ABCF->ab-angle a

Percentage Accurate: 18.6% → 56.2%

Time: 31.9s

Alternatives: 13

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.6% 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: 56.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{t\_1}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C))) (t_1 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 5e-242)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 5e+130)
       (*
        (sqrt
         (*
          F
          (/
           (+ A (+ C (hypot (- A C) B_m)))
           (fma -4.0 (* A C) (pow B_m 2.0)))))
        t_1)
       (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ t_1 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double t_1 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-242) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+130) {
		tmp = sqrt((F * ((A + (C + hypot((A - C), B_m))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * t_1;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (t_1 / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	t_1 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-242)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+130)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(Float64(A - C), B_m))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(t_1 / 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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-242], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+130], N[(N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e-242

   1. Initial program 20.4%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in A around -inf 27.7%

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

      1. *-commutative 27.7%

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

   7. Simplified 27.7%

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

   if 4.9999999999999998e-242 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999996e130

   1. Initial program 37.5%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in F around 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. distribute-rgt-neg-in 45.6%

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

   7. Simplified 58.0%

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

   if 4.9999999999999996e130 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in A around 0 6.1%

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

      1. mul-1-neg 6.1%

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

      2. *-commutative 6.1%

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

      3. distribute-rgt-neg-in 6.1%

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

      4. +-commutative 6.1%

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

      5. unpow2 6.1%

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

      6. unpow2 6.1%

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

      7. hypot-define 23.4%

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

   7. Simplified 23.4%

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

      1. pow1/2 23.4%

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

      2. *-commutative 23.4%

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

      3. hypot-undefine 6.1%

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

      4. unpow2 6.1%

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

      5. unpow2 6.1%

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

      6. +-commutative 6.1%

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

      7. unpow-prod-down 8.2%

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

      8. pow1/2 8.2%

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

      9. +-commutative 8.2%

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

      10. unpow2 8.2%

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

      11. unpow2 8.2%

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

      12. hypot-undefine 34.8%

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

      13. pow1/2 34.8%

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

   9. Applied egg-rr 34.8%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+17}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_1\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C))) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e-127)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e+17)
       (/ (sqrt (* (* F t_1) (* 2.0 (+ A (+ C (hypot B_m (- A C))))))) (- t_1))
       (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (- (sqrt 2.0)) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-127) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e+17) {
		tmp = sqrt(((F * t_1) * (2.0 * (A + (C + hypot(B_m, (A - C))))))) / -t_1;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-127)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e+17)
		tmp = Float64(sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))))) / Float64(-t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], 5e-127], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+17], N[(N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999997e-127

   1. Initial program 19.8%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in A around -inf 26.8%

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

      1. *-commutative 26.8%

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

   7. Simplified 26.8%

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

   if 4.9999999999999997e-127 < (pow.f64 B #s(literal 2 binary64)) < 1e17

   1. Initial program 55.3%

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

   3. Simplified 69.2%

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

   if 1e17 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.9%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in A around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. *-commutative 8.7%

