ABCF->ab-angle b

Percentage Accurate: 19.0% → 54.4%

Time: 30.6s

Alternatives: 16

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 54.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A - \mathsf{hypot}\left(B\_m, A\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ t_2 := \left(4 \cdot A\right) \cdot C - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(\sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot t\_2\right)\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_0}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)} \cdot \sqrt{2 \cdot t\_1}}{-t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_4\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B\_m}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot t\_0}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- A (hypot B_m A)))
        (t_1 (fma C (* A -4.0) (pow B_m 2.0)))
        (t_2 (- (* (* 4.0 A) C) (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))
            (* 2.0 (* F t_2))))
          t_2))
        (t_4 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (* F (/ t_0 (fma -4.0 (* A C) (pow B_m 2.0))))) (- (sqrt 2.0)))
     (if (<= t_3 -5e-213)
       (/
        (* (sqrt (* F (- A (- (hypot B_m (- A C)) C)))) (sqrt (* 2.0 t_1)))
        (- t_1))
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* (* F t_4) (* 2.0 (+ A (+ A (/ (* (pow B_m 2.0) -0.5) C))))))
          (- t_4))
         (* (/ (sqrt 2.0) B_m) (- (sqrt (* F t_0)))))))))

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 = A - hypot(B_m, A);
	double t_1 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double t_2 = ((4.0 * A) * C) - pow(B_m, 2.0);
	double t_3 = sqrt(((sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * t_2)))) / t_2;
	double t_4 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt((F * (t_0 / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_3 <= -5e-213) {
		tmp = (sqrt((F * (A - (hypot(B_m, (A - C)) - C)))) * sqrt((2.0 * t_1))) / -t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_4) * (2.0 * (A + (A + ((pow(B_m, 2.0) * -0.5) / C)))))) / -t_4;
	} else {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * t_0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(A - hypot(B_m, A))
	t_1 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	t_2 = Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))) - Float64(A + C)) * Float64(2.0 * Float64(F * t_2)))) / t_2)
	t_4 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(t_0 / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_3 <= -5e-213)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C)))) * sqrt(Float64(2.0 * t_1))) / Float64(-t_1));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_4) * Float64(2.0 * Float64(A + Float64(A + Float64(Float64((B_m ^ 2.0) * -0.5) / C)))))) / Float64(-t_4));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * t_0))));
	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[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(F * N[(t$95$0 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -5e-213], N[(N[(N[Sqrt[N[(F * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * t$95$4), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.2%

