ABCF->ab-angle a

Percentage Accurate: 18.8% → 63.1%

Time: 34.5s

Alternatives: 16

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: 18.8% 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: 63.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ t_5 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_4}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t\_5\right)} \cdot \sqrt{t\_4}}{-t\_5}\\ \mathbf{elif}\;t\_3 \leq 10^{+196}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\frac{{B\_m}^{2}}{A} \cdot -0.5 + 2 \cdot C\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\right)\right)} \cdot \sqrt{C}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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))
        (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))
        (t_4 (+ A (+ C (hypot B_m (- A C)))))
        (t_5 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (* F (/ t_4 (fma -4.0 (* A C) (pow B_m 2.0))))) (- (sqrt 2.0)))
     (if (<= t_3 -5e-205)
       (/ (* (sqrt (* F (* 2.0 t_5))) (sqrt t_4)) (- t_5))
       (if (<= t_3 1e+196)
         (/ (sqrt (* t_1 (+ (* (/ (pow B_m 2.0) A) -0.5) (* 2.0 C)))) t_2)
         (if (<= t_3 INFINITY)
           (/
            (*
             (sqrt (* 2.0 (* 2.0 (* F (fma B_m B_m (* C (* A -4.0)))))))
             (sqrt C))
            t_2)
           (*
            (/ (sqrt 2.0) B_m)
            (* (sqrt (+ C (hypot B_m C))) (- (sqrt 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 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 t_4 = A + (C + hypot(B_m, (A - C)));
	double t_5 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt((F * (t_4 / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_3 <= -5e-205) {
		tmp = (sqrt((F * (2.0 * t_5))) * sqrt(t_4)) / -t_5;
	} else if (t_3 <= 1e+196) {
		tmp = sqrt((t_1 * (((pow(B_m, 2.0) / A) * -0.5) + (2.0 * C)))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * (2.0 * (F * fma(B_m, B_m, (C * (A * -4.0))))))) * sqrt(C)) / t_2;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(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)
	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)
	t_4 = Float64(A + Float64(C + hypot(B_m, Float64(A - C))))
	t_5 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(t_4 / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_3 <= -5e-205)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * t_5))) * sqrt(t_4)) / Float64(-t_5));
	elseif (t_3 <= 1e+196)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(Float64((B_m ^ 2.0) / A) * -0.5) + Float64(2.0 * C)))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(C * Float64(A * -4.0))))))) * sqrt(C)) / t_2);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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]}, Block[{t$95$4 = N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(F * N[(t$95$4 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -5e-205], N[(N[(N[Sqrt[N[(F * N[(2.0 * t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] / (-t$95$5)), $MachinePrecision], If[LessEqual[t$95$3, 1e+196], N[(N[Sqrt[N[(t$95$1 * N[(N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in F around 0 18.9%

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

      1. mul-1-neg 18.9%

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

   5. Simplified 68.0%

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

   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))) < -5.00000000000000001e-205

   1. Initial program 93.4%

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

   3. Simplified 93.5%

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

      1. pow1/2 93.5%

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

      2. associate-*r* 93.5%

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

      3. associate-+r+ 93.4%

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

      4. hypot-undefine 93.4%

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

      5. unpow2 93.4%

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

      6. unpow2 93.4%

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

      7. +-commutative 93.4%

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

      8. unpow-prod-down 95.6%

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

      9. *-commutative 95.6%

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

      10. pow1/2 95.6%

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

   6. Applied egg-rr 95.7%

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

      1. unpow1/2 95.7%

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

      2. associate-*l* 95.7%

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

      3. hypot-undefine 95.7%

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

      4. unpow2 95.7%

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

      5. unpow2 95.7%

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

      6. +-commutative 95.7%

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

      7. unpow2 95.7%

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

      8. unpow2 95.7%

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

      9. hypot-undefine 95.7%

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

   8. Simplified 95.7%

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

   if -5.00000000000000001e-205 < (/.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))) < 9.9999999999999995e195

   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 -inf 31.9%

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

   if 9.9999999999999995e195 < (/.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 8.6%

