ABCF->ab-angle a

Percentage Accurate: 19.5% → 61.4%

Time: 21.5s

Alternatives: 19

Speedup: 6.0×

• Could not converge on a ground truth

  1. A = -1.925410279500884e-45
  2. B = 6.633405607018215e-61
  3. C = 5.495567459388563e+93
  4. F = 1.5305352666613396e+113

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 61.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-179}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{e^{\log 2 \cdot 0.5}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 (- INFINITY))
     (*
      (sqrt
       (*
        F
        (/ (+ (+ A C) (hypot B_m (- A C))) (fma -4.0 (* A C) (pow B_m 2.0)))))
      (- (sqrt 2.0)))
     (if (<= t_2 -1e-179)
       t_2
       (if (<= t_2 INFINITY)
         (/
          (*
           (sqrt (* 2.0 (* F t_0)))
           (sqrt (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
          (- t_0))
         (*
          (* (sqrt (+ C (hypot C B_m))) (sqrt F))
          (/ (exp (* (log 2.0) 0.5)) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sqrt((F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_2 <= -1e-179) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * (F * t_0))) * sqrt(((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / -t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (exp((log(2.0) * 0.5)) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_2 <= -1e-179)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(exp(Float64(log(2.0) * 0.5)) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * 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] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, -1e-179], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.2%

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

   3. Taylor expanded in F around 0 23.6%

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

   5. Simplified 66.2%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-179

   1. Initial program 99.1%

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

   if -1e-179 < (/.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 18.7%

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

   3. Simplified 35.6%

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

      1. associate-*r* 35.6%

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

      2. associate-+r+ 34.2%

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

      3. hypot-undefine 18.7%

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

      4. unpow2 18.7%

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

      5. unpow2 18.7%

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

      6. +-commutative 18.7%

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

      7. sqrt-prod 21.7%

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

      8. *-commutative 21.7%

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

      9. associate-+l+ 23.4%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in A around -inf 42.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 2.1%

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

      1. mul-1-neg 2.1%

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

      2. *-commutative 2.1%

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

      3. *-commutative 2.1%

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

   5. Simplified 2.1%

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

      1. sqrt-prod 2.0%

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

      2. +-commutative 2.0%

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

      3. unpow2 2.0%

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

      4. unpow2 2.0%

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

      5. hypot-define 25.9%

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

   7. Applied egg-rr 25.9%

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

      1. pow1/2 25.9%

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

      2. pow-to-exp 25.9%

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

   9. Applied egg-rr 25.9%

      \[\leadsto -\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\color{blue}{e^{\log 2 \cdot 0.5}}}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{e^{\log 2 \cdot 0.5}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot {C}^{2}\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2000000000:\\ \;\;\;\;\sqrt{\frac{C \cdot F}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-2\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

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) 2e-303)
   (/
    (sqrt (* (* A -16.0) (* F (pow C 2.0))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2000000000.0)
     (* (sqrt (/ (* C F) (fma -4.0 (* A C) (pow B_m 2.0)))) (- 2.0))
     (if (<= (pow B_m 2.0) 5e+283)
       (/ (sqrt (* 2.0 (* F (+ C (hypot C B_m))))) (- B_m))
       (* (/ (sqrt 2.0) B_m) (* (sqrt 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 (pow(B_m, 2.0) <= 2e-303) {
		tmp = sqrt(((A * -16.0) * (F * pow(C, 2.0)))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (pow(B_m, 2.0) <= 2000000000.0) {
		tmp = sqrt(((C * F) / fma(-4.0, (A * C), pow(B_m, 2.0)))) * -2.0;
	} else if (pow(B_m, 2.0) <= 5e+283) {
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -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 ^ 2.0) <= 2e-303)
		tmp = Float64(sqrt(Float64(Float64(A * -16.0) * Float64(F * (C ^ 2.0)))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif ((B_m ^ 2.0) <= 2000000000.0)
		tmp = Float64(sqrt(Float64(Float64(C * F) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))) * Float64(-2.0));
	elseif ((B_m ^ 2.0) <= 5e+283)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-303], N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[Power[C, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2000000000.0], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-2.0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+283], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999986e-303

   1. Initial program 23.5%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in B around 0 29.8%

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

      1. associate-*r* 29.8%

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

   7. Simplified 29.8%

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

   8. Taylor expanded in A around -inf 19.9%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 19.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   10. Simplified 19.9%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 1.99999999999999986e-303 < (pow.f64 B #s(literal 2 binary64)) < 2e9

   1. Initial program 29.8%

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

   3. Simplified 43.0%

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

      1. associate-*r* 43.0%

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

      2. associate-+r+ 41.5%

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

      3. hypot-undefine 29.8%

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

      4. unpow2 29.8%

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

      5. unpow2 29.8%

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

      6. +-commutative 29.8%

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

      7. sqrt-prod 34.6%

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

      8. *-commutative 34.6%

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

      9. associate-+l+ 35.5%

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

   6. Applied egg-rr 48.9%

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

   7. Taylor expanded in A around -inf 31.1%

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

   8. Taylor expanded in F around 0 17.1%

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

      1. mul-1-neg 17.1%

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

      2. *-commutative 17.1%

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

      3. unpow2 17.1%

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

      4. rem-square-sqrt 17.3%

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

      5. *-commutative 17.3%

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

      6. fma-define 17.3%

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

      7. *-commutative 17.3%

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

   10. Simplified 17.3%

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

   if 2e9 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e283

   1. Initial program 22.4%

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

   3. Taylor expanded in A around 0 17.2%

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

      1. mul-1-neg 17.2%

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

      2. *-commutative 17.2%

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

      3. *-commutative 17.2%

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

   5. Simplified 17.2%

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

      1. neg-sub0 17.2%

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

      2. associate-*r/ 17.2%

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

      3. pow1/2 17.3%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F\right)}^{0.5}} \cdot \sqrt{2}}{B} \]

      4. *-commutative 17.3%

         \[\leadsto 0 - \frac{{\color{blue}{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}}^{0.5} \cdot \sqrt{2}}{B} \]

      5. pow1/2 17.3%

         \[\leadsto 0 - \frac{{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

      6. pow-prod-down 17.4%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right) \cdot 2\right)}^{0.5}}}{B} \]