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

   3. Taylor expanded in F around 0 21.0%

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

      1. mul-1-neg 21.0%

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

      2. *-commutative 21.0%

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

      3. associate-/l* 37.9%

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

      4. associate--l+ 37.9%

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

      5. unpow2 37.9%

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

      6. unpow2 37.9%

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

      7. hypot-undefine 57.5%

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

      8. cancel-sign-sub-inv 57.5%

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

   5. Simplified 57.5%

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

   6. Taylor expanded in C around 0 40.6%

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

      1. mul-1-neg 40.6%

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

      2. +-commutative 40.6%

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

      3. unpow2 40.6%

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

      4. unpow2 40.6%

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

      5. hypot-define 50.8%

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

   8. Simplified 50.8%

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

   9. Taylor expanded in F around 0 23.6%

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

       1. fma-define 23.6%

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

       2. associate-/l* 40.6%

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

       3. +-commutative 40.6%

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

       4. unpow2 40.6%

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

       5. unpow2 40.6%

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

       6. hypot-undefine 50.8%

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

   11. Simplified 50.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.99999999999999977e-213

   1. Initial program 96.5%

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

   3. Simplified 72.6%

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

      1. pow1/2 72.6%

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

      2. associate-*r* 99.6%

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

      3. unpow-prod-down 99.3%

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

      4. associate-+r- 99.3%

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

      5. hypot-undefine 99.3%

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

      6. unpow2 99.3%

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

      7. unpow2 99.3%

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

      8. +-commutative 99.3%

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

      9. unpow2 99.3%

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

      10. unpow2 99.3%

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

      11. hypot-define 99.3%

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

      12. pow1/2 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. unpow1/2 99.3%

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

      2. associate-+r- 99.3%

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

      3. hypot-undefine 99.3%

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

      4. unpow2 99.3%

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

      5. unpow2 99.3%

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

      6. +-commutative 99.3%

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

      7. unpow2 99.3%

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

      8. unpow2 99.3%

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

      9. hypot-undefine 99.3%

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

   8. Simplified 99.3%

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

   if -4.99999999999999977e-213 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 17.1%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in C around inf 29.4%

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

      1. associate-*r/ 29.4%

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

      2. mul-1-neg 29.4%

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

   7. Simplified 29.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. +-commutative 1.8%

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

      3. unpow2 1.8%

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

      4. unpow2 1.8%

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

      5. hypot-define 16.6%

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

   5. Simplified 16.6%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{A - \mathsf{hypot}\left(B, A\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A - \mathsf{hypot}\left(B\_m, A\right)\\ t_1 := \left(4 \cdot A\right) \cdot C - {B\_m}^{2}\\ t_2 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := F \cdot t\_2\\ t_4 := \frac{\sqrt{\left(\sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot t\_1\right)\right)}}{t\_1}\\ t_5 := -t\_2\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_0}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)\right)}}{t\_5}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \left(2 \cdot \left(A + \left(A + \frac{{B\_m}^{2} \cdot -0.5}{C}\right)\right)\right)}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot t\_0}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- A (hypot B_m A)))
        (t_1 (- (* (* 4.0 A) C) (pow B_m 2.0)))
        (t_2 (fma B_m B_m (* A (* C -4.0))))
        (t_3 (* F t_2))
        (t_4
         (/
          (sqrt
           (*
            (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))
            (* 2.0 (* F t_1))))
          t_1))
        (t_5 (- t_2)))
   (if (<= t_4 (- INFINITY))
     (* (sqrt (* F (/ t_0 (fma -4.0 (* A C) (pow B_m 2.0))))) (- (sqrt 2.0)))
     (if (<= t_4 -5e-213)
       (/ (sqrt (* t_3 (* 2.0 (- A (- (hypot B_m (- A C)) C))))) t_5)
       (if (<= t_4 INFINITY)
         (/
          (sqrt (* t_3 (* 2.0 (+ A (+ A (/ (* (pow B_m 2.0) -0.5) C))))))
          t_5)
         (* (/ (sqrt 2.0) B_m) (- (sqrt (* F t_0)))))))))

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 = A - hypot(B_m, A);
	double t_1 = ((4.0 * A) * C) - pow(B_m, 2.0);
	double t_2 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_3 = F * t_2;
	double t_4 = sqrt(((sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * t_1)))) / t_1;
	double t_5 = -t_2;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = sqrt((F * (t_0 / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_4 <= -5e-213) {
		tmp = sqrt((t_3 * (2.0 * (A - (hypot(B_m, (A - C)) - C))))) / t_5;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * (2.0 * (A + (A + ((pow(B_m, 2.0) * -0.5) / C)))))) / t_5;
	} else {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * t_0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(A - hypot(B_m, A))
	t_1 = Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0))
	t_2 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_3 = Float64(F * t_2)
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))) - Float64(A + C)) * Float64(2.0 * Float64(F * t_1)))) / t_1)
	t_5 = Float64(-t_2)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(t_0 / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_4 <= -5e-213)
		tmp = Float64(sqrt(Float64(t_3 * Float64(2.0 * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C))))) / t_5);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_3 * Float64(2.0 * Float64(A + Float64(A + Float64(Float64((B_m ^ 2.0) * -0.5) / C)))))) / t_5);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * t_0))));
	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[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(F * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = (-t$95$2)}, If[LessEqual[t$95$4, (-Infinity)], N[(N[Sqrt[N[(F * N[(t$95$0 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, -5e-213], N[(N[Sqrt[N[(t$95$3 * N[(2.0 * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$3 * N[(2.0 * N[(A + N[(A + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.2%

