ABCF->ab-angle b

Percentage Accurate: 20.7% → 48.7%

Time: 20.0s

Alternatives: 10

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: 20.7% 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: 48.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+292}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(A, C \cdot -4, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e-40)
     (/
      (sqrt (* (* t_0 F) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
      (- t_0))
     (if (<= (pow B_m 2.0) 1e+292)
       (-
        (sqrt
         (*
          2.0
          (*
           F
           (/
            (- (+ A C) (hypot B_m (- A C)))
            (fma A (* C -4.0) (pow B_m 2.0)))))))
       (- (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) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-40) {
		tmp = sqrt(((t_0 * F) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e+292) {
		tmp = -sqrt((2.0 * (F * (((A + C) - hypot(B_m, (A - C))) / fma(A, (C * -4.0), pow(B_m, 2.0))))));
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-40)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e+292)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) - hypot(B_m, Float64(A - C))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)))))));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-40], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+292], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999965e-40

   1. Initial program 29.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in C around inf 36.8%

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

      1. mul-1-neg 36.8%

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

   7. Simplified 36.8%

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

   if 4.99999999999999965e-40 < (pow.f64 B #s(literal 2 binary64)) < 1e292

   1. Initial program 27.5%

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

   3. Taylor expanded in F around 0 33.4%

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

   5. Simplified 46.4%

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

      1. sqrt-unprod 46.5%

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

      2. associate-/l* 59.6%

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

      3. associate-+r- 60.5%

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

      4. fma-undefine 60.5%

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

      5. *-commutative 60.5%

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

      6. associate-*r* 60.5%

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

      7. fma-define 60.5%

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

   7. Applied egg-rr 60.5%

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

   if 1e292 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.8%

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

   3. Taylor expanded in F around 0 5.1%

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

   5. Simplified 9.2%

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

      1. sqrt-unprod 9.2%

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

      2. associate-/l* 10.4%

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

      3. associate-+r- 10.4%

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

      4. fma-undefine 10.4%

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

      5. *-commutative 10.4%

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

      6. associate-*r* 10.4%

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

      7. fma-define 10.6%

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

   7. Applied egg-rr 10.6%

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

   8. Taylor expanded in B around inf 31.0%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+292}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(A \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+292}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(A, C \cdot -4, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e-40)
     (/ (sqrt (* (* t_0 F) (* A 4.0))) (- t_0))
     (if (<= (pow B_m 2.0) 1e+292)
       (-
        (sqrt
         (*
          2.0
          (*
           F
           (/
            (- (+ A C) (hypot B_m (- A C)))
            (fma A (* C -4.0) (pow B_m 2.0)))))))
       (- (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) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-40) {
		tmp = sqrt(((t_0 * F) * (A * 4.0))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e+292) {
		tmp = -sqrt((2.0 * (F * (((A + C) - hypot(B_m, (A - C))) / fma(A, (C * -4.0), pow(B_m, 2.0))))));
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-40)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(A * 4.0))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e+292)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) - hypot(B_m, Float64(A - C))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)))))));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-40], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+292], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999965e-40