      7. +-commutative 17.4%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      8. unpow2 17.4%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      9. unpow2 17.4%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      10. hypot-define 17.9%

          \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \color{blue}{\mathsf{hypot}\left(C, B\right)}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

   7. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{0 - \frac{{\left(\left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right) \cdot 2\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 17.9%

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

      2. distribute-neg-frac2 17.9%

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

      3. unpow1/2 17.9%

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

   9. Simplified 17.9%

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

   if 5.0000000000000004e283 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

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

   3. Taylor expanded in A around 0 3.2%

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

      1. mul-1-neg 3.2%

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

      2. *-commutative 3.2%

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

      3. *-commutative 3.2%

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

   5. Simplified 3.2%

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

      1. sqrt-prod 3.3%

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

      2. +-commutative 3.3%

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

      3. unpow2 3.3%

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

      4. unpow2 3.3%

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

      5. hypot-define 39.4%

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

   7. Applied egg-rr 39.4%

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

   8. Taylor expanded in C around 0 35.0%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot {C}^{2}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2000000000:\\ \;\;\;\;\sqrt{\frac{C \cdot F}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-2\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 200000:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \sqrt{2 \cdot C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 200000.0)
     (/ (* (sqrt (* 2.0 (* F t_0))) (sqrt (* 2.0 C))) (- t_0))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 200000.0) {
		tmp = (sqrt((2.0 * (F * t_0))) * sqrt((2.0 * C))) / -t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 200000.0)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * sqrt(Float64(2.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 200000.0], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e5

   1. Initial program 26.8%

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

   3. Simplified 38.6%

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

      1. associate-*r* 38.6%

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

      2. associate-+r+ 37.3%

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

      3. hypot-undefine 26.8%

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

      4. unpow2 26.8%

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

      5. unpow2 26.8%

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

      6. +-commutative 26.8%

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

      7. sqrt-prod 28.7%

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

      8. *-commutative 28.7%

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

      9. associate-+l+ 29.4%

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

   6. Applied egg-rr 45.2%

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

   7. Taylor expanded in A around -inf 28.0%

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

   if 2e5 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.2%

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

   3. Taylor expanded in A around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. *-commutative 9.5%

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

      3. *-commutative 9.5%

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

   5. Simplified 9.5%

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

      1. sqrt-prod 9.6%

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

      2. +-commutative 9.6%

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

      3. unpow2 9.6%

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

      4. unpow2 9.6%

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

      5. hypot-define 30.9%

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

   7. Applied egg-rr 30.9%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 200000:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 200000:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot t\_0\right)} \cdot \frac{\sqrt{2 \cdot C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 200000.0)
     (* (sqrt (* F (* 2.0 t_0))) (/ (sqrt (* 2.0 C)) (- t_0)))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 200000.0) {
		tmp = sqrt((F * (2.0 * t_0))) * (sqrt((2.0 * C)) / -t_0);
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 200000.0)
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * t_0))) * Float64(sqrt(Float64(2.0 * C)) / Float64(-t_0)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 200000.0], N[(N[Sqrt[N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e5

   1. Initial program 26.8%

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

   3. Simplified 38.6%

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

      1. associate-*r* 38.6%

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

      2. associate-+r+ 37.3%

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

      3. hypot-undefine 26.8%

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

      4. unpow2 26.8%

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

      5. unpow2 26.8%

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

      6. +-commutative 26.8%

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

      7. sqrt-prod 28.7%

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

      8. *-commutative 28.7%

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

      9. associate-+l+ 29.4%

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

   6. Applied egg-rr 45.2%

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

   7. Taylor expanded in A around -inf 28.0%

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

      1. associate-/l* 28.0%

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

      2. associate-*l* 28.0%

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

      3. *-commutative 28.0%

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

   9. Applied egg-rr 28.0%

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

   if 2e5 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.2%

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

   3. Taylor expanded in A around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. *-commutative 9.5%

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

      3. *-commutative 9.5%

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

   5. Simplified 9.5%

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

      1. sqrt-prod 9.6%

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

      2. +-commutative 9.6%

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

      3. unpow2 9.6%

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

      4. unpow2 9.6%

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

      5. hypot-define 30.9%

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

   7. Applied egg-rr 30.9%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 200000:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \frac{\sqrt{2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2000000000:\\ \;\;\;\;\sqrt{\frac{C \cdot F}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-2\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

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) 2000000000.0)
   (* (sqrt (/ (* C F) (fma -4.0 (* A C) (pow B_m 2.0)))) (- 2.0))
   (if (<= (pow B_m 2.0) 5e+283)
     (/ (sqrt (* 2.0 (* F (+ C (hypot C B_m))))) (- B_m))
     (* (/ (sqrt 2.0) B_m) (* (sqrt 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 (pow(B_m, 2.0) <= 2000000000.0) {
		tmp = sqrt(((C * F) / fma(-4.0, (A * C), pow(B_m, 2.0)))) * -2.0;
	} else if (pow(B_m, 2.0) <= 5e+283) {
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -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 ^ 2.0) <= 2000000000.0)
		tmp = Float64(sqrt(Float64(Float64(C * F) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))) * Float64(-2.0));
	elseif ((B_m ^ 2.0) <= 5e+283)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2000000000.0], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-2.0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+283], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e9

   1. Initial program 26.5%

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

   3. Simplified 38.3%

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

      1. associate-*r* 38.3%

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

      2. associate-+r+ 37.0%

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

      3. hypot-undefine 26.5%

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

      4. unpow2 26.5%

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

      5. unpow2 26.5%

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

      6. +-commutative 26.5%

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

      7. sqrt-prod 28.5%

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

      8. *-commutative 28.5%

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

      9. associate-+l+ 29.2%

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

   6. Applied egg-rr 44.9%

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

   7. Taylor expanded in A around -inf 27.8%

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

   8. Taylor expanded in F around 0 14.0%

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

      1. mul-1-neg 14.0%

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

      2. *-commutative 14.0%

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

      3. unpow2 14.0%

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

      4. rem-square-sqrt 14.1%

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

      5. *-commutative 14.1%

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

      6. fma-define 14.1%

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

      7. *-commutative 14.1%

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

   10. Simplified 14.1%

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

   if 2e9 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e283

   1. Initial program 22.4%

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

   3. Taylor expanded in A around 0 17.2%

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

      1. mul-1-neg 17.2%

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

      2. *-commutative 17.2%

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

      3. *-commutative 17.2%

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

   5. Simplified 17.2%

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

      1. neg-sub0 17.2%

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

      2. associate-*r/ 17.2%

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

      3. pow1/2 17.3%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F\right)}^{0.5}} \cdot \sqrt{2}}{B} \]