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

      3. distribute-rgt-neg-in 8.7%

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

      4. +-commutative 8.7%

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

      5. unpow2 8.7%

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

      6. unpow2 8.7%

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

      7. hypot-define 23.8%

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

   7. Simplified 23.8%

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

      1. pow1/2 23.8%

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

      2. *-commutative 23.8%

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

      3. hypot-undefine 8.7%

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

      4. unpow2 8.7%

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

      5. unpow2 8.7%

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

      6. +-commutative 8.7%

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

      7. unpow-prod-down 10.4%

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

      8. pow1/2 10.4%

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

      9. +-commutative 10.4%

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

      10. unpow2 10.4%

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

      11. unpow2 10.4%

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

      12. hypot-undefine 33.2%

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

      13. pow1/2 33.2%

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

   9. Applied egg-rr 33.2%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+17}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_3 := C + \mathsf{hypot}\left(C, B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot t\_3}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t\_3} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C)))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_3 (+ C (hypot C B_m))))
   (if (<= (pow B_m 2.0) 5e-127)
     (/ (sqrt (* t_2 (* 2.0 C))) t_1)
     (if (<= (pow B_m 2.0) 5e+20)
       (/ (sqrt (* t_2 t_3)) t_1)
       (* (* (sqrt t_3) (sqrt F)) (/ (- (sqrt 2.0)) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_3 = C + hypot(C, B_m);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-127) {
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e+20) {
		tmp = sqrt((t_2 * t_3)) / t_1;
	} else {
		tmp = (sqrt(t_3) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double t_2 = 2.0 * ((Math.pow(B_m, 2.0) - t_0) * F);
	double t_3 = C + Math.hypot(C, B_m);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-127) {
		tmp = Math.sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (Math.pow(B_m, 2.0) <= 5e+20) {
		tmp = Math.sqrt((t_2 * t_3)) / t_1;
	} else {
		tmp = (Math.sqrt(t_3) * Math.sqrt(F)) * (-Math.sqrt(2.0) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = 4.0 * (A * C)
	t_1 = t_0 - math.pow(B_m, 2.0)
	t_2 = 2.0 * ((math.pow(B_m, 2.0) - t_0) * F)
	t_3 = C + math.hypot(C, B_m)
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-127:
		tmp = math.sqrt((t_2 * (2.0 * C))) / t_1
	elif math.pow(B_m, 2.0) <= 5e+20:
		tmp = math.sqrt((t_2 * t_3)) / t_1
	else:
		tmp = (math.sqrt(t_3) * math.sqrt(F)) * (-math.sqrt(2.0) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_3 = Float64(C + hypot(C, B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-127)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * C))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e+20)
		tmp = Float64(sqrt(Float64(t_2 * t_3)) / t_1);
	else
		tmp = Float64(Float64(sqrt(t_3) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = 4.0 * (A * C);
	t_1 = t_0 - (B_m ^ 2.0);
	t_2 = 2.0 * (((B_m ^ 2.0) - t_0) * F);
	t_3 = C + hypot(C, B_m);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-127)
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	elseif ((B_m ^ 2.0) <= 5e+20)
		tmp = sqrt((t_2 * t_3)) / t_1;
	else
		tmp = (sqrt(t_3) * sqrt(F)) * (-sqrt(2.0) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-127], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+20], N[(N[Sqrt[N[(t$95$2 * t$95$3), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[t$95$3], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999997e-127

   1. Initial program 19.8%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in A around -inf 26.8%

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

      1. *-commutative 26.8%

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

   7. Simplified 26.8%

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

   if 4.9999999999999997e-127 < (pow.f64 B #s(literal 2 binary64)) < 5e20

   1. Initial program 53.4%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in A around 0 41.3%

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

      1. +-commutative 41.3%

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

      2. unpow2 41.3%

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

      3. unpow2 41.3%

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

      4. hypot-define 44.8%

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

   7. Simplified 44.8%

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

   if 5e20 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.0%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in A around 0 8.8%