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

   3. Taylor expanded in A around -inf 14.3%

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

      1. pow1/2 14.3%

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

      2. associate-*r* 14.3%

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

      3. unpow-prod-down 44.7%

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

      4. *-commutative 44.7%

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

      5. *-commutative 44.7%

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

      6. pow1/2 44.7%

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

   5. Applied egg-rr 44.7%

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

      1. unpow1/2 44.7%

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

      2. associate-*l* 44.7%

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

      3. unpow2 44.7%

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

      4. fmm-undef 44.7%

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

      5. distribute-rgt-neg-in 44.7%

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

      6. distribute-lft-neg-in 44.7%

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

      7. metadata-eval 44.7%

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

   7. Simplified 44.7%

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

   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 1.8%

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

      1. mul-1-neg 1.8%

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

      2. unpow2 1.8%

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

      3. unpow2 1.8%

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

      4. hypot-define 19.5%

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

   5. Simplified 19.5%

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

      1. pow1/2 19.5%

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

      2. *-commutative 19.5%

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

      3. unpow-prod-down 24.5%

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

      4. pow1/2 24.5%

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

      5. pow1/2 24.5%

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

   7. Applied egg-rr 24.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 43.8%

   \[\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{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\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 -5 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \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 10^{+196}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\frac{{B}^{2}}{A} \cdot -0.5 + 2 \cdot C\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{2 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)} \cdot \sqrt{C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C - {B\_m}^{2}\\ t_2 := \frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot t\_0\right)\right)} \cdot \sqrt{C}}{t\_1}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-271}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{\left(2 \cdot C\right) \cdot t\_0}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+23}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (- (* (* 4.0 A) C) (pow B_m 2.0)))
        (t_2 (/ (* (sqrt (* 2.0 (* 2.0 (* F t_0)))) (sqrt C)) t_1)))
   (if (<= (pow B_m 2.0) 1e-271)
     t_2
     (if (<= (pow B_m 2.0) 2e-102)
       (/ (* (sqrt (* 2.0 F)) (sqrt (* (* 2.0 C) t_0))) t_1)
       (if (<= (pow B_m 2.0) 1e+23)
         t_2
         (*
          (/ (sqrt 2.0) B_m)
          (* (sqrt (+ C (hypot B_m C))) (- (sqrt 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 t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = ((4.0 * A) * C) - pow(B_m, 2.0);
	double t_2 = (sqrt((2.0 * (2.0 * (F * t_0)))) * sqrt(C)) / t_1;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-271) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 2e-102) {
		tmp = (sqrt((2.0 * F)) * sqrt(((2.0 * C) * t_0))) / t_1;
	} else if (pow(B_m, 2.0) <= 1e+23) {
		tmp = t_2;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(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)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0))
	t_2 = Float64(Float64(sqrt(Float64(2.0 * Float64(2.0 * Float64(F * t_0)))) * sqrt(C)) / t_1)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-271)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 2e-102)
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) * sqrt(Float64(Float64(2.0 * C) * t_0))) / t_1);
	elseif ((B_m ^ 2.0) <= 1e+23)
		tmp = t_2;
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-271], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-102], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+23], t$95$2, N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999963e-272 or 1.99999999999999987e-102 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999992e22

   1. Initial program 22.0%

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

   3. Taylor expanded in A around -inf 23.5%

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

      1. pow1/2 23.7%

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

      2. associate-*r* 23.7%

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

      3. unpow-prod-down 26.1%

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

      4. *-commutative 26.1%

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

      5. *-commutative 26.1%

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

      6. pow1/2 26.1%

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

   5. Applied egg-rr 26.1%

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

      1. unpow1/2 26.1%

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

      2. associate-*l* 26.1%

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

      3. unpow2 26.1%

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

      4. fmm-undef 26.1%

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

      5. distribute-rgt-neg-in 26.1%

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

      6. distribute-lft-neg-in 26.1%

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

      7. metadata-eval 26.1%

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

   7. Simplified 26.1%

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

   if 9.99999999999999963e-272 < (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999987e-102

   1. Initial program 5.1%

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

   3. Taylor expanded in A around -inf 15.6%

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

      1. add-cbrt-cube 11.9%

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

      2. pow1/3 11.2%

         \[\leadsto \frac{-\color{blue}{{\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right) \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)}^{0.3333333333333333}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied egg-rr 11.2%

      \[\leadsto \frac{-\color{blue}{{\left({\left(2 \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. unpow1/3 12.0%

         \[\leadsto \frac{-\color{blue}{\sqrt[3]{{\left(2 \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)\right)}^{1.5}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. associate-*r* 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\color{blue}{\left(\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\color{blue}{\left(\left(2 \cdot F\right) \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)} \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \left(\color{blue}{B \cdot B} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. fmm-undef 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -C \cdot \left(4 \cdot A\right)\right)}\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. distribute-lft-neg-in 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(\left(-4\right) \cdot A\right)}\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(\color{blue}{-4} \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Simplified 12.0%

      \[\leadsto \frac{-\color{blue}{\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   8. Step-by-step derivation