      4. *-commutative 17.3%

         \[\leadsto 0 - \frac{{\color{blue}{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}}^{0.5} \cdot \sqrt{2}}{B} \]

      5. pow1/2 17.3%

         \[\leadsto 0 - \frac{{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

      6. pow-prod-down 17.4%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right) \cdot 2\right)}^{0.5}}}{B} \]

      7. +-commutative 17.4%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      8. unpow2 17.4%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      9. unpow2 17.4%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      10. hypot-define 17.9%

          \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \color{blue}{\mathsf{hypot}\left(C, B\right)}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

   7. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{0 - \frac{{\left(\left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right) \cdot 2\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 17.9%

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

      2. distribute-neg-frac2 17.9%

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

      3. unpow1/2 17.9%

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

   9. Simplified 17.9%

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

   if 5.0000000000000004e283 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

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

   3. Taylor expanded in A around 0 3.2%

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

      1. mul-1-neg 3.2%

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

      2. *-commutative 3.2%

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

      3. *-commutative 3.2%

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

   5. Simplified 3.2%

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

      1. sqrt-prod 3.3%

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

      2. +-commutative 3.3%

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

      3. unpow2 3.3%

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

      4. unpow2 3.3%

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

      5. hypot-define 39.4%

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

   7. Applied egg-rr 39.4%

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

   8. Taylor expanded in C around 0 35.0%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2000000000:\\ \;\;\;\;\sqrt{\frac{C \cdot F}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-2\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 200000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 200000.0)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 200000.0) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 200000.0)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 200000.0], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e5

   1. Initial program 26.8%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in A around -inf 27.3%

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

      1. *-commutative 27.3%

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

   7. Simplified 27.3%

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

   if 2e5 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.2%

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

   3. Taylor expanded in A around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. *-commutative 9.5%

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

      3. *-commutative 9.5%

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

   5. Simplified 9.5%

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

      1. sqrt-prod 9.6%

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

      2. +-commutative 9.6%

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

      3. unpow2 9.6%

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

      4. unpow2 9.6%

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

      5. hypot-define 30.9%

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

   7. Applied egg-rr 30.9%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 200000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 370:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 9.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

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 (* A (* C -4.0)))))
   (if (<= B_m 370.0)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (if (<= B_m 9.2e+141)
       (/ (sqrt (* 2.0 (* F (+ C (hypot C B_m))))) (- B_m))
       (* (/ (sqrt 2.0) B_m) (* (sqrt 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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 370.0) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else if (B_m <= 9.2e+141) {
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -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)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 370.0)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	elseif (B_m <= 9.2e+141)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 370.0], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 9.2e+141], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 370

   1. Initial program 20.8%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in A around -inf 19.5%

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

      1. *-commutative 19.5%

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

   7. Simplified 19.5%

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

   if 370 < B < 9.2000000000000006e141

   1. Initial program 24.0%

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

   3. Taylor expanded in A around 0 35.7%

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

      1. mul-1-neg 35.7%

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

      2. *-commutative 35.7%

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

      3. *-commutative 35.7%

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

   5. Simplified 35.7%

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

      1. neg-sub0 35.7%

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

      2. associate-*r/ 35.8%

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

      3. pow1/2 35.8%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F\right)}^{0.5}} \cdot \sqrt{2}}{B} \]

      4. *-commutative 35.8%

         \[\leadsto 0 - \frac{{\color{blue}{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}}^{0.5} \cdot \sqrt{2}}{B} \]

      5. pow1/2 35.8%

         \[\leadsto 0 - \frac{{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

      6. pow-prod-down 36.0%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right) \cdot 2\right)}^{0.5}}}{B} \]

      7. +-commutative 36.0%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      8. unpow2 36.0%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      9. unpow2 36.0%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      10. hypot-define 36.5%

          \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \color{blue}{\mathsf{hypot}\left(C, B\right)}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

   7. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{0 - \frac{{\left(\left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right) \cdot 2\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 36.5%

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

      2. distribute-neg-frac2 36.5%

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

      3. unpow1/2 36.5%

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

   9. Simplified 36.5%

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

   if 9.2000000000000006e141 < B

   1. Initial program 0.2%

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

   3. Taylor expanded in A around 0 5.3%

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

      1. mul-1-neg 5.3%

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

      2. *-commutative 5.3%

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

      3. *-commutative 5.3%

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

   5. Simplified 5.3%

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

      1. sqrt-prod 5.6%

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

      2. +-commutative 5.6%

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

      3. unpow2 5.6%

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

      4. unpow2 5.6%

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

      5. hypot-define 76.9%

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

   7. Applied egg-rr 76.9%

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

   8. Taylor expanded in C around 0 71.1%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 370:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B\_m \leq 3700000:\\ \;\;\;\;\sqrt{\frac{C \cdot F}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-2\right)\\ \mathbf{elif}\;B\_m \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