   1. Initial program 29.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in A around -inf 36.5%

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

   if 4.99999999999999965e-40 < (pow.f64 B #s(literal 2 binary64)) < 1e292

   1. Initial program 27.5%

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

   3. Taylor expanded in F around 0 33.4%

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

   5. Simplified 46.4%

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

      1. sqrt-unprod 46.5%

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

      2. associate-/l* 59.6%

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

      3. associate-+r- 60.5%

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

      4. fma-undefine 60.5%

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

      5. *-commutative 60.5%

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

      6. associate-*r* 60.5%

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

      7. fma-define 60.5%

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

   7. Applied egg-rr 60.5%

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

   if 1e292 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.8%

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

   3. Taylor expanded in F around 0 5.1%

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

   5. Simplified 9.2%

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

      1. sqrt-unprod 9.2%

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

      2. associate-/l* 10.4%

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

      3. associate-+r- 10.4%

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

      4. fma-undefine 10.4%

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

      5. *-commutative 10.4%

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

      6. associate-*r* 10.4%

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

      7. fma-define 10.6%

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

   7. Applied egg-rr 10.6%

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

   8. Taylor expanded in B around inf 31.0%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+292}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(A \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{A - \mathsf{hypot}\left(A, B\_m\right)}{{B\_m}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e-7)
     (/ (sqrt (* (* t_0 F) (* A 4.0))) (- t_0))
     (if (<= (pow B_m 2.0) 2e+266)
       (- (sqrt (* 2.0 (* F (/ (- A (hypot A B_m)) (pow B_m 2.0))))))
       (- (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) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-7) {
		tmp = sqrt(((t_0 * F) * (A * 4.0))) / -t_0;
	} else if (pow(B_m, 2.0) <= 2e+266) {
		tmp = -sqrt((2.0 * (F * ((A - hypot(A, B_m)) / pow(B_m, 2.0)))));
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-7)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(A * 4.0))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 2e+266)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(A - hypot(A, B_m)) / (B_m ^ 2.0))))));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-7], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+266], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999995e-8

   1. Initial program 30.6%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in A around -inf 36.0%

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

   if 9.9999999999999995e-8 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e266

   1. Initial program 24.6%

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

   3. Taylor expanded in F around 0 31.7%

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

   5. Simplified 43.3%

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

      1. sqrt-unprod 43.5%

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

      2. associate-/l* 56.6%

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

      3. associate-+r- 57.7%

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

      4. fma-undefine 57.7%

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

      5. *-commutative 57.7%

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

      6. associate-*r* 57.7%

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

      7. fma-define 57.7%

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

   7. Applied egg-rr 57.7%

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

   8. Taylor expanded in C around 0 25.9%

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

      1. associate-/l* 30.2%

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

      2. unpow2 30.2%

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

      3. unpow2 30.2%

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

      4. hypot-define 33.6%

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

   10. Simplified 33.6%

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

   if 2.0000000000000001e266 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.4%

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

   3. Taylor expanded in F around 0 7.5%

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

   5. Simplified 11.5%

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

      1. sqrt-unprod 11.4%

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

      2. associate-/l* 14.1%

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

      3. associate-+r- 14.1%

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

      4. fma-undefine 14.1%

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

      5. *-commutative 14.1%

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

      6. associate-*r* 14.1%

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

      7. fma-define 14.3%

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

   7. Applied egg-rr 14.3%

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

   8. Taylor expanded in B around inf 32.5%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{A - \mathsf{hypot}\left(A, B\right)}{{B}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A \cdot 4\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{A - \mathsf{hypot}\left(A, B\_m\right)}{{B\_m}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-7)
   (/
    (sqrt (* (* (fma B_m B_m (* A (* C -4.0))) F) (* A 4.0)))
    (* C (* A 4.0)))
   (if (<= (pow B_m 2.0) 2e+266)
     (- (sqrt (* 2.0 (* F (/ (- A (hypot A B_m)) (pow B_m 2.0))))))
     (- (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) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-7) {
		tmp = sqrt(((fma(B_m, B_m, (A * (C * -4.0))) * F) * (A * 4.0))) / (C * (A * 4.0));
	} else if (pow(B_m, 2.0) <= 2e+266) {
		tmp = -sqrt((2.0 * (F * ((A - hypot(A, B_m)) / pow(B_m, 2.0)))));
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-7)
		tmp = Float64(sqrt(Float64(Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) * F) * Float64(A * 4.0))) / Float64(C * Float64(A * 4.0)));
	elseif ((B_m ^ 2.0) <= 2e+266)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(A - hypot(A, B_m)) / (B_m ^ 2.0))))));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / 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-7], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+266], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999995e-8

   1. Initial program 30.6%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in A around -inf 36.0%

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

   6. Taylor expanded in B around 0 35.3%

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

      1. associate-*r* 35.3%

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

   8. Simplified 35.3%

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

   if 9.9999999999999995e-8 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e266

   1. Initial program 24.6%

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

   3. Taylor expanded in F around 0 31.7%

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

   5. Simplified 43.3%

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

      1. sqrt-unprod 43.5%

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

      2. associate-/l* 56.6%

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

      3. associate-+r- 57.7%

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

      4. fma-undefine 57.7%

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

      5. *-commutative 57.7%

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

      6. associate-*r* 57.7%

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

      7. fma-define 57.7%

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

   7. Applied egg-rr 57.7%

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

   8. Taylor expanded in C around 0 25.9%

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

      1. associate-/l* 30.2%

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

      2. unpow2 30.2%

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

      3. unpow2 30.2%

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

      4. hypot-define 33.6%

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

   10. Simplified 33.6%

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

   if 2.0000000000000001e266 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.4%