      1. pow1/3 11.2%

         \[\leadsto \frac{-\color{blue}{{\left({\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow-pow 15.7%

         \[\leadsto \frac{-\color{blue}{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval 15.7%

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

      4. pow1/2 15.6%

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

      5. associate-*l* 11.9%

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

      6. *-commutative 11.9%

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

      7. sqrt-prod 18.4%

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

      8. *-commutative 18.4%

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

   9. Applied egg-rr 18.4%

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

   if 9.9999999999999992e22 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.6%

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

   3. Taylor expanded in A around 0 10.9%

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

      1. mul-1-neg 10.9%

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

      2. unpow2 10.9%

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

      3. unpow2 10.9%

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

      4. hypot-define 28.6%

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

   5. Simplified 28.6%

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

      1. pow1/2 28.6%

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

      2. *-commutative 28.6%

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

      3. unpow-prod-down 34.8%

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

      4. pow1/2 34.8%

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

      5. pow1/2 34.8%

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

   7. Applied egg-rr 34.8%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-271}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)} \cdot \sqrt{C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+23}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)} \cdot \sqrt{C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-271}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot t\_0\right)\right)} \cdot \sqrt{C}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{\left(2 \cdot C\right) \cdot t\_0}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+23}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)} \cdot \sqrt{2 \cdot C}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (- (* (* 4.0 A) C) (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 1e-271)
     (/ (* (sqrt (* 2.0 (* 2.0 (* F t_0)))) (sqrt C)) t_1)
     (if (<= (pow B_m 2.0) 2e-102)
       (/ (* (sqrt (* 2.0 F)) (sqrt (* (* 2.0 C) t_0))) t_1)
       (if (<= (pow B_m 2.0) 1e+23)
         (/ (* (sqrt (* t_0 (* 2.0 F))) (sqrt (* 2.0 C))) t_1)
         (*
          (/ (sqrt 2.0) B_m)
          (* (sqrt (+ C (hypot B_m C))) (- (sqrt 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 t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = ((4.0 * A) * C) - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-271) {
		tmp = (sqrt((2.0 * (2.0 * (F * t_0)))) * sqrt(C)) / t_1;
	} else if (pow(B_m, 2.0) <= 2e-102) {
		tmp = (sqrt((2.0 * F)) * sqrt(((2.0 * C) * t_0))) / t_1;
	} else if (pow(B_m, 2.0) <= 1e+23) {
		tmp = (sqrt((t_0 * (2.0 * F))) * sqrt((2.0 * C))) / t_1;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(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)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-271)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(2.0 * Float64(F * t_0)))) * sqrt(C)) / t_1);
	elseif ((B_m ^ 2.0) <= 2e-102)
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) * sqrt(Float64(Float64(2.0 * C) * t_0))) / t_1);
	elseif ((B_m ^ 2.0) <= 1e+23)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) * sqrt(Float64(2.0 * C))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-271], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-102], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+23], N[(N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999963e-272

   1. Initial program 20.9%

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

   3. Taylor expanded in A around -inf 24.8%

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

      1. pow1/2 25.0%

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

      2. associate-*r* 25.0%

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

      3. unpow-prod-down 26.1%

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

      4. *-commutative 26.1%

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

      5. *-commutative 26.1%

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

      6. pow1/2 26.1%

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

   5. Applied egg-rr 26.1%

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

      1. unpow1/2 26.1%

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

      2. associate-*l* 26.1%

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

      3. unpow2 26.1%

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

      4. fmm-undef 26.1%

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

      5. distribute-rgt-neg-in 26.1%

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

      6. distribute-lft-neg-in 26.1%

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

      7. metadata-eval 26.1%

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

   7. Simplified 26.1%

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

   if 9.99999999999999963e-272 < (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999987e-102

   1. Initial program 5.1%

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

   3. Taylor expanded in A around -inf 15.6%

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

      1. add-cbrt-cube 11.9%

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

      2. pow1/3 11.2%

         \[\leadsto \frac{-\color{blue}{{\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right) \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)}^{0.3333333333333333}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied egg-rr 11.2%

      \[\leadsto \frac{-\color{blue}{{\left({\left(2 \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. unpow1/3 12.0%

         \[\leadsto \frac{-\color{blue}{\sqrt[3]{{\left(2 \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)\right)}^{1.5}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. associate-*r* 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\color{blue}{\left(\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\color{blue}{\left(\left(2 \cdot F\right) \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)} \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \left(\color{blue}{B \cdot B} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. fmm-undef 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -C \cdot \left(4 \cdot A\right)\right)}\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. distribute-lft-neg-in 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(\left(-4\right) \cdot A\right)}\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval 12.0%

         \[\leadsto \frac{-\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(\color{blue}{-4} \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Simplified 12.0%

      \[\leadsto \frac{-\color{blue}{\sqrt[3]{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   8. Step-by-step derivation