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 (<= B_m 3e-120)
   (/
    (sqrt (* (* -8.0 (* F (* A C))) (+ C (hypot B_m C))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 3700000.0)
     (* (sqrt (/ (* C F) (fma -4.0 (* A C) (pow B_m 2.0)))) (- 2.0))
     (if (<= B_m 3.4e+143)
       (/ (sqrt (* 2.0 (* F (+ C (hypot C B_m))))) (- B_m))
       (* (/ (sqrt 2.0) B_m) (* (sqrt 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 <= 3e-120) {
		tmp = sqrt(((-8.0 * (F * (A * C))) * (C + hypot(B_m, C)))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (B_m <= 3700000.0) {
		tmp = sqrt(((C * F) / fma(-4.0, (A * C), pow(B_m, 2.0)))) * -2.0;
	} else if (B_m <= 3.4e+143) {
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -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 <= 3e-120)
		tmp = Float64(sqrt(Float64(Float64(-8.0 * Float64(F * Float64(A * C))) * Float64(C + hypot(B_m, C)))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif (B_m <= 3700000.0)
		tmp = Float64(sqrt(Float64(Float64(C * F) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))) * Float64(-2.0));
	elseif (B_m <= 3.4e+143)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3e-120], N[(N[Sqrt[N[(N[(-8.0 * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3700000.0], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-2.0)), $MachinePrecision], If[LessEqual[B$95$m, 3.4e+143], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.00000000000000011e-120

   1. Initial program 19.7%

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

   3. Simplified 27.6%

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

   5. Taylor expanded in B around 0 19.5%

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

      1. associate-*r* 19.5%

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

   7. Simplified 19.5%

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

   8. Taylor expanded in A around 0 8.5%

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

      1. associate-*r* 8.5%

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

      2. unpow2 8.5%

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

      3. unpow2 8.5%

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

      4. hypot-undefine 9.1%

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

   10. Simplified 9.1%

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

   11. Taylor expanded in A around 0 8.5%

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

       1. associate-*r* 8.5%

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

       2. unpow2 8.5%

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

       3. unpow2 8.5%

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

       4. hypot-undefine 9.1%

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

       5. associate-*l* 12.2%

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

       6. associate-*r* 12.2%

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

       7. associate-*r* 12.6%

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

   13. Simplified 12.6%

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

   if 3.00000000000000011e-120 < B < 3.7e6

   1. Initial program 30.1%

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

   3. Simplified 40.6%

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

      1. associate-*r* 40.6%

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

      2. associate-+r+ 39.7%

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

      3. hypot-undefine 30.1%

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

      4. unpow2 30.1%

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

      5. unpow2 30.1%

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

      6. +-commutative 30.1%

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

      7. sqrt-prod 34.5%

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

      8. *-commutative 34.5%

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

      9. associate-+l+ 35.3%

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

   6. Applied egg-rr 49.0%

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

   7. Taylor expanded in A around -inf 25.5%

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

   8. Taylor expanded in F around 0 16.7%

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

      1. mul-1-neg 16.7%

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

      2. *-commutative 16.7%

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

      3. unpow2 16.7%

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

      4. rem-square-sqrt 17.0%

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

      5. *-commutative 17.0%

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

      6. fma-define 17.0%

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

      7. *-commutative 17.0%

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

   10. Simplified 17.0%

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

   if 3.7e6 < B < 3.39999999999999982e143

   1. Initial program 24.0%

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

   3. Taylor expanded in A around 0 35.7%

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

      1. mul-1-neg 35.7%

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

      2. *-commutative 35.7%

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

      3. *-commutative 35.7%

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

   5. Simplified 35.7%

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

      1. neg-sub0 35.7%

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

      2. associate-*r/ 35.8%

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

      3. pow1/2 35.8%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F\right)}^{0.5}} \cdot \sqrt{2}}{B} \]

      4. *-commutative 35.8%

         \[\leadsto 0 - \frac{{\color{blue}{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}}^{0.5} \cdot \sqrt{2}}{B} \]

      5. pow1/2 35.8%

         \[\leadsto 0 - \frac{{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

      6. pow-prod-down 36.0%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right) \cdot 2\right)}^{0.5}}}{B} \]

      7. +-commutative 36.0%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      8. unpow2 36.0%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      9. unpow2 36.0%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      10. hypot-define 36.5%

          \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \color{blue}{\mathsf{hypot}\left(C, B\right)}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

   7. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{0 - \frac{{\left(\left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right) \cdot 2\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 36.5%

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

      2. distribute-neg-frac2 36.5%

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

      3. unpow1/2 36.5%

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

   9. Simplified 36.5%

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

   if 3.39999999999999982e143 < B

   1. Initial program 0.2%

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

   3. Taylor expanded in A around 0 5.3%

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

      1. mul-1-neg 5.3%

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

      2. *-commutative 5.3%

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

      3. *-commutative 5.3%

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

   5. Simplified 5.3%

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

      1. sqrt-prod 5.6%

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

      2. +-commutative 5.6%

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

      3. unpow2 5.6%

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

      4. unpow2 5.6%

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

      5. hypot-define 76.9%

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

   7. Applied egg-rr 76.9%

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

   8. Taylor expanded in C around 0 71.1%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3700000:\\ \;\;\;\;\sqrt{\frac{C \cdot F}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-2\right)\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(C + \mathsf{hypot}\left(B\_m, C\right)\right) \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B\_m \leq 48000:\\ \;\;\;\;\sqrt{\frac{C \cdot F}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-2\right)\\ \mathbf{elif}\;B\_m \leq 1.4 \cdot 10^{+142}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

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 (<= B_m 2.1e-149)
   (/
    (sqrt (* -8.0 (* (+ C (hypot B_m C)) (* A (* C F)))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 48000.0)
     (* (sqrt (/ (* C F) (fma -4.0 (* A C) (pow B_m 2.0)))) (- 2.0))
     (if (<= B_m 1.4e+142)
       (/ (sqrt (* 2.0 (* F (+ C (hypot C B_m))))) (- B_m))
       (* (/ (sqrt 2.0) B_m) (* (sqrt 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 <= 2.1e-149) {
		tmp = sqrt((-8.0 * ((C + hypot(B_m, C)) * (A * (C * F))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (B_m <= 48000.0) {
		tmp = sqrt(((C * F) / fma(-4.0, (A * C), pow(B_m, 2.0)))) * -2.0;
	} else if (B_m <= 1.4e+142) {
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -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 <= 2.1e-149)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(C + hypot(B_m, C)) * Float64(A * Float64(C * F))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif (B_m <= 48000.0)
		tmp = Float64(sqrt(Float64(Float64(C * F) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))) * Float64(-2.0));
	elseif (B_m <= 1.4e+142)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.1e-149], N[(N[Sqrt[N[(-8.0 * N[(N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 48000.0], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-2.0)), $MachinePrecision], If[LessEqual[B$95$m, 1.4e+142], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.10000000000000011e-149