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

      1. mul-1-neg 8.8%

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

      2. *-commutative 8.8%

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

      3. distribute-rgt-neg-in 8.8%

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

      4. +-commutative 8.8%

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

      5. unpow2 8.8%

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

      6. unpow2 8.8%

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

      7. hypot-define 24.0%

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

   7. Simplified 24.0%

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

      1. pow1/2 24.0%

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

      2. *-commutative 24.0%

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

      3. hypot-undefine 8.8%

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

      4. unpow2 8.8%

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

      5. unpow2 8.8%

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

      6. +-commutative 8.8%

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

      7. unpow-prod-down 10.5%

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

      8. pow1/2 10.5%

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

      9. +-commutative 10.5%

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

      10. unpow2 10.5%

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

      11. unpow2 10.5%

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

      12. hypot-undefine 33.5%

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

      13. pow1/2 33.5%

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

   9. Applied egg-rr 33.5%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-255}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot {C}^{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{B\_m \cdot {\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-255)
   (/
    (sqrt (* -16.0 (* A (* F (pow C 2.0)))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (if (<= (pow B_m 2.0) 2e+252)
     (/
      (* B_m (pow (* 2.0 (* F (+ C (hypot C B_m)))) 0.5))
      (- (fma B_m B_m (* (* A C) -4.0))))
     (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 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 tmp;
	if (pow(B_m, 2.0) <= 1e-255) {
		tmp = sqrt((-16.0 * (A * (F * pow(C, 2.0))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+252) {
		tmp = (B_m * pow((2.0 * (F * (C + hypot(C, B_m)))), 0.5)) / -fma(B_m, B_m, ((A * C) * -4.0));
	} else {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / 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)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-255)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(F * (C ^ 2.0))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+252)
		tmp = Float64(Float64(B_m * (Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m)))) ^ 0.5)) / Float64(-fma(B_m, B_m, Float64(Float64(A * C) * -4.0))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / 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], 1e-255], N[(N[Sqrt[N[(-16.0 * N[(A * N[(F * N[Power[C, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+252], N[(N[(B$95$m * N[Power[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e-255

   1. Initial program 20.1%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in A around -inf 26.4%

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

   6. Taylor expanded in B around 0 25.5%

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

      1. *-commutative 25.5%

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

      2. *-commutative 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in C around inf 18.3%

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

   if 1e-255 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000002e252

   1. Initial program 32.9%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in A around 0 20.1%

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

      1. distribute-frac-neg 20.1%

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

   7. Applied egg-rr 22.6%

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

   if 2.0000000000000002e252 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in B around inf 24.6%

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

      1. mul-1-neg 24.6%

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

      2. distribute-rgt-neg-in 24.6%

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

   7. Simplified 24.6%

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

      1. pow1 24.6%

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

      2. distribute-rgt-neg-out 24.6%

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

      3. pow1/2 24.6%

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

      4. pow1/2 24.6%

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

      5. pow-prod-down 24.6%

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

   9. Applied egg-rr 24.6%

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

       1. unpow1 24.6%

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

       2. unpow1/2 24.6%

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

       3. associate-*l/ 24.6%

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

   11. Simplified 24.6%

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

       1. pow1/2 24.6%

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

       2. div-inv 24.7%

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

       3. unpow-prod-down 34.0%

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

       4. pow1/2 34.0%

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

       5. *-commutative 34.0%

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

   13. Applied egg-rr 34.0%

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

       1. unpow1/2 34.0%

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

   15. Simplified 34.0%

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

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-255}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot {C}^{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{B \cdot {\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + 4 \cdot \left({B\_m}^{2} \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{B\_m \cdot {\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-171)
   (/
    (sqrt (* C (+ (* -16.0 (* A (* C F))) (* 4.0 (* (pow B_m 2.0) F)))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (if (<= (pow B_m 2.0) 2e+252)
     (/
      (* B_m (pow (* 2.0 (* F (+ C (hypot C B_m)))) 0.5))
      (- (fma B_m B_m (* (* A C) -4.0))))
     (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 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 tmp;
	if (pow(B_m, 2.0) <= 2e-171) {
		tmp = sqrt((C * ((-16.0 * (A * (C * F))) + (4.0 * (pow(B_m, 2.0) * F))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+252) {
		tmp = (B_m * pow((2.0 * (F * (C + hypot(C, B_m)))), 0.5)) / -fma(B_m, B_m, ((A * C) * -4.0));
	} else {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / 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)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-171)
		tmp = Float64(sqrt(Float64(C * Float64(Float64(-16.0 * Float64(A * Float64(C * F))) + Float64(4.0 * Float64((B_m ^ 2.0) * F))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+252)
		tmp = Float64(Float64(B_m * (Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m)))) ^ 0.5)) / Float64(-fma(B_m, B_m, Float64(Float64(A * C) * -4.0))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / 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-171], N[(N[Sqrt[N[(C * N[(N[(-16.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+252], N[(N[(B$95$m * N[Power[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e-171