      1. pow1/3 11.2%

         \[\leadsto \frac{-\color{blue}{{\left({\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow-pow 15.7%

         \[\leadsto \frac{-\color{blue}{{\left(\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right) \cdot \left(C \cdot 2\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval 15.7%

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

      4. pow1/2 15.6%

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

      5. associate-*l* 11.9%

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

      6. *-commutative 11.9%

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

      7. sqrt-prod 18.4%

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

      8. *-commutative 18.4%

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

   9. Applied egg-rr 18.4%

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

   if 1.99999999999999987e-102 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999992e22

   1. Initial program 25.5%

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

   3. Taylor expanded in A around -inf 19.2%

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

      1. sqrt-prod 26.0%

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

      2. *-commutative 26.0%

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

      3. *-commutative 26.0%

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

      4. *-commutative 26.0%

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

   5. Applied egg-rr 26.0%

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

      1. associate-*r* 26.0%

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

      2. unpow2 26.0%

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

      3. fmm-undef 26.0%

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

      4. distribute-rgt-neg-in 26.0%

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

      5. distribute-lft-neg-in 26.0%

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

      6. metadata-eval 26.0%

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

   7. Simplified 26.0%

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

   if 9.9999999999999992e22 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.6%

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

   3. Taylor expanded in A around 0 10.9%

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

      1. mul-1-neg 10.9%

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

      2. unpow2 10.9%

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

      3. unpow2 10.9%

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

      4. hypot-define 28.6%

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

   5. Simplified 28.6%

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

      1. pow1/2 28.6%

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

      2. *-commutative 28.6%

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

      3. unpow-prod-down 34.8%

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

      4. pow1/2 34.8%

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

      5. pow1/2 34.8%

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

   7. Applied egg-rr 34.8%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-271}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)} \cdot \sqrt{C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \sqrt{2 \cdot C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.8% accurate, 1.2× speedup

Math

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

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 (<= (pow B_m 2.0) 1e+23)
   (/
    (* (sqrt (* 2.0 (* 2.0 (* F (fma B_m B_m (* C (* A -4.0))))))) (sqrt C))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (* (/ (sqrt 2.0) B_m) (* (sqrt (+ C (hypot B_m C))) (- (sqrt 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 (pow(B_m, 2.0) <= 1e+23) {
		tmp = (sqrt((2.0 * (2.0 * (F * fma(B_m, B_m, (C * (A * -4.0))))))) * sqrt(C)) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(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 ((B_m ^ 2.0) <= 1e+23)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(C * Float64(A * -4.0))))))) * sqrt(C)) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+23], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999992e22

   1. Initial program 18.7%

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

   3. Taylor expanded in A around -inf 22.0%

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

      1. pow1/2 22.1%

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

      2. associate-*r* 22.1%

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

      3. unpow-prod-down 21.8%

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

      4. *-commutative 21.8%

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

      5. *-commutative 21.8%

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

      6. pow1/2 21.8%

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

   5. Applied egg-rr 21.8%

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

      1. unpow1/2 21.8%

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

      2. associate-*l* 21.8%

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

      3. unpow2 21.8%

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

      4. fmm-undef 21.8%

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

      5. distribute-rgt-neg-in 21.8%

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

      6. distribute-lft-neg-in 21.8%

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

      7. metadata-eval 21.8%

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

   7. Simplified 21.8%

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

   if 9.9999999999999992e22 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.6%

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

   3. Taylor expanded in A around 0 10.9%

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

      1. mul-1-neg 10.9%

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

      2. unpow2 10.9%

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

      3. unpow2 10.9%

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

      4. hypot-define 28.6%

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

   5. Simplified 28.6%

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

      1. pow1/2 28.6%

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

      2. *-commutative 28.6%

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

      3. unpow-prod-down 34.8%

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

      4. pow1/2 34.8%

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

      5. pow1/2 34.8%

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

   7. Applied egg-rr 34.8%

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

4. Final simplification 27.4%

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

Alternative 5: 57.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 2e-60)
     (/
      (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 (+ C (hypot B_m C))) (- (sqrt 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 t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-60) {
		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((C + hypot(B_m, C))) * -sqrt(F));
	}
	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 (Math.pow(B_m, 2.0) <= 2e-60) {
		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((C + Math.hypot(B_m, C))) * -Math.sqrt(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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-60:
		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((C + math.hypot(B_m, C))) * -math.sqrt(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)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-60)
		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) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-60)
		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((C + hypot(B_m, C))) * -sqrt(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-60], 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[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-60