   1. Initial program 19.0%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in B around 0 18.9%

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

      1. associate-*r* 18.9%

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

   7. Simplified 18.9%

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

   8. Taylor expanded in A around 0 8.6%

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

      1. associate-*r* 8.6%

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

      2. unpow2 8.6%

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

      3. unpow2 8.6%

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

      4. hypot-undefine 9.3%

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

   10. Simplified 9.3%

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

       1. distribute-frac-neg2 9.3%

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

       2. associate-*r* 11.9%

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

   12. Applied egg-rr 11.9%

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

   if 2.10000000000000011e-149 < B < 48000

   1. Initial program 32.2%

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

   3. Simplified 44.5%

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

      1. associate-*r* 44.5%

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

      2. associate-+r+ 43.6%

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

      3. hypot-undefine 32.2%

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

      4. unpow2 32.2%

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

      5. unpow2 32.2%

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

      6. +-commutative 32.2%

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

      7. sqrt-prod 39.5%

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

      8. *-commutative 39.5%

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

      9. associate-+l+ 40.3%

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

   6. Applied egg-rr 55.0%

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

   7. Taylor expanded in A around -inf 28.6%

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

   8. Taylor expanded in F around 0 14.0%

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

      1. mul-1-neg 14.0%

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

      2. *-commutative 14.0%

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

      3. unpow2 14.0%

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

      4. rem-square-sqrt 14.2%

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

      5. *-commutative 14.2%

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

      6. fma-define 14.2%

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

      7. *-commutative 14.2%

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

   10. Simplified 14.2%

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

   if 48000 < B < 1.4e142

   1. Initial program 24.0%

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

   3. Taylor expanded in A around 0 35.7%

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

      1. mul-1-neg 35.7%

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

      2. *-commutative 35.7%

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

      3. *-commutative 35.7%

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

   5. Simplified 35.7%

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

      1. neg-sub0 35.7%

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

      2. associate-*r/ 35.8%

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

      3. pow1/2 35.8%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F\right)}^{0.5}} \cdot \sqrt{2}}{B} \]

      4. *-commutative 35.8%

         \[\leadsto 0 - \frac{{\color{blue}{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}}^{0.5} \cdot \sqrt{2}}{B} \]

      5. pow1/2 35.8%

         \[\leadsto 0 - \frac{{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

      6. pow-prod-down 36.0%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right) \cdot 2\right)}^{0.5}}}{B} \]

      7. +-commutative 36.0%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      8. unpow2 36.0%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      9. unpow2 36.0%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      10. hypot-define 36.5%

          \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \color{blue}{\mathsf{hypot}\left(C, B\right)}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

   7. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{0 - \frac{{\left(\left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right) \cdot 2\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 36.5%

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

      2. distribute-neg-frac2 36.5%

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

      3. unpow1/2 36.5%

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

   9. Simplified 36.5%

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

   if 1.4e142 < B

   1. Initial program 0.2%

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

   3. Taylor expanded in A around 0 5.3%

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

      1. mul-1-neg 5.3%

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

      2. *-commutative 5.3%

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

      3. *-commutative 5.3%

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

   5. Simplified 5.3%

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

      1. sqrt-prod 5.6%

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

      2. +-commutative 5.6%

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

      3. unpow2 5.6%

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

      4. unpow2 5.6%

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

      5. hypot-define 76.9%

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

   7. Applied egg-rr 76.9%

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

   8. Taylor expanded in C around 0 71.1%

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

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 48000:\\ \;\;\;\;\sqrt{\frac{C \cdot F}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-2\right)\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+142}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 9.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (<= F 9.2e+52)
   (/ (sqrt (* 2.0 (* F (+ C (hypot C B_m))))) (- B_m))
   (* (sqrt 2.0) (/ (sqrt 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 (F <= 9.2e+52) {
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	} else {
		tmp = sqrt(2.0) * (sqrt(F) / -sqrt(B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 9.2e+52) {
		tmp = Math.sqrt((2.0 * (F * (C + Math.hypot(C, B_m))))) / -B_m;
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt(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 F <= 9.2e+52:
		tmp = math.sqrt((2.0 * (F * (C + math.hypot(C, B_m))))) / -B_m
	else:
		tmp = math.sqrt(2.0) * (math.sqrt(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 (F <= 9.2e+52)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(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 (F <= 9.2e+52)
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	else
		tmp = sqrt(2.0) * (sqrt(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[F, 9.2e+52], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 9.1999999999999999e52

   1. Initial program 19.5%

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

   3. Taylor expanded in A around 0 8.0%

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

      1. mul-1-neg 8.0%

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

      2. *-commutative 8.0%

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

      3. *-commutative 8.0%

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

   5. Simplified 8.0%

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

      1. neg-sub0 8.0%

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

      2. associate-*r/ 8.1%

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

      3. pow1/2 8.3%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F\right)}^{0.5}} \cdot \sqrt{2}}{B} \]

      4. *-commutative 8.3%

         \[\leadsto 0 - \frac{{\color{blue}{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}}^{0.5} \cdot \sqrt{2}}{B} \]

      5. pow1/2 8.3%

         \[\leadsto 0 - \frac{{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

      6. pow-prod-down 8.3%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right) \cdot 2\right)}^{0.5}}}{B} \]

      7. +-commutative 8.3%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      8. unpow2 8.3%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      9. unpow2 8.3%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      10. hypot-define 19.0%

          \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \color{blue}{\mathsf{hypot}\left(C, B\right)}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

   7. Applied egg-rr 19.0%

      \[\leadsto \color{blue}{0 - \frac{{\left(\left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right) \cdot 2\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 19.0%