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

   3. Taylor expanded in F around 0 21.0%

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

      1. mul-1-neg 21.0%

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

      2. *-commutative 21.0%

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

      3. associate-/l* 37.9%

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

      4. associate--l+ 37.9%

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

      5. unpow2 37.9%

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

      6. unpow2 37.9%

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

      7. hypot-undefine 57.5%

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

      8. cancel-sign-sub-inv 57.5%

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

   5. Simplified 57.5%

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

   6. Taylor expanded in C around 0 40.6%

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

      1. mul-1-neg 40.6%

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

      2. +-commutative 40.6%

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

      3. unpow2 40.6%

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

      4. unpow2 40.6%

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

      5. hypot-define 50.8%

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

   8. Simplified 50.8%

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

   9. Taylor expanded in F around 0 23.6%

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

       1. fma-define 23.6%

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

       2. associate-/l* 40.6%

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

       3. +-commutative 40.6%

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

       4. unpow2 40.6%

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

       5. unpow2 40.6%

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

       6. hypot-undefine 50.8%

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

   11. Simplified 50.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.99999999999999977e-213

   1. Initial program 96.5%

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

   3. Simplified 96.5%

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

   if -4.99999999999999977e-213 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 17.1%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in C around inf 29.4%

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

      1. associate-*r/ 29.4%

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

      2. mul-1-neg 29.4%

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

   7. Simplified 29.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. +-commutative 1.8%

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

      3. unpow2 1.8%

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

      4. unpow2 1.8%

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

      5. hypot-define 16.6%

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

   5. Simplified 16.6%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{A - \mathsf{hypot}\left(B, A\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 50000000000:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(F \cdot \left(2 \cdot t\_0\right)\right) \cdot \left(2 \cdot A\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;-{\left(\sqrt{\frac{\sqrt{2 \cdot \left(F \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)}}{B\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (pow B_m 2.0) (* -4.0 (* A C)))))
   (if (<= (pow B_m 2.0) 2e-318)
     (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))
     (if (<= (pow B_m 2.0) 50000000000.0)
       (/ -1.0 (/ t_0 (sqrt (* (* F (* 2.0 t_0)) (* 2.0 A)))))
       (if (<= (pow B_m 2.0) 2e+139)
         (-
          (pow
           (sqrt (/ (sqrt (* 2.0 (* F (* -0.5 (/ (pow B_m 2.0) C))))) B_m))
           2.0))
         (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m 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) {
	double t_0 = pow(B_m, 2.0) + (-4.0 * (A * C));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-318) {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 50000000000.0) {
		tmp = -1.0 / (t_0 / sqrt(((F * (2.0 * t_0)) * (2.0 * A))));
	} else if (pow(B_m, 2.0) <= 2e+139) {
		tmp = -pow(sqrt((sqrt((2.0 * (F * (-0.5 * (pow(B_m, 2.0) / C))))) / B_m)), 2.0);
	} else {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) + (-4.0 * (A * C));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-318) {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 50000000000.0) {
		tmp = -1.0 / (t_0 / Math.sqrt(((F * (2.0 * t_0)) * (2.0 * A))));
	} else if (Math.pow(B_m, 2.0) <= 2e+139) {
		tmp = -Math.pow(Math.sqrt((Math.sqrt((2.0 * (F * (-0.5 * (Math.pow(B_m, 2.0) / C))))) / B_m)), 2.0);
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (A - Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) + (-4.0 * (A * C))
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-318:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 50000000000.0:
		tmp = -1.0 / (t_0 / math.sqrt(((F * (2.0 * t_0)) * (2.0 * A))))
	elif math.pow(B_m, 2.0) <= 2e+139:
		tmp = -math.pow(math.sqrt((math.sqrt((2.0 * (F * (-0.5 * (math.pow(B_m, 2.0) / C))))) / B_m)), 2.0)
	else:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (A - math.hypot(B_m, A))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(Float64(F * Float64(2.0 * t_0)) * Float64(2.0 * A)))));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = Float64(-(sqrt(Float64(sqrt(Float64(2.0 * Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))) / B_m)) ^ 2.0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) + (-4.0 * (A * C));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = -1.0 / (t_0 / sqrt(((F * (2.0 * t_0)) * (2.0 * A))));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = -(sqrt((sqrt((2.0 * (F * (-0.5 * ((B_m ^ 2.0) / C))))) / B_m)) ^ 2.0);
	else
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-318], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 50000000000.0], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+139], (-N[Power[N[Sqrt[N[(N[Sqrt[N[(2.0 * N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), 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]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000024e-318