   1. Initial program 17.4%

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

   3. Taylor expanded in A around -inf 22.5%

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

   if 1.9999999999999999e-60 < (pow.f64 B #s(literal 2 binary64))

   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 0 11.2%

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

      1. mul-1-neg 11.2%

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

      2. unpow2 11.2%

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

      3. unpow2 11.2%

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

      4. hypot-define 27.1%

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

   5. Simplified 27.1%

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

      1. pow1/2 27.1%

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

      2. *-commutative 27.1%

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

      3. unpow-prod-down 32.7%

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

      4. pow1/2 32.7%

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

      5. pow1/2 32.7%

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

   7. Applied egg-rr 32.7%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+45}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 1e+45)
     (/
      (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 (pow(B_m, 2.0) <= 1e+45) {
		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 ** 2.0d0) <= 1d+45) 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 (Math.pow(B_m, 2.0) <= 1e+45) {
		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 math.pow(B_m, 2.0) <= 1e+45:
		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 ^ 2.0) <= 1e+45)
		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) / B_m) * Float64(sqrt(F) * Float64(-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 ^ 2.0) <= 1e+45)
		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[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+45], 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 (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999993e44

   1. Initial program 18.9%

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

   3. Taylor expanded in A around -inf 22.6%

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

   if 9.9999999999999993e44 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.0%

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

   3. Taylor expanded in A around 0 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. unpow2 10.4%

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

      3. unpow2 10.4%

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

      4. hypot-define 28.9%

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

   5. Simplified 28.9%

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

      1. pow1/2 28.9%

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

      2. *-commutative 28.9%

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

      3. unpow-prod-down 35.4%

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

      4. pow1/2 35.4%

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

      5. pow1/2 35.4%

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

   7. Applied egg-rr 35.4%

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

   8. Taylor expanded in C around 0 33.2%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+45}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{+45}:\\ \;\;\;\;\frac{2 \cdot \sqrt{C \cdot \left(F \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 1e+45)
   (/
    (* 2.0 (sqrt (* C (* F (- (pow B_m 2.0) (* 4.0 (* A C)))))))
    (- (* (* 4.0 A) C) (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 tmp;
	if (pow(B_m, 2.0) <= 1e+45) {
		tmp = (2.0 * sqrt((C * (F * (pow(B_m, 2.0) - (4.0 * (A * C))))))) / (((4.0 * A) * C) - 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) :: tmp
    if ((b_m ** 2.0d0) <= 1d+45) then
        tmp = (2.0d0 * sqrt((c * (f * ((b_m ** 2.0d0) - (4.0d0 * (a * c))))))) / (((4.0d0 * a) * c) - (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 tmp;
	if (Math.pow(B_m, 2.0) <= 1e+45) {
		tmp = (2.0 * Math.sqrt((C * (F * (Math.pow(B_m, 2.0) - (4.0 * (A * C))))))) / (((4.0 * A) * C) - 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):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+45:
		tmp = (2.0 * math.sqrt((C * (F * (math.pow(B_m, 2.0) - (4.0 * (A * C))))))) / (((4.0 * A) * C) - 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)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+45)
		tmp = Float64(Float64(2.0 * sqrt(Float64(C * Float64(F * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C))))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-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 ^ 2.0) <= 1e+45)
		tmp = (2.0 * sqrt((C * (F * ((B_m ^ 2.0) - (4.0 * (A * C))))))) / (((4.0 * A) * C) - (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_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+45], N[(N[(2.0 * N[Sqrt[N[(C * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - 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 (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999993e44

   1. Initial program 18.9%

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

   3. Taylor expanded in A around -inf 22.6%

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

   4. Taylor expanded in F around 0 22.6%

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

   if 9.9999999999999993e44 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.0%