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

      2. distribute-neg-frac2 19.0%

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

      3. unpow1/2 18.9%

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

   9. Simplified 18.9%

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

   if 9.1999999999999999e52 < F

   1. Initial program 16.6%

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

   3. Taylor expanded in B around inf 15.9%

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

      1. mul-1-neg 15.9%

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

      2. *-commutative 15.9%

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

   5. Simplified 15.9%

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

      1. sqrt-div 16.6%

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

   7. Applied egg-rr 16.6%

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

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 9.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.22 \cdot 10^{+44}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\sqrt{\frac{B\_m}{F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.22e+44)
   (/ (sqrt (* 2.0 (* F (+ C (hypot C B_m))))) (- B_m))
   (* (sqrt 2.0) (/ -1.0 (sqrt (/ B_m F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.22e+44) {
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	} else {
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / 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 tmp;
	if (F <= 1.22e+44) {
		tmp = Math.sqrt((2.0 * (F * (C + Math.hypot(C, B_m))))) / -B_m;
	} else {
		tmp = Math.sqrt(2.0) * (-1.0 / Math.sqrt((B_m / F)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.22e+44:
		tmp = math.sqrt((2.0 * (F * (C + math.hypot(C, B_m))))) / -B_m
	else:
		tmp = math.sqrt(2.0) * (-1.0 / math.sqrt((B_m / F)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.22e+44)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-1.0 / sqrt(Float64(B_m / F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.22e+44)
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	else
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.22e+44], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.22e44

   1. Initial program 19.1%

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

   3. Taylor expanded in A around 0 7.5%

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

      1. mul-1-neg 7.5%

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

      2. *-commutative 7.5%

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

      3. *-commutative 7.5%

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

   5. Simplified 7.5%

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

      1. neg-sub0 7.5%

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

      2. associate-*r/ 7.5%

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

      3. pow1/2 7.7%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F\right)}^{0.5}} \cdot \sqrt{2}}{B} \]

      4. *-commutative 7.7%

         \[\leadsto 0 - \frac{{\color{blue}{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}}^{0.5} \cdot \sqrt{2}}{B} \]

      5. pow1/2 7.7%

         \[\leadsto 0 - \frac{{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

      6. pow-prod-down 7.7%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right) \cdot 2\right)}^{0.5}}}{B} \]

      7. +-commutative 7.7%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      8. unpow2 7.7%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      9. unpow2 7.7%

         \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

      10. hypot-define 18.6%

          \[\leadsto 0 - \frac{{\left(\left(F \cdot \left(C + \color{blue}{\mathsf{hypot}\left(C, B\right)}\right)\right) \cdot 2\right)}^{0.5}}{B} \]

   7. Applied egg-rr 18.6%

      \[\leadsto \color{blue}{0 - \frac{{\left(\left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right) \cdot 2\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 18.6%

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

      2. distribute-neg-frac2 18.6%

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

      3. unpow1/2 18.5%

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

   9. Simplified 18.5%

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

   if 1.22e44 < F

   1. Initial program 17.3%

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

   3. Taylor expanded in B around inf 16.5%

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

      1. mul-1-neg 16.5%

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

      2. *-commutative 16.5%

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

   5. Simplified 16.5%

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

      1. add-cbrt-cube 12.3%

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

      2. pow1/3 11.7%

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

      3. add-sqr-sqrt 11.7%

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

      4. pow1 11.7%

         \[\leadsto -\sqrt{2} \cdot {\left(\color{blue}{{\left(\frac{F}{B}\right)}^{1}} \cdot \sqrt{\frac{F}{B}}\right)}^{0.3333333333333333} \]

      5. pow1/2 12.1%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{1} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 12.1%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left({\left(\frac{F}{B}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 12.1%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 12.1%

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

      1. unpow1/3 12.7%

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

   9. Simplified 12.7%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\sqrt[3]{{\left(\frac{F}{B}\right)}^{1.5}}} \]
   10. Step-by-step derivation

       1. pow1/3 12.1%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left({\left(\frac{F}{B}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 16.9%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 16.9%

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

       4. pow1/2 16.5%

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

       5. clear-num 16.5%

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

       6. sqrt-div 17.1%

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

       7. metadata-eval 17.1%

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

   11. Applied egg-rr 17.1%

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

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.22 \cdot 10^{+44}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\sqrt{\frac{B}{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{F \cdot \left(B\_m + C\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\sqrt{\frac{B\_m}{F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.5e+29)
   (* (sqrt (* F (+ B_m C))) (/ (sqrt 2.0) (- B_m)))
   (* (sqrt 2.0) (/ -1.0 (sqrt (/ B_m F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.5e+29) {
		tmp = sqrt((F * (B_m + C))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 1.5d+29) then
        tmp = sqrt((f * (b_m + c))) * (sqrt(2.0d0) / -b_m)
    else
        tmp = sqrt(2.0d0) * ((-1.0d0) / sqrt((b_m / f)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.5e+29) {
		tmp = Math.sqrt((F * (B_m + C))) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = Math.sqrt(2.0) * (-1.0 / Math.sqrt((B_m / F)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.5e+29:
		tmp = math.sqrt((F * (B_m + C))) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = math.sqrt(2.0) * (-1.0 / math.sqrt((B_m / F)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.5e+29)
		tmp = Float64(sqrt(Float64(F * Float64(B_m + C))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-1.0 / sqrt(Float64(B_m / F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.5e+29)
		tmp = sqrt((F * (B_m + C))) * (sqrt(2.0) / -B_m);
	else
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.5e+29], N[(N[Sqrt[N[(F * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.5e29

   1. Initial program 19.1%

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

   3. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. *-commutative 7.0%

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

      3. *-commutative 7.0%

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

   5. Simplified 7.0%

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

   6. Taylor expanded in C around 0 15.3%

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

      1. distribute-rgt-out 15.3%

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

   8. Simplified 15.3%

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

   if 1.5e29 < F

   1. Initial program 17.3%

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

   3. Taylor expanded in B around inf 16.6%

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

      1. mul-1-neg 16.6%

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

      2. *-commutative 16.6%

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

   5. Simplified 16.6%

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

      1. add-cbrt-cube 12.6%

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

      2. pow1/3 12.0%

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

      3. add-sqr-sqrt 12.0%

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

      4. pow1 12.0%

         \[\leadsto -\sqrt{2} \cdot {\left(\color{blue}{{\left(\frac{F}{B}\right)}^{1}} \cdot \sqrt{\frac{F}{B}}\right)}^{0.3333333333333333} \]