   1. Initial program 13.6%

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

   3. Taylor expanded in F around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. *-commutative 12.7%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 19.0%

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

      8. cancel-sign-sub-inv 19.0%

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

   5. Simplified 19.0%

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

   6. Taylor expanded in A around -inf 26.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 2.0000024e-318 < (pow.f64 B #s(literal 2 binary64)) < 5e10

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf 24.4%

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

      1. clear-num 24.3%

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

      2. inv-pow 24.3%

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

      3. associate-*l* 24.5%

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

      4. associate-*r* 24.5%

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

      5. associate-*l* 24.5%

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

   5. Applied egg-rr 24.5%

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

      1. unpow-1 24.5%

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

      2. cancel-sign-sub-inv 24.5%

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

      3. metadata-eval 24.5%

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

      4. cancel-sign-sub-inv 24.5%

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

      5. metadata-eval 24.5%

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

   7. Simplified 24.5%

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

   if 5e10 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e139

   1. Initial program 9.6%

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

   3. Taylor expanded in A around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. unpow2 2.3%

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

      3. unpow2 2.3%

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

      4. hypot-define 3.2%

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

   5. Simplified 3.2%

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

      1. add-sqr-sqrt 2.4%

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

      2. pow2 2.4%

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

      3. associate-*l/ 2.4%

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

      4. pow1/2 2.4%

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

      5. pow1/2 2.5%

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

      6. pow-prod-down 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. unpow1/2 2.4%

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

   9. Simplified 2.4%

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

   10. Taylor expanded in C around inf 8.7%

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

   if 2.00000000000000007e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.7%

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

   3. Taylor expanded in C around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. hypot-define 25.1%

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

   5. Simplified 25.1%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 50000000000:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} + -4 \cdot \left(A \cdot C\right)}{\sqrt{\left(F \cdot \left(2 \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(2 \cdot A\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;-{\left(\sqrt{\frac{\sqrt{2 \cdot \left(F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{B}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 50000000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)))
   (if (<= (pow B_m 2.0) 2e-318)
     (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))
     (if (<= (pow B_m 2.0) 50000000000.0)
       (/
        (sqrt (* (* 2.0 A) (* 2.0 (* -4.0 (* A (* C F))))))
        (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (if (<= (pow B_m 2.0) 2e+139)
         (* t_0 (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))
         (* t_0 (- (sqrt (* F (- A (hypot B_m 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) {
	double t_0 = sqrt(2.0) / B_m;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-318) {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 50000000000.0) {
		tmp = sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+139) {
		tmp = t_0 * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-318) {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 50000000000.0) {
		tmp = Math.sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 2e+139) {
		tmp = t_0 * -Math.sqrt((F * (-0.5 * (Math.pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * -Math.sqrt((F * (A - Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-318:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 50000000000.0:
		tmp = math.sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 2e+139:
		tmp = t_0 * -math.sqrt((F * (-0.5 * (math.pow(B_m, 2.0) / C))))
	else:
		tmp = t_0 * -math.sqrt((F * (A - math.hypot(B_m, A))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * A) * Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = t_0 * -sqrt((F * (-0.5 * ((B_m ^ 2.0) / C))));
	else
		tmp = t_0 * -sqrt((F * (A - hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-318], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 50000000000.0], N[(N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+139], N[(t$95$0 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$0 * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000024e-318