   1. Initial program 19.9%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in A around -inf 26.1%

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

   6. Taylor expanded in B around 0 26.3%

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

      1. *-commutative 26.3%

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

      2. *-commutative 26.3%

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

   8. Simplified 26.3%

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

   9. Taylor expanded in C around 0 26.3%

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

   if 2e-171 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000002e252

   1. Initial program 35.9%

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

   3. Simplified 36.1%

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

   5. Taylor expanded in A around 0 19.6%

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

      1. distribute-frac-neg 19.6%

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

   7. Applied egg-rr 22.7%

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

   if 2.0000000000000002e252 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in B around inf 24.6%

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

      1. mul-1-neg 24.6%

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

      2. distribute-rgt-neg-in 24.6%

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

   7. Simplified 24.6%

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

      1. pow1 24.6%

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

      2. distribute-rgt-neg-out 24.6%

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

      3. pow1/2 24.6%

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

      4. pow1/2 24.6%

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

      5. pow-prod-down 24.6%

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

   9. Applied egg-rr 24.6%

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

       1. unpow1 24.6%

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

       2. unpow1/2 24.6%

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

       3. associate-*l/ 24.6%

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

   11. Simplified 24.6%

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

       1. pow1/2 24.6%

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

       2. div-inv 24.7%

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

       3. unpow-prod-down 34.0%

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

       4. pow1/2 34.0%

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

       5. *-commutative 34.0%

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

   13. Applied egg-rr 34.0%

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

       1. unpow1/2 34.0%

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

   15. Simplified 34.0%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + 4 \cdot \left({B}^{2} \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{B \cdot {\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{B\_m \cdot {\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C))))
   (if (<= (pow B_m 2.0) 5e-127)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 2e+252)
       (/
        (* B_m (pow (* 2.0 (* F (+ C (hypot C B_m)))) 0.5))
        (- (fma B_m B_m (* (* A C) -4.0))))
       (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-127) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+252) {
		tmp = (B_m * pow((2.0 * (F * (C + hypot(C, B_m)))), 0.5)) / -fma(B_m, B_m, ((A * C) * -4.0));
	} else {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-127)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+252)
		tmp = Float64(Float64(B_m * (Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m)))) ^ 0.5)) / Float64(-fma(B_m, B_m, Float64(Float64(A * C) * -4.0))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / 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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-127], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+252], N[(N[(B$95$m * N[Power[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999997e-127

   1. Initial program 19.8%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in A around -inf 26.8%

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

      1. *-commutative 26.8%

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

   7. Simplified 26.8%

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

   if 4.9999999999999997e-127 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000002e252

   1. Initial program 37.3%

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

   3. Simplified 37.2%

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

   5. Taylor expanded in A around 0 19.7%

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

      1. distribute-frac-neg 19.7%

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

   7. Applied egg-rr 22.9%

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

   if 2.0000000000000002e252 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in B around inf 24.6%