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

   3. Taylor expanded in A around 0 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. unpow2 10.4%

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

      3. unpow2 10.4%

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

      4. hypot-define 28.9%

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

   5. Simplified 28.9%

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

      1. pow1/2 28.9%

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

      2. *-commutative 28.9%

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

      3. unpow-prod-down 35.4%

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

      4. pow1/2 35.4%

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

      5. pow1/2 35.4%

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

   7. Applied egg-rr 35.4%

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

   8. Taylor expanded in C around 0 33.2%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+45}:\\ \;\;\;\;\frac{2 \cdot \sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{-4 \cdot \left(A \cdot \left(-C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 4e+39)
   (/
    (sqrt (* (* 2.0 (* (- (pow B_m 2.0) (* (* 4.0 A) C)) F)) (* 2.0 C)))
    (* -4.0 (* A (- C))))
   (* (/ (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 (pow(B_m, 2.0) <= 4e+39) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * (2.0 * C))) / (-4.0 * (A * -C));
	} 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) :: tmp
    if ((b_m ** 2.0d0) <= 4d+39) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - ((4.0d0 * a) * c)) * f)) * (2.0d0 * c))) / ((-4.0d0) * (a * -c))
    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 tmp;
	if (Math.pow(B_m, 2.0) <= 4e+39) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * (2.0 * C))) / (-4.0 * (A * -C));
	} 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 math.pow(B_m, 2.0) <= 4e+39:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * (2.0 * C))) / (-4.0 * (A * -C))
	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 ^ 2.0) <= 4e+39)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(2.0 * C))) / Float64(-4.0 * Float64(A * Float64(-C))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-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 ^ 2.0) <= 4e+39)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - ((4.0 * A) * C)) * F)) * (2.0 * C))) / (-4.0 * (A * -C));
	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[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+39], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(-4.0 * N[(A * (-C)), $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 (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999976e39

   1. Initial program 18.5%

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

   3. Taylor expanded in A around -inf 22.2%

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

   4. Taylor expanded in B around 0 22.0%

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

   if 3.99999999999999976e39 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.7%

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

   3. Taylor expanded in A around 0 11.2%

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

      1. mul-1-neg 11.2%

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

      2. unpow2 11.2%

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

      3. unpow2 11.2%

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

      4. hypot-define 29.3%

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

   5. Simplified 29.3%

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

      1. pow1/2 29.3%

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

      2. *-commutative 29.3%

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

      3. unpow-prod-down 35.7%

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

      4. pow1/2 35.7%

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

      5. pow1/2 35.7%

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

   7. Applied egg-rr 35.7%

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

   8. Taylor expanded in C around 0 32.8%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{-4 \cdot \left(A \cdot \left(-C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.4 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{-4 \cdot \left(A \cdot \left(-C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 1.4e+20)
   (/
    (sqrt (* (* 2.0 (* (- (pow B_m 2.0) (* (* 4.0 A) C)) F)) (* 2.0 C)))
    (* -4.0 (* A (- C))))
   (/ (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 <= 1.4e+20) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * (2.0 * C))) / (-4.0 * (A * -C));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.4d+20) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - ((4.0d0 * a) * c)) * f)) * (2.0d0 * c))) / ((-4.0d0) * (a * -c))
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.4e+20) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * (2.0 * C))) / (-4.0 * (A * -C));
	} 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 <= 1.4e+20:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * (2.0 * C))) / (-4.0 * (A * -C))
	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 <= 1.4e+20)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(2.0 * C))) / Float64(-4.0 * Float64(A * Float64(-C))));
	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 <= 1.4e+20)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - ((4.0 * A) * C)) * F)) * (2.0 * C))) / (-4.0 * (A * -C));
	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, 1.4e+20], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(-4.0 * N[(A * (-C)), $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 < 1.4e20

   1. Initial program 17.7%

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

   3. Taylor expanded in A around -inf 17.6%

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

   4. Taylor expanded in B around 0 17.0%

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

   if 1.4e20 < B

   1. Initial program 13.9%

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

   3. Taylor expanded in B around inf 43.1%

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

      1. mul-1-neg 43.1%

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

   5. Simplified 43.1%

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

      1. sqrt-unprod 43.3%

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

      2. pow1/2 43.3%

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

   7. Applied egg-rr 43.3%

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

      1. pow1/2 43.3%

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

      2. associate-*l/ 43.3%

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

      3. sqrt-div 66.0%

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

   9. Applied egg-rr 66.0%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{-4 \cdot \left(A \cdot \left(-C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 5.5e+19)
   (/
    (sqrt (* (* 2.0 C) (* 2.0 (* -4.0 (* A (* C F))))))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (/ (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 <= 5.5e+19) {
		tmp = sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 5.5d+19) then
        tmp = sqrt(((2.0d0 * c) * (2.0d0 * ((-4.0d0) * (a * (c * f)))))) / (((4.0d0 * a) * c) - (b_m ** 2.0d0))
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.5e+19) {
		tmp = Math.sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} 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 <= 5.5e+19:
		tmp = math.sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	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 <= 5.5e+19)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	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 <= 5.5e+19)
		tmp = sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	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, 5.5e+19], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $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 < 5.5e19