   1. Initial program 13.6%

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

   3. Taylor expanded in F around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. *-commutative 12.7%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 19.0%

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

      8. cancel-sign-sub-inv 19.0%

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

   5. Simplified 19.0%

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

   6. Taylor expanded in A around -inf 26.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 2.0000024e-318 < (pow.f64 B #s(literal 2 binary64)) < 5e10

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf 24.4%

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

   4. Taylor expanded in B around 0 19.0%

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

   if 5e10 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e139

   1. Initial program 9.6%

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

   3. Taylor expanded in A around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. unpow2 2.3%

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

      3. unpow2 2.3%

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

      4. hypot-define 3.2%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in C around inf 13.4%

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

   if 2.00000000000000007e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.7%

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

   3. Taylor expanded in C around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. hypot-define 25.1%

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

   5. Simplified 25.1%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 50000000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 50000000000:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)))
   (if (<= (pow B_m 2.0) 2e-318)
     (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))
     (if (<= (pow B_m 2.0) 50000000000.0)
       (/
        (sqrt (* -8.0 (* (* A C) (* F (+ A A)))))
        (- (fma C (* A -4.0) (pow B_m 2.0))))
       (if (<= (pow B_m 2.0) 2e+139)
         (* t_0 (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))
         (* t_0 (- (sqrt (* F (- A (hypot B_m 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) {
	double t_0 = sqrt(2.0) / B_m;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-318) {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 50000000000.0) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+139) {
		tmp = t_0 * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(-fma(C, Float64(A * -4.0), (B_m ^ 2.0))));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-318], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 50000000000.0], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+139], N[(t$95$0 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$0 * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000024e-318

   1. Initial program 13.6%

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

   3. Taylor expanded in F around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. *-commutative 12.7%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 19.0%

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

      8. cancel-sign-sub-inv 19.0%

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

   5. Simplified 19.0%

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

   6. Taylor expanded in A around -inf 26.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 2.0000024e-318 < (pow.f64 B #s(literal 2 binary64)) < 5e10

   1. Initial program 23.6%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in C around inf 20.3%

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

      1. associate-*r* 20.3%

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

      2. *-commutative 20.3%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(\color{blue}{\left(C \cdot A\right)} \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

      3. mul-1-neg 20.3%

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

   7. Simplified 20.3%

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

   if 5e10 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e139

   1. Initial program 9.6%

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

   3. Taylor expanded in A around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. unpow2 2.3%

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

      3. unpow2 2.3%

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

      4. hypot-define 3.2%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in C around inf 13.4%