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

      1. mul-1-neg 24.6%

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

      2. distribute-rgt-neg-in 24.6%

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

   7. Simplified 24.6%

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

      1. pow1 24.6%

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

      2. distribute-rgt-neg-out 24.6%

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

      3. pow1/2 24.6%

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

      4. pow1/2 24.6%

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

      5. pow-prod-down 24.6%

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

   9. Applied egg-rr 24.6%

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

       1. unpow1 24.6%

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

       2. unpow1/2 24.6%

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

       3. associate-*l/ 24.6%

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

   11. Simplified 24.6%

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

       1. pow1/2 24.6%

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

       2. div-inv 24.7%

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

       3. unpow-prod-down 34.0%

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

       4. pow1/2 34.0%

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

       5. *-commutative 34.0%

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

   13. Applied egg-rr 34.0%

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

       1. unpow1/2 34.0%

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

   15. Simplified 34.0%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{B \cdot {\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-255}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot {C}^{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-255)
   (/
    (sqrt (* -16.0 (* A (* F (pow C 2.0)))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (if (<= (pow B_m 2.0) 2e+252)
     (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (+ C (hypot C B_m)))))
     (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 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 tmp;
	if (pow(B_m, 2.0) <= 1e-255) {
		tmp = sqrt((-16.0 * (A * (F * pow(C, 2.0))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+252) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C + hypot(C, B_m))));
	} else {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-255) {
		tmp = Math.sqrt((-16.0 * (A * (F * Math.pow(C, 2.0))))) / ((4.0 * (A * C)) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 2e+252) {
		tmp = (-Math.sqrt(2.0) / B_m) * Math.sqrt((F * (C + Math.hypot(C, B_m))));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.sqrt((1.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-255:
		tmp = math.sqrt((-16.0 * (A * (F * math.pow(C, 2.0))))) / ((4.0 * (A * C)) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 2e+252:
		tmp = (-math.sqrt(2.0) / B_m) * math.sqrt((F * (C + math.hypot(C, B_m))))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.sqrt((1.0 / 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)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-255)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(F * (C ^ 2.0))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+252)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(C + hypot(C, B_m)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-255)
		tmp = sqrt((-16.0 * (A * (F * (C ^ 2.0))))) / ((4.0 * (A * C)) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 2e+252)
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C + hypot(C, B_m))));
	else
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-255], N[(N[Sqrt[N[(-16.0 * N[(A * N[(F * N[Power[C, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+252], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e-255

   1. Initial program 20.1%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in A around -inf 26.4%

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

   6. Taylor expanded in B around 0 25.5%

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

      1. *-commutative 25.5%

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

      2. *-commutative 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in C around inf 18.3%

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

   if 1e-255 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000002e252

   1. Initial program 32.9%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in A around 0 20.1%

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

      1. mul-1-neg 20.1%

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

      2. *-commutative 20.1%

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

      3. distribute-rgt-neg-in 20.1%

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

      4. +-commutative 20.1%

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

      5. unpow2 20.1%

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

      6. unpow2 20.1%

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

      7. hypot-define 22.5%

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

   7. Simplified 22.5%

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

   if 2.0000000000000002e252 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in B around inf 24.6%

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

      1. mul-1-neg 24.6%

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

      2. distribute-rgt-neg-in 24.6%

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

   7. Simplified 24.6%

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

      1. pow1 24.6%

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

      2. distribute-rgt-neg-out 24.6%

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

      3. pow1/2 24.6%

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

      4. pow1/2 24.6%

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

      5. pow-prod-down 24.6%

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

   9. Applied egg-rr 24.6%

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

       1. unpow1 24.6%

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

       2. unpow1/2 24.6%

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

       3. associate-*l/ 24.6%

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

   11. Simplified 24.6%

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

       1. pow1/2 24.6%

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

       2. div-inv 24.7%

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

       3. unpow-prod-down 34.0%

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

       4. pow1/2 34.0%

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

       5. *-commutative 34.0%

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

   13. Applied egg-rr 34.0%

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

       1. unpow1/2 34.0%

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

   15. Simplified 34.0%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-255}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot {C}^{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C))))
   (if (<= (pow B_m 2.0) 2e-240)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (- (sqrt 2.0)) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-240) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-240) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt((C + Math.hypot(C, B_m))) * Math.sqrt(F)) * (-Math.sqrt(2.0) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = 4.0 * (A * C)
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-240:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt((C + math.hypot(C, B_m))) * math.sqrt(F)) * (-math.sqrt(2.0) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-240)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = 4.0 * (A * C);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-240)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-240], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-240