   1. Initial program 17.7%

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

   3. Taylor expanded in A around -inf 17.6%

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

   4. Taylor expanded in B around 0 17.2%

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

   if 5.5e19 < B

   1. Initial program 13.9%

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

   3. Taylor expanded in B around inf 43.1%

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

      1. mul-1-neg 43.1%

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

   5. Simplified 43.1%

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

      1. sqrt-unprod 43.3%

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

      2. pow1/2 43.3%

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

   7. Applied egg-rr 43.3%

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

      1. pow1/2 43.3%

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

      2. associate-*l/ 43.3%

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

      3. sqrt-div 66.0%

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

   9. Applied egg-rr 66.0%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.2 \cdot 10^{+26}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{{B\_m}^{2} - 4 \cdot \left(A \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 6.2e+26)
   (* -2.0 (sqrt (/ (* C F) (- (pow B_m 2.0) (* 4.0 (* A C))))))
   (/ (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 <= 6.2e+26) {
		tmp = -2.0 * sqrt(((C * F) / (pow(B_m, 2.0) - (4.0 * (A * C)))));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 6.2d+26) then
        tmp = (-2.0d0) * sqrt(((c * f) / ((b_m ** 2.0d0) - (4.0d0 * (a * c)))))
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6.2e+26) {
		tmp = -2.0 * Math.sqrt(((C * F) / (Math.pow(B_m, 2.0) - (4.0 * (A * C)))));
	} 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 <= 6.2e+26:
		tmp = -2.0 * math.sqrt(((C * F) / (math.pow(B_m, 2.0) - (4.0 * (A * C)))))
	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 <= 6.2e+26)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(C * F) / Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C))))));
	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 <= 6.2e+26)
		tmp = -2.0 * sqrt(((C * F) / ((B_m ^ 2.0) - (4.0 * (A * C)))));
	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, 6.2e+26], N[(-2.0 * N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $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 < 6.1999999999999999e26

   1. Initial program 18.0%

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

   3. Taylor expanded in A around -inf 17.9%

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

   4. Taylor expanded in F around 0 11.9%

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

   if 6.1999999999999999e26 < B

   1. Initial program 12.4%

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

   3. Taylor expanded in B around inf 44.4%

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

      1. mul-1-neg 44.4%

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

   5. Simplified 44.4%

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

      1. sqrt-unprod 44.6%

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

      2. pow1/2 44.6%

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

   7. Applied egg-rr 44.6%

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

      1. pow1/2 44.6%

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

      2. associate-*l/ 44.6%

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

      3. sqrt-div 68.2%

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

   9. Applied egg-rr 68.2%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{+26}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 7 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(B\_m, C\right)\right) \cdot \left(2 \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

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 (<= C 7e-147)
   (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))
   (/ (sqrt (* (+ C (hypot B_m C)) (* 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) {
	double tmp;
	if (C <= 7e-147) {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	} else {
		tmp = sqrt(((C + hypot(B_m, C)) * (2.0 * F))) / -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 (C <= 7e-147) {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	} else {
		tmp = Math.sqrt(((C + Math.hypot(B_m, C)) * (2.0 * F))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 7e-147:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	else:
		tmp = math.sqrt(((C + math.hypot(B_m, C)) * (2.0 * F))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 7e-147)
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	else
		tmp = Float64(sqrt(Float64(Float64(C + hypot(B_m, C)) * Float64(2.0 * F))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 7e-147)
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	else
		tmp = sqrt(((C + hypot(B_m, C)) * (2.0 * F))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 7e-147], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 7.00000000000000007e-147