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

   if 2.00000000000000007e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.7%

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

   3. Taylor expanded in C around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. hypot-define 25.1%

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

   5. Simplified 25.1%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 50000000000:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 50000000000:\\ \;\;\;\;\frac{2 \cdot \sqrt{A \cdot \left(F \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)))
   (if (<= (pow B_m 2.0) 2e-318)
     (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))
     (if (<= (pow B_m 2.0) 50000000000.0)
       (/
        (* 2.0 (sqrt (* A (* F (- (pow B_m 2.0) (* 4.0 (* A C)))))))
        (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (if (<= (pow B_m 2.0) 2e+139)
         (* t_0 (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))
         (* t_0 (- (sqrt (* F (- A (hypot B_m 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) {
	double t_0 = sqrt(2.0) / B_m;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-318) {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 50000000000.0) {
		tmp = (2.0 * sqrt((A * (F * (pow(B_m, 2.0) - (4.0 * (A * C))))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+139) {
		tmp = t_0 * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-318) {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 50000000000.0) {
		tmp = (2.0 * Math.sqrt((A * (F * (Math.pow(B_m, 2.0) - (4.0 * (A * C))))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 2e+139) {
		tmp = t_0 * -Math.sqrt((F * (-0.5 * (Math.pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * -Math.sqrt((F * (A - Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-318:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 50000000000.0:
		tmp = (2.0 * math.sqrt((A * (F * (math.pow(B_m, 2.0) - (4.0 * (A * C))))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 2e+139:
		tmp = t_0 * -math.sqrt((F * (-0.5 * (math.pow(B_m, 2.0) / C))))
	else:
		tmp = t_0 * -math.sqrt((F * (A - math.hypot(B_m, A))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = Float64(Float64(2.0 * sqrt(Float64(A * Float64(F * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C))))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = (2.0 * sqrt((A * (F * ((B_m ^ 2.0) - (4.0 * (A * C))))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = t_0 * -sqrt((F * (-0.5 * ((B_m ^ 2.0) / C))));
	else
		tmp = t_0 * -sqrt((F * (A - hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-318], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 50000000000.0], N[(N[(2.0 * N[Sqrt[N[(A * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+139], N[(t$95$0 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$0 * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000024e-318

   1. Initial program 13.6%

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

   3. Taylor expanded in F around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. *-commutative 12.7%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 19.0%

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

      8. cancel-sign-sub-inv 19.0%

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

   5. Simplified 19.0%

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

   6. Taylor expanded in A around -inf 26.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 2.0000024e-318 < (pow.f64 B #s(literal 2 binary64)) < 5e10

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf 24.4%

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

   4. Taylor expanded in F around 0 24.4%

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

   if 5e10 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e139

   1. Initial program 9.6%

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

   3. Taylor expanded in A around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. unpow2 2.3%

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

      3. unpow2 2.3%

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

      4. hypot-define 3.2%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in C around inf 13.4%

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

   if 2.00000000000000007e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.7%

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

   3. Taylor expanded in C around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. hypot-define 25.1%

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

   5. Simplified 25.1%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 50000000000:\\ \;\;\;\;\frac{2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B\_m}\\ t_1 := 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 50000000000:\\ \;\;\;\;\sqrt{\left(2 \cdot A\right) \cdot \left(F \cdot \left(2 \cdot \left({B\_m}^{2} - t\_1\right)\right)\right)} \cdot \frac{1}{t\_1 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)) (t_1 (* 4.0 (* A C))))
   (if (<= (pow B_m 2.0) 2e-318)
     (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))
     (if (<= (pow B_m 2.0) 50000000000.0)
       (*
        (sqrt (* (* 2.0 A) (* F (* 2.0 (- (pow B_m 2.0) t_1)))))
        (/ 1.0 (- t_1 (pow B_m 2.0))))
       (if (<= (pow B_m 2.0) 2e+139)
         (* t_0 (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))
         (* t_0 (- (sqrt (* F (- A (hypot B_m 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) {
	double t_0 = sqrt(2.0) / B_m;
	double t_1 = 4.0 * (A * C);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-318) {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 50000000000.0) {
		tmp = sqrt(((2.0 * A) * (F * (2.0 * (pow(B_m, 2.0) - t_1))))) * (1.0 / (t_1 - pow(B_m, 2.0)));
	} else if (pow(B_m, 2.0) <= 2e+139) {
		tmp = t_0 * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double t_1 = 4.0 * (A * C);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-318) {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 50000000000.0) {
		tmp = Math.sqrt(((2.0 * A) * (F * (2.0 * (Math.pow(B_m, 2.0) - t_1))))) * (1.0 / (t_1 - Math.pow(B_m, 2.0)));
	} else if (Math.pow(B_m, 2.0) <= 2e+139) {
		tmp = t_0 * -Math.sqrt((F * (-0.5 * (Math.pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * -Math.sqrt((F * (A - Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	t_1 = 4.0 * (A * C)
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-318:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 50000000000.0:
		tmp = math.sqrt(((2.0 * A) * (F * (2.0 * (math.pow(B_m, 2.0) - t_1))))) * (1.0 / (t_1 - math.pow(B_m, 2.0)))
	elif math.pow(B_m, 2.0) <= 2e+139:
		tmp = t_0 * -math.sqrt((F * (-0.5 * (math.pow(B_m, 2.0) / C))))
	else:
		tmp = t_0 * -math.sqrt((F * (A - math.hypot(B_m, A))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	t_1 = Float64(4.0 * Float64(A * C))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * A) * Float64(F * Float64(2.0 * Float64((B_m ^ 2.0) - t_1))))) * Float64(1.0 / Float64(t_1 - (B_m ^ 2.0))));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	t_1 = 4.0 * (A * C);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = sqrt(((2.0 * A) * (F * (2.0 * ((B_m ^ 2.0) - t_1))))) * (1.0 / (t_1 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = t_0 * -sqrt((F * (-0.5 * ((B_m ^ 2.0) / C))));
	else
		tmp = t_0 * -sqrt((F * (A - hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-318], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 50000000000.0], N[(N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * N[(F * N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+139], N[(t$95$0 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$0 * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000024e-318