   1. Initial program 20.2%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in A around -inf 27.4%

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

      1. *-commutative 27.4%

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

   7. Simplified 27.4%

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

   if 1.9999999999999999e-240 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 19.1%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in A around 0 13.2%

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

      1. mul-1-neg 13.2%

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

      2. *-commutative 13.2%

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

      3. distribute-rgt-neg-in 13.2%

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

      4. +-commutative 13.2%

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

      5. unpow2 13.2%

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

      6. unpow2 13.2%

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

      7. hypot-define 24.1%

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

   7. Simplified 24.1%

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

      1. pow1/2 24.2%

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

      2. *-commutative 24.2%

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

      3. hypot-undefine 13.2%

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

      4. unpow2 13.2%

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

      5. unpow2 13.2%

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

      6. +-commutative 13.2%

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

      7. unpow-prod-down 14.9%

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

      8. pow1/2 14.9%

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

      9. +-commutative 14.9%

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

      10. unpow2 14.9%

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

      11. unpow2 14.9%

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

      12. hypot-undefine 31.4%

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

      13. pow1/2 31.4%

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

   9. Applied egg-rr 31.4%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.05 \cdot 10^{+128}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.05e+128)
   (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (+ C (hypot C B_m)))))
   (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 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 tmp;
	if (B_m <= 1.05e+128) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C + hypot(C, B_m))));
	} else {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.05e+128) {
		tmp = (-Math.sqrt(2.0) / B_m) * Math.sqrt((F * (C + Math.hypot(C, B_m))));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.sqrt((1.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.05e+128:
		tmp = (-math.sqrt(2.0) / B_m) * math.sqrt((F * (C + math.hypot(C, B_m))))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.sqrt((1.0 / 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)
	tmp = 0.0
	if (B_m <= 1.05e+128)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(C + hypot(C, B_m)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.05e+128)
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C + hypot(C, B_m))));
	else
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.05e+128], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.05e128

   1. Initial program 22.5%

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

   3. Simplified 22.9%

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

   5. Taylor expanded in A around 0 10.8%

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

      1. mul-1-neg 10.8%

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

      2. *-commutative 10.8%

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

      3. distribute-rgt-neg-in 10.8%

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

      4. +-commutative 10.8%

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

      5. unpow2 10.8%

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

      6. unpow2 10.8%

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

      7. hypot-define 12.4%

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

   7. Simplified 12.4%

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

   if 1.05e128 < B

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in B around inf 49.7%

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

      1. mul-1-neg 49.7%

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

      2. distribute-rgt-neg-in 49.7%

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

   7. Simplified 49.7%

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

      1. pow1 49.7%

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

      2. distribute-rgt-neg-out 49.7%

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

      3. pow1/2 49.7%

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

      4. pow1/2 49.7%

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

      5. pow-prod-down 49.6%

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

   9. Applied egg-rr 49.6%

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

       1. unpow1 49.6%

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

       2. unpow1/2 49.6%

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

       3. associate-*l/ 49.8%

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

   11. Simplified 49.8%

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

       1. pow1/2 49.8%

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

       2. div-inv 49.8%

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

       3. unpow-prod-down 71.0%

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

       4. pow1/2 71.0%

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

       5. *-commutative 71.0%

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

   13. Applied egg-rr 71.0%

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

       1. unpow1/2 71.0%

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

   15. Simplified 71.0%

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

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.05 \cdot 10^{+128}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 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((2.0 * F)) * -sqrt((1.0 / 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((2.0d0 * f)) * -sqrt((1.0d0 / 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((2.0 * F)) * -Math.sqrt((1.0 / 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((2.0 * F)) * -math.sqrt((1.0 / 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(2.0 * F)) * Float64(-sqrt(Float64(1.0 / 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((2.0 * F)) * -sqrt((1.0 / B_m));
end