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

   3. Taylor expanded in F around 0 7.5%

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

   5. Simplified 11.5%

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

      1. sqrt-unprod 11.4%

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

      2. associate-/l* 14.1%

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

      3. associate-+r- 14.1%

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

      4. fma-undefine 14.1%

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

      5. *-commutative 14.1%

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

      6. associate-*r* 14.1%

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

      7. fma-define 14.3%

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

   7. Applied egg-rr 14.3%

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

   8. Taylor expanded in B around inf 32.5%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A \cdot 4\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{A - \mathsf{hypot}\left(A, B\right)}{{B}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A \cdot 4\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-7)
   (/
    (sqrt (* (* (fma B_m B_m (* A (* C -4.0))) F) (* A 4.0)))
    (* C (* A 4.0)))
   (if (<= (pow B_m 2.0) 2e+306)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m A))))))
     (- (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) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-7) {
		tmp = sqrt(((fma(B_m, B_m, (A * (C * -4.0))) * F) * (A * 4.0))) / (C * (A * 4.0));
	} else if (pow(B_m, 2.0) <= 2e+306) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-7)
		tmp = Float64(sqrt(Float64(Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) * F) * Float64(A * 4.0))) / Float64(C * Float64(A * 4.0)));
	elseif ((B_m ^ 2.0) <= 2e+306)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / 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-7], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+306], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999995e-8

   1. Initial program 30.6%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in A around -inf 36.0%

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

   6. Taylor expanded in B around 0 35.3%

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

      1. associate-*r* 35.3%

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

   8. Simplified 35.3%

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

   if 9.9999999999999995e-8 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000003e306

   1. Initial program 24.9%

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

   3. Taylor expanded in C around 0 19.4%

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

      1. mul-1-neg 19.4%

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

      2. +-commutative 19.4%

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

      3. unpow2 19.4%

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

      4. unpow2 19.4%

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

      5. hypot-define 21.2%

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

   5. Simplified 21.2%

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

   if 2.00000000000000003e306 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0 1.8%

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

   5. Simplified 6.2%

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

      1. sqrt-unprod 6.2%

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

      2. associate-/l* 7.5%

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

      3. associate-+r- 7.5%

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

      4. fma-undefine 7.5%

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

      5. *-commutative 7.5%

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

      6. associate-*r* 7.5%

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

      7. fma-define 7.8%

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

   7. Applied egg-rr 7.8%

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

   8. Taylor expanded in B around inf 30.0%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A \cdot 4\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A \cdot 4\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+132}:\\ \;\;\;\;-\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{\mathsf{fma}\left(-2, F, 2 \cdot \left(F \cdot \frac{A + C}{B\_m}\right)\right)}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-109)
   (/
    (sqrt (* (* (fma B_m B_m (* A (* C -4.0))) F) (* A 4.0)))
    (* C (* A 4.0)))
   (if (<= (pow B_m 2.0) 5e+132)
     (- (sqrt (/ F (- C))))
     (- (sqrt (/ (fma -2.0 F (* 2.0 (* F (/ (+ A C) B_m)))) 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-109) {
		tmp = sqrt(((fma(B_m, B_m, (A * (C * -4.0))) * F) * (A * 4.0))) / (C * (A * 4.0));
	} else if (pow(B_m, 2.0) <= 5e+132) {
		tmp = -sqrt((F / -C));
	} else {
		tmp = -sqrt((fma(-2.0, F, (2.0 * (F * ((A + C) / B_m)))) / 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-109)
		tmp = Float64(sqrt(Float64(Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) * F) * Float64(A * 4.0))) / Float64(C * Float64(A * 4.0)));
	elseif ((B_m ^ 2.0) <= 5e+132)
		tmp = Float64(-sqrt(Float64(F / Float64(-C))));
	else
		tmp = Float64(-sqrt(Float64(fma(-2.0, F, Float64(2.0 * Float64(F * Float64(Float64(A + C) / B_m)))) / 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-109], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+132], (-N[Sqrt[N[(F / (-C)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(N[(-2.0 * F + N[(2.0 * N[(F * N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999999e-110

   1. Initial program 31.3%

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

   3. Simplified 42.2%

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

   5. Taylor expanded in A around -inf 37.6%

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

   6. Taylor expanded in B around 0 36.9%

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

      1. associate-*r* 36.9%

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

   8. Simplified 36.9%

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

   if 9.9999999999999999e-110 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000001e132