   1. Initial program 13.6%

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

   3. Taylor expanded in F around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. *-commutative 12.7%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 19.0%

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

      8. cancel-sign-sub-inv 19.0%

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

   5. Simplified 19.0%

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

   6. Taylor expanded in A around -inf 26.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 2.0000024e-318 < (pow.f64 B #s(literal 2 binary64)) < 5e10

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf 24.4%

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

      1. div-inv 24.3%

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

      2. associate-*r* 24.3%

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

      3. associate-*l* 24.3%

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

      4. associate-*l* 24.4%

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

   5. Applied egg-rr 24.4%

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

   if 5e10 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e139

   1. Initial program 9.6%

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

   3. Taylor expanded in A around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. unpow2 2.3%

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

      3. unpow2 2.3%

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

      4. hypot-define 3.2%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in C around inf 13.4%

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

   if 2.00000000000000007e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.7%

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

   3. Taylor expanded in C around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. hypot-define 25.1%

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

   5. Simplified 25.1%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 50000000000:\\ \;\;\;\;\sqrt{\left(2 \cdot A\right) \cdot \left(F \cdot \left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B\_m}\\ t_1 := {B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 50000000000:\\ \;\;\;\;\frac{-1}{\frac{t\_1}{\sqrt{\left(F \cdot \left(2 \cdot t\_1\right)\right) \cdot \left(2 \cdot A\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)) (t_1 (+ (pow B_m 2.0) (* -4.0 (* A C)))))
   (if (<= (pow B_m 2.0) 2e-318)
     (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))
     (if (<= (pow B_m 2.0) 50000000000.0)
       (/ -1.0 (/ t_1 (sqrt (* (* F (* 2.0 t_1)) (* 2.0 A)))))
       (if (<= (pow B_m 2.0) 2e+139)
         (* t_0 (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))
         (* t_0 (- (sqrt (* F (- A (hypot B_m 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) {
	double t_0 = sqrt(2.0) / B_m;
	double t_1 = pow(B_m, 2.0) + (-4.0 * (A * C));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-318) {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 50000000000.0) {
		tmp = -1.0 / (t_1 / sqrt(((F * (2.0 * t_1)) * (2.0 * A))));
	} else if (pow(B_m, 2.0) <= 2e+139) {
		tmp = t_0 * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double t_1 = Math.pow(B_m, 2.0) + (-4.0 * (A * C));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-318) {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 50000000000.0) {
		tmp = -1.0 / (t_1 / Math.sqrt(((F * (2.0 * t_1)) * (2.0 * A))));
	} else if (Math.pow(B_m, 2.0) <= 2e+139) {
		tmp = t_0 * -Math.sqrt((F * (-0.5 * (Math.pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * -Math.sqrt((F * (A - Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	t_1 = math.pow(B_m, 2.0) + (-4.