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Simplified 19.8%

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

5. Taylor expanded in B around inf 15.1%

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

   1. mul-1-neg 15.1%

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

   2. distribute-rgt-neg-in 15.1%

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

7. Simplified 15.1%

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

   1. pow1 15.1%

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

   2. distribute-rgt-neg-out 15.1%

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

   3. pow1/2 15.3%

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

   4. pow1/2 15.3%

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

   5. pow-prod-down 15.3%

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

9. Applied egg-rr 15.3%

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

    1. unpow1 15.3%

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

    2. unpow1/2 15.1%

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

    3. associate-*l/ 15.1%

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

11. Simplified 15.1%

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

    1. pow1/2 15.3%

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

    2. div-inv 15.3%

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

    3. unpow-prod-down 18.1%

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

    4. pow1/2 18.1%

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

    5. *-commutative 18.1%

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

13. Applied egg-rr 18.1%

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

    1. unpow1/2 18.1%

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

15. Simplified 18.1%

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

16. Final simplification 18.1%

    \[\leadsto \sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right) \]
17. Add Preprocessing

Alternative 11: 35.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Simplified 19.8%

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

5. Taylor expanded in B around inf 15.1%

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

   1. mul-1-neg 15.1%

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

   2. distribute-rgt-neg-in 15.1%

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

7. Simplified 15.1%

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

   1. pow1 15.1%

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

   2. distribute-rgt-neg-out 15.1%

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

   3. pow1/2 15.3%

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

   4. pow1/2 15.3%

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

   5. pow-prod-down 15.3%

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

9. Applied egg-rr 15.3%

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

    1. unpow1 15.3%

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

    2. unpow1/2 15.1%

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

    3. associate-*l/ 15.1%

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

11. Simplified 15.1%

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

    1. pow1/2 15.3%

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

    2. associate-/l* 15.3%

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

    3. metadata-eval 15.3%

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

    4. metadata-eval 15.3%

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

    5. sqrt-pow2 15.2%

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

    6. unpow-prod-down 18.0%

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

    7. pow1/2 18.0%

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

    8. pow1/2 18.0%

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

    9. sqrt-pow2 18.1%

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

    10. metadata-eval 18.1%

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

    11. metadata-eval 18.1%

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

13. Applied egg-rr 18.1%

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

14. Final simplification 18.1%

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

Alternative 12: 27.0% accurate, 5.9× speedup

Math

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

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(Float64(2.0 * 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[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision])

Derivation

1. Initial program 19.5%

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

3. Simplified 19.8%

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

5. Taylor expanded in B around inf 15.1%

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

   1. mul-1-neg 15.1%

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

   2. distribute-rgt-neg-in 15.1%

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

7. Simplified 15.1%

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

   1. pow1 15.1%

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

   2. distribute-rgt-neg-out 15.1%

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

   3. pow1/2 15.3%

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

   4. pow1/2 15.3%

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

   5. pow-prod-down 15.3%

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

9. Applied egg-rr 15.3%

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

    1. unpow1 15.3%

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

    2. unpow1/2 15.1%

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

    3. associate-*l/ 15.1%

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

11. Simplified 15.1%

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

    1. pow1/2 15.3%

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

    2. *-commutative 15.3%

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

13. Applied egg-rr 15.3%

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

14. Final simplification 15.3%

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

Alternative 13: 27.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Simplified 19.8%

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

5. Taylor expanded in B around inf 15.1%

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

   1. mul-1-neg 15.1%

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

   2. distribute-rgt-neg-in 15.1%

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

7. Simplified 15.1%

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

   1. pow1 15.1%

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

   2. distribute-rgt-neg-out 15.1%

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

   3. pow1/2 15.3%

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

   4. pow1/2 15.3%

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

   5. pow-prod-down 15.3%

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

9. Applied egg-rr 15.3%

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

    1. unpow1 15.3%

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

    2. unpow1/2 15.1%

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

    3. associate-*l/ 15.1%

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

11. Simplified 15.1%

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

12. Final simplification 15.1%

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

Reproduce

herbie shell --seed 2024060 
(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))))