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

       1. clear-num 22.8%

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

       2. sqrt-div 22.9%

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

       3. metadata-eval 22.9%

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

   11. Applied egg-rr 22.9%

       \[\leadsto -\color{blue}{\frac{1}{\sqrt{\frac{B}{F \cdot 2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\frac{B + -4 \cdot \left(A \cdot \frac{C}{B}\right)}{\sqrt{\left(2 \cdot F\right) \cdot \left(B + C \cdot \left(1 + \frac{C}{B} \cdot 0.5\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{B}{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.3% accurate, 6.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* 2.0 (/ 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 * (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 * (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 * (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 * (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(Float64(2.0 * Float64(F / 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 * (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[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 18.2%

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

3. Taylor expanded in B around inf 14.2%

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

   1. mul-1-neg 14.2%

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

   2. distribute-rgt-neg-in 14.2%

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

5. Simplified 14.2%

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

   1. pow1 14.2%

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

   2. distribute-rgt-neg-out 14.2%

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

   3. pow1/2 14.4%

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

   4. pow1/2 14.4%

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

   5. pow-prod-down 14.4%

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

7. Applied egg-rr 14.4%

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

   1. unpow1 14.4%

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

   2. unpow1/2 14.2%

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

   3. associate-*l/ 14.3%

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

9. Simplified 14.3%

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

    1. *-un-lft-identity 14.3%

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

    2. associate-/l* 14.3%

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

11. Applied egg-rr 14.3%

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

    1. *-lft-identity 14.3%

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

    2. associate-*r/ 14.3%

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

    3. associate-*l/ 14.2%

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

    4. *-commutative 14.2%

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

13. Simplified 14.2%

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

14. Final simplification 14.2%

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

Alternative 12: 26.3% accurate, 6.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (/ (* 2.0 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 * 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 * 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 * 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 * 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(Float64(Float64(2.0 * F) / 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 * 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[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 18.2%

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

3. Taylor expanded in B around inf 14.2%

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

   1. mul-1-neg 14.2%

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

   2. distribute-rgt-neg-in 14.2%

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

5. Simplified 14.2%

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

   1. pow1 14.2%

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

   2. distribute-rgt-neg-out 14.2%

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

   3. pow1/2 14.4%

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

   4. pow1/2 14.4%

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

   5. pow-prod-down 14.4%

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

7. Applied egg-rr 14.4%

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

   1. unpow1 14.4%

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

   2. unpow1/2 14.2%

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

   3. associate-*l/ 14.3%

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

9. Simplified 14.3%

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

10. Final simplification 14.3%

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

Reproduce

herbie shell --seed 2024075 
(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))))