   1. Initial program 14.9%

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

   3. Taylor expanded in B around inf 14.0%

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

      1. mul-1-neg 14.0%

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

   5. Simplified 14.0%

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

      1. sqrt-unprod 14.1%

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

      2. pow1/2 14.3%

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

   7. Applied egg-rr 14.3%

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

      1. pow1/2 14.1%

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

      2. associate-*l/ 14.1%

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

      3. sqrt-div 19.9%

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

   9. Applied egg-rr 19.9%

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

   if 7.00000000000000007e-147 < C

   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 0 4.5%

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

      1. mul-1-neg 4.5%

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

      2. unpow2 4.5%

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

      3. unpow2 4.5%

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

      4. hypot-define 11.7%

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

   5. Simplified 11.7%

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

      1. associate-*l/ 11.6%

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

      2. pow1/2 11.6%

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

      3. pow1/2 11.8%

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

      4. pow-prod-down 11.8%

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

   7. Applied egg-rr 11.8%

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

      1. unpow1/2 11.7%

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

      2. associate-*r* 11.7%

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

   9. Simplified 11.7%

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

4. Final simplification 16.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 7 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.5% accurate, 3.1× speedup

Math

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

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 16.9%

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

3. Taylor expanded in B around inf 11.7%

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

   1. mul-1-neg 11.7%

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

5. Simplified 11.7%

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

   1. sqrt-unprod 11.8%

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

   2. pow1/2 12.0%

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

7. Applied egg-rr 12.0%

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

   1. pow1/2 11.8%

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

   2. associate-*l/ 11.8%

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

   3. sqrt-div 16.2%

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

9. Applied egg-rr 16.2%

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

10. Final simplification 16.2%

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

Alternative 14: 27.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 5.5 \cdot 10^{+151}:\\ \;\;\;\;-{\left(2 \cdot \frac{F}{B\_m}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 5.5e+151)
   (- (pow (* 2.0 (/ F B_m)) 0.5))
   (* -2.0 (/ (sqrt (* C F)) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 5.5e+151) {
		tmp = -pow((2.0 * (F / B_m)), 0.5);
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 5.5d+151) then
        tmp = -((2.0d0 * (f / b_m)) ** 0.5d0)
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 5.5e+151) {
		tmp = -Math.pow((2.0 * (F / B_m)), 0.5);
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 5.5e+151:
		tmp = -math.pow((2.0 * (F / B_m)), 0.5)
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 5.5e+151)
		tmp = Float64(-(Float64(2.0 * Float64(F / B_m)) ^ 0.5));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / 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 (C <= 5.5e+151)
		tmp = -((2.0 * (F / B_m)) ^ 0.5);
	else
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 5.5e+151], (-N[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 5.4999999999999994e151

   1. Initial program 19.7%

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

   3. Taylor expanded in B around inf 13.4%

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

      1. mul-1-neg 13.4%

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

   5. Simplified 13.4%

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

      1. sqrt-unprod 13.5%

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

      2. pow1/2 13.7%

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

   7. Applied egg-rr 13.7%

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

   if 5.4999999999999994e151 < C

   1. Initial program 1.5%

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

   3. Taylor expanded in A around -inf 18.7%

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

   4. Taylor expanded in B around inf 4.1%

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

      1. associate-*l/ 4.1%

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

      2. *-lft-identity 4.1%

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

   6. Simplified 4.1%

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

4. Final simplification 12.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 5.5 \cdot 10^{+151}:\\ \;\;\;\;-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -{\left(2 \cdot \frac{F}{B\_m}\right)}^{0.5} \end{array} \]

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 (- (pow (* 2.0 (/ F B_m)) 0.5)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -pow((2.0 * (F / B_m)), 0.5);
}

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 = -((2.0d0 * (f / b_m)) ** 0.5d0)
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.pow((2.0 * (F / B_m)), 0.5);
}

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.pow((2.0 * (F / B_m)), 0.5)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-(Float64(2.0 * Float64(F / B_m)) ^ 0.5))
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 = -((2.0 * (F / B_m)) ^ 0.5);
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[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision])

Derivation

1. Initial program 16.9%

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

3. Taylor expanded in B around inf 11.7%

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

   1. mul-1-neg 11.7%

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

5. Simplified 11.7%

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

   1. sqrt-unprod 11.8%

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

   2. pow1/2 12.0%

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

7. Applied egg-rr 12.0%

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

8. Final simplification 12.0%

   \[\leadsto -{\left(2 \cdot \frac{F}{B}\right)}^{0.5} \]
9. Add Preprocessing

Alternative 16: 27.1% 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 16.9%

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

3. Taylor expanded in B around inf 11.7%

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

   1. mul-1-neg 11.7%

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

5. Simplified 11.7%

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

   1. pow1 11.7%

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

   2. sqrt-unprod 11.8%

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

7. Applied egg-rr 11.8%

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

   1. unpow1 11.8%

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

9. Simplified 11.8%

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

10. Final simplification 11.8%

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

Reproduce

herbie shell --seed 2024095 
(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))))