   1. Initial program 25.9%

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

   3. Taylor expanded in F around 0 29.2%

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

   5. Simplified 41.3%

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

      1. sqrt-unprod 41.3%

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

      2. associate-/l* 46.9%

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

      3. associate-+r- 48.3%

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

      4. fma-undefine 48.3%

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

      5. *-commutative 48.3%

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

      6. associate-*r* 48.3%

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

      7. fma-define 48.3%

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

   7. Applied egg-rr 48.3%

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

   8. Taylor expanded in A around -inf 24.3%

      \[\leadsto -\sqrt{\color{blue}{-1 \cdot \frac{F}{C}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 24.3%

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

      2. mul-1-neg 24.3%

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

   10. Simplified 24.3%

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

   if 5.0000000000000001e132 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.4%

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

   3. Taylor expanded in F around 0 15.4%

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

   5. Simplified 22.7%

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

      1. sqrt-unprod 22.7%

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

      2. associate-/l* 31.7%

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

      3. associate-+r- 31.6%

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

      4. fma-undefine 31.6%

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

      5. *-commutative 31.6%

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

      6. associate-*r* 31.6%

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

      7. fma-define 31.7%

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

   7. Applied egg-rr 31.7%

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

   8. Taylor expanded in B around inf 26.2%

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

      1. fma-define 26.2%

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

      2. associate-/l* 31.3%

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

   10. Simplified 31.3%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A \cdot 4\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+132}:\\ \;\;\;\;-\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{\mathsf{fma}\left(-2, F, 2 \cdot \left(F \cdot \frac{A + C}{B}\right)\right)}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+132}:\\ \;\;\;\;-\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{\mathsf{fma}\left(-2, F, 2 \cdot \left(F \cdot \frac{A + C}{B\_m}\right)\right)}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+132)
   (- (sqrt (/ F (- C))))
   (- (sqrt (/ (fma -2.0 F (* 2.0 (* F (/ (+ A C) B_m)))) 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) <= 5e+132) {
		tmp = -sqrt((F / -C));
	} else {
		tmp = -sqrt((fma(-2.0, F, (2.0 * (F * ((A + C) / B_m)))) / 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) <= 5e+132)
		tmp = Float64(-sqrt(Float64(F / Float64(-C))));
	else
		tmp = Float64(-sqrt(Float64(fma(-2.0, F, Float64(2.0 * Float64(F * Float64(Float64(A + C) / B_m)))) / 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], 5e+132], (-N[Sqrt[N[(F / (-C)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(N[(-2.0 * F + N[(2.0 * N[(F * N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000001e132

   1. Initial program 29.5%

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

   3. Taylor expanded in F around 0 21.0%

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

   5. Simplified 29.5%

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

      1. sqrt-unprod 29.6%

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

      2. associate-/l* 31.8%

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

      3. associate-+r- 35.4%

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

      4. fma-undefine 35.4%

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

      5. *-commutative 35.4%

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

      6. associate-*r* 35.4%

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

      7. fma-define 35.4%

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

   7. Applied egg-rr 35.4%

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

   8. Taylor expanded in A around -inf 18.9%

      \[\leadsto -\sqrt{\color{blue}{-1 \cdot \frac{F}{C}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 18.9%

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

      2. mul-1-neg 18.9%

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

   10. Simplified 18.9%

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

   if 5.0000000000000001e132 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.4%