      5. pow1/2 12.4%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{1} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 12.4%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left({\left(\frac{F}{B}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 12.4%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 12.4%

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

      1. unpow1/3 13.0%

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

   9. Simplified 13.0%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\sqrt[3]{{\left(\frac{F}{B}\right)}^{1.5}}} \]
   10. Step-by-step derivation

       1. pow1/3 12.4%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left({\left(\frac{F}{B}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 17.0%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 17.0%

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

       4. pow1/2 16.6%

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

       5. clear-num 16.6%

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

       6. sqrt-div 17.2%

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

       7. metadata-eval 17.2%

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

   11. Applied egg-rr 17.2%

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

4. Final simplification 16.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\sqrt{\frac{B}{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{B\_m \cdot F} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\sqrt{\frac{B\_m}{F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.15e-15)
   (* (sqrt (* B_m F)) (/ (sqrt 2.0) (- B_m)))
   (* (sqrt 2.0) (/ -1.0 (sqrt (/ B_m F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.15e-15) {
		tmp = sqrt((B_m * F)) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 1.15d-15) then
        tmp = sqrt((b_m * f)) * (sqrt(2.0d0) / -b_m)
    else
        tmp = sqrt(2.0d0) * ((-1.0d0) / sqrt((b_m / f)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.15e-15) {
		tmp = Math.sqrt((B_m * F)) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = Math.sqrt(2.0) * (-1.0 / Math.sqrt((B_m / F)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.15e-15:
		tmp = math.sqrt((B_m * F)) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = math.sqrt(2.0) * (-1.0 / math.sqrt((B_m / F)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.15e-15)
		tmp = Float64(sqrt(Float64(B_m * F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-1.0 / sqrt(Float64(B_m / F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.15e-15)
		tmp = sqrt((B_m * F)) * (sqrt(2.0) / -B_m);
	else
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.15e-15], N[(N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.14999999999999995e-15

   1. Initial program 18.4%

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

   3. Taylor expanded in A around 0 6.4%

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

      1. mul-1-neg 6.4%

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

      2. *-commutative 6.4%

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

      3. *-commutative 6.4%

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

   5. Simplified 6.4%

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

   6. Taylor expanded in C around 0 16.1%

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

   if 1.14999999999999995e-15 < F

   1. Initial program 18.2%

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

   3. Taylor expanded in B around inf 16.8%

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

      1. mul-1-neg 16.8%

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

      2. *-commutative 16.8%

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

   5. Simplified 16.8%

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

      1. add-cbrt-cube 12.8%

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

      2. pow1/3 12.1%

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

      3. add-sqr-sqrt 12.1%

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

      4. pow1 12.1%

         \[\leadsto -\sqrt{2} \cdot {\left(\color{blue}{{\left(\frac{F}{B}\right)}^{1}} \cdot \sqrt{\frac{F}{B}}\right)}^{0.3333333333333333} \]

      5. pow1/2 12.4%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{1} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 12.4%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left({\left(\frac{F}{B}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 12.4%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 12.4%

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

      1. unpow1/3 13.1%

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

   9. Simplified 13.1%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\sqrt[3]{{\left(\frac{F}{B}\right)}^{1.5}}} \]
   10. Step-by-step derivation

       1. pow1/3 12.4%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left({\left(\frac{F}{B}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 17.1%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 17.1%

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

       4. pow1/2 16.8%

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

       5. clear-num 16.8%

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

       6. sqrt-div 17.3%

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

       7. metadata-eval 17.3%

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

   11. Applied egg-rr 17.3%

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

4. Final simplification 16.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{B \cdot F} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\sqrt{\frac{B}{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 3.3 \cdot 10^{+192}:\\ \;\;\;\;-\sqrt{2 \cdot \left|\frac{F}{B\_m}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{-B\_m} \cdot \sqrt{C \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 3.3e+192)
   (- (sqrt (* 2.0 (fabs (/ F B_m)))))
   (* (/ 2.0 (- B_m)) (sqrt (* C 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 (C <= 3.3e+192) {
		tmp = -sqrt((2.0 * fabs((F / B_m))));
	} else {
		tmp = (2.0 / -B_m) * sqrt((C * F));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 3.3d+192) then
        tmp = -sqrt((2.0d0 * abs((f / b_m))))
    else
        tmp = (2.0d0 / -b_m) * sqrt((c * f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 3.3e+192) {
		tmp = -Math.sqrt((2.0 * Math.abs((F / B_m))));
	} else {
		tmp = (2.0 / -B_m) * Math.sqrt((C * F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 3.3e+192:
		tmp = -math.sqrt((2.0 * math.fabs((F / B_m))))
	else:
		tmp = (2.0 / -B_m) * math.sqrt((C * 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 (C <= 3.3e+192)
		tmp = Float64(-sqrt(Float64(2.0 * abs(Float64(F / B_m)))));
	else
		tmp = Float64(Float64(2.0 / Float64(-B_m)) * sqrt(Float64(C * F)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 3.3e+192)
		tmp = -sqrt((2.0 * abs((F / B_m))));
	else
		tmp = (2.0 / -B_m) * sqrt((C * F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 3.3e+192], (-N[Sqrt[N[(2.0 * N[Abs[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(2.0 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 3.3000000000000001e192