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

   3. Taylor expanded in F around 0 15.4%

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

   5. Simplified 22.7%

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

      1. sqrt-unprod 22.7%

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

      2. associate-/l* 31.7%

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

      3. associate-+r- 31.6%

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

      4. fma-undefine 31.6%

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

      5. *-commutative 31.6%

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

      6. associate-*r* 31.6%

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

      7. fma-define 31.7%

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

   7. Applied egg-rr 31.7%

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

   8. Taylor expanded in B around inf 26.2%

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

      1. fma-define 26.2%

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

      2. associate-/l* 31.3%

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

   10. Simplified 31.3%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+132}:\\ \;\;\;\;-\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{\mathsf{fma}\left(-2, F, 2 \cdot \left(F \cdot \frac{A + C}{B}\right)\right)}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.85 \cdot 10^{+66}:\\ \;\;\;\;-\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.85e+66) (- (sqrt (/ F (- C)))) (- (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) {
	double tmp;
	if (B_m <= 1.85e+66) {
		tmp = -sqrt((F / -C));
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.85d+66) then
        tmp = -sqrt((f / -c))
    else
        tmp = -sqrt(((-2.0d0) * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.85e+66) {
		tmp = -Math.sqrt((F / -C));
	} else {
		tmp = -Math.sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.85e+66:
		tmp = -math.sqrt((F / -C))
	else:
		tmp = -math.sqrt((-2.0 * (F / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.85e+66)
		tmp = Float64(-sqrt(Float64(F / Float64(-C))));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.85e+66)
		tmp = -sqrt((F / -C));
	else
		tmp = -sqrt((-2.0 * (F / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.85e+66], (-N[Sqrt[N[(F / (-C)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 1.85e66

   1. Initial program 24.7%

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

   3. Taylor expanded in F around 0 18.9%

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

   5. Simplified 26.5%

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

      1. sqrt-unprod 26.6%

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

      2. associate-/l* 29.3%

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

      3. associate-+r- 32.0%

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

      4. fma-undefine 32.0%

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

      5. *-commutative 32.0%

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

      6. associate-*r* 32.0%

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

      7. fma-define 32.1%

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

   7. Applied egg-rr 32.1%

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

   8. Taylor expanded in A around -inf 15.7%

      \[\leadsto -\sqrt{\color{blue}{-1 \cdot \frac{F}{C}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 15.7%

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

      2. mul-1-neg 15.7%

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

   10. Simplified 15.7%

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

   if 1.85e66 < B

   1. Initial program 10.3%

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

   3. Taylor expanded in F around 0 19.4%

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

   5. Simplified 29.7%

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

      1. sqrt-unprod 29.7%

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

      2. associate-/l* 43.9%

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

      3. associate-+r- 43.9%

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

      4. fma-undefine 43.9%

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

      5. *-commutative 43.9%

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

      6. associate-*r* 43.9%

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

      7. fma-define 44.0%

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

   7. Applied egg-rr 44.0%

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

   8. Taylor expanded in B around inf 63.9%

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

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.85 \cdot 10^{+66}:\\ \;\;\;\;-\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.1% accurate, 6.0× speedup

Math

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

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 (- C)))))

C

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

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 / -c))
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 / -C));
}

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 / -C))

Julia

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in F around 0 19.0%

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

5. Simplified 27.0%

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

   1. sqrt-unprod 27.1%

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

   2. associate-/l* 31.8%

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

   3. associate-+r- 34.0%

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

   4. fma-undefine 34.0%

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

   5. *-commutative 34.0%

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

   6. associate-*r* 34.0%

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

   7. fma-define 34.1%

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

7. Applied egg-rr 34.1%

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

8. Taylor expanded in A around -inf 14.3%

   \[\leadsto -\sqrt{\color{blue}{-1 \cdot \frac{F}{C}}} \]
9. Step-by-step derivation

   1. associate-*r/ 14.3%

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

   2. mul-1-neg 14.3%

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

10. Simplified 14.3%

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

11. Final simplification 14.3%

    \[\leadsto -\sqrt{\frac{F}{-C}} \]
12. Add Preprocessing

Alternative 10: 1.8% accurate, 6.0× speedup

Math

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

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 (- A)))))

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 / -A));
}

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 / -a))
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 / -A));
}

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 / -A))

Julia

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in F around 0 19.0%

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

5. Simplified 27.0%

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

   1. sqrt-unprod 27.1%

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

   2. associate-/l* 31.8%

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

   3. associate-+r- 34.0%

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

   4. fma-undefine 34.0%

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

   5. *-commutative 34.0%

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

   6. associate-*r* 34.0%

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

   7. fma-define 34.1%

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

7. Applied egg-rr 34.1%

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

8. Taylor expanded in C around -inf 10.5%

   \[\leadsto -\sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
9. Step-by-step derivation

   1. associate-*r/ 10.5%

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

   2. mul-1-neg 10.5%

      \[\leadsto -\sqrt{\frac{\color{blue}{-F}}{A}} \]

10. Simplified 10.5%

    \[\leadsto -\sqrt{\color{blue}{\frac{-F}{A}}} \]

11. Final simplification 10.5%

    \[\leadsto -\sqrt{\frac{F}{-A}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))