   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 B around inf 15.0%

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

      1. mul-1-neg 15.0%

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

      2. *-commutative 15.0%

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

   5. Simplified 15.0%

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

      1. *-commutative 15.0%

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

      2. pow1/2 15.2%

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

      3. pow1/2 15.2%

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

      4. pow-prod-down 15.3%

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

   7. Applied egg-rr 15.3%

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

      1. unpow1/2 15.1%

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

   9. Simplified 15.1%

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

       1. add-sqr-sqrt 15.1%

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

       2. sqrt-unprod 18.7%

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

       3. pow2 18.7%

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

   11. Applied egg-rr 18.7%

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

       1. unpow2 18.7%

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

       2. rem-sqrt-square 31.4%

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

   13. Simplified 31.4%

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

   if 3.3000000000000001e192 < C

   1. Initial program 2.4%

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

   3. Taylor expanded in A around 0 0.9%

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

      1. mul-1-neg 0.9%

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

      2. *-commutative 0.9%

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

      3. *-commutative 0.9%

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

   5. Simplified 0.9%

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

      1. sqrt-prod 0.9%

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

      2. +-commutative 0.9%

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

      3. unpow2 0.9%

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

      4. unpow2 0.9%

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

      5. hypot-define 16.4%

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

   7. Applied egg-rr 16.4%

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

   8. Taylor expanded in C around inf 6.3%

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

      1. associate-*r* 6.3%

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

      2. neg-mul-1 6.3%

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

      3. unpow2 6.3%

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

      4. rem-square-sqrt 6.3%

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

   10. Simplified 6.3%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3.3 \cdot 10^{+192}:\\ \;\;\;\;-\sqrt{2 \cdot \left|\frac{F}{B}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{-B} \cdot \sqrt{C \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.6 \cdot 10^{+189}:\\ \;\;\;\;-{\left(2 \cdot \frac{F}{B\_m}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{-B\_m} \cdot \sqrt{C \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 1.6e+189)
   (- (pow (* 2.0 (/ F B_m)) 0.5))
   (* (/ 2.0 (- B_m)) (sqrt (* C 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 (C <= 1.6e+189) {
		tmp = -pow((2.0 * (F / B_m)), 0.5);
	} else {
		tmp = (2.0 / -B_m) * sqrt((C * F));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 1.6d+189) then
        tmp = -((2.0d0 * (f / b_m)) ** 0.5d0)
    else
        tmp = (2.0d0 / -b_m) * sqrt((c * f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 1.6e+189) {
		tmp = -Math.pow((2.0 * (F / B_m)), 0.5);
	} else {
		tmp = (2.0 / -B_m) * Math.sqrt((C * F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 1.6e+189:
		tmp = -math.pow((2.0 * (F / B_m)), 0.5)
	else:
		tmp = (2.0 / -B_m) * math.sqrt((C * 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 (C <= 1.6e+189)
		tmp = Float64(-(Float64(2.0 * Float64(F / B_m)) ^ 0.5));
	else
		tmp = Float64(Float64(2.0 / Float64(-B_m)) * sqrt(Float64(C * F)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 1.6e+189)
		tmp = -((2.0 * (F / B_m)) ^ 0.5);
	else
		tmp = (2.0 / -B_m) * sqrt((C * F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 1.6e+189], (-N[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), N[(N[(2.0 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.6e189

   1. Initial program 20.2%

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

   3. Taylor expanded in B around inf 15.1%

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

      1. mul-1-neg 15.1%

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

      2. *-commutative 15.1%

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

   5. Simplified 15.1%

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

      1. *-commutative 15.1%

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

      2. pow1/2 15.3%

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

      3. pow1/2 15.3%

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

      4. pow-prod-down 15.3%

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

   7. Applied egg-rr 15.3%

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

   if 1.6e189 < C

   1. Initial program 2.5%

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

   3. Taylor expanded in A around 0 1.0%

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

      1. mul-1-neg 1.0%

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

      2. *-commutative 1.0%

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

      3. *-commutative 1.0%

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

   5. Simplified 1.0%

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

      1. sqrt-prod 1.0%

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

      2. +-commutative 1.0%

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

      3. unpow2 1.0%

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

      4. unpow2 1.0%

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

      5. hypot-define 15.9%

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

   7. Applied egg-rr 15.9%

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

   8. Taylor expanded in C around inf 6.2%

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

      1. associate-*r* 6.2%

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

      2. neg-mul-1 6.2%

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

      3. unpow2 6.2%

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

      4. rem-square-sqrt 6.2%

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

   10. Simplified 6.2%

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

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.6 \cdot 10^{+189}:\\ \;\;\;\;-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{-B} \cdot \sqrt{C \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.7% 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 18.3%

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

3. Taylor expanded in B around inf 14.1%

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

   1. mul-1-neg 14.1%

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

   2. *-commutative 14.1%

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

5. Simplified 14.1%

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

   1. *-commutative 14.1%

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

   2. pow1/2 14.3%

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

   3. pow1/2 14.3%

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

   4. pow-prod-down 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. Final simplification 14.3%

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

Alternative 17: 27.7% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

3. Taylor expanded in B around inf 14.1%

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

   1. mul-1-neg 14.1%

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

   2. *-commutative 14.1%

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

5. Simplified 14.1%

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

   1. *-commutative 14.1%

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

   2. pow1/2 14.3%

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

   3. pow1/2 14.3%

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

   4. pow-prod-down 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. unpow1/2 14.1%

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

9. Simplified 14.1%

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

10. Final simplification 14.1%

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

Alternative 18: 27.7% accurate, 6.0× speedup

Math

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

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 (* F (/ 2.0 B_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

3. Taylor expanded in B around inf 14.1%

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

   1. mul-1-neg 14.1%

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

   2. *-commutative 14.1%

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

5. Simplified 14.1%

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

   1. *-commutative 14.1%

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

   2. pow1/2 14.3%

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

   3. pow1/2 14.3%

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

   4. pow-prod-down 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. unpow1/2 14.1%

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

9. Simplified 14.1%

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

    1. associate-*l/ 14.1%

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

11. Applied egg-rr 14.1%

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

    1. associate-/l* 14.1%

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

13. Simplified 14.1%

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

Alternative 19: 2.4% 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 sqrt(Float64(2.0 * Float64(F / B_m)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

3. Taylor expanded in B around inf 14.1%

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

   1. mul-1-neg 14.1%

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

   2. *-commutative 14.1%

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

5. Simplified 14.1%

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

   1. *-commutative 14.1%

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

   2. pow1/2 14.3%

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

   3. pow1/2 14.3%

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

   4. pow-prod-down 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. unpow1/2 14.1%

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

9. Simplified 14.1%

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

    1. add-sqr-sqrt 0.5%

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

    2. sqrt-unprod 1.6%

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

    3. sqr-neg 1.6%

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

    4. add-sqr-sqrt 1.6%

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

    5. *-un-lft-identity 1.6%

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

    6. *-commutative 1.6%

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

11. Applied egg-rr 1.6%

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

    1. *-lft-identity 1.6%

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

13. Simplified 1.6%

    \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024186 
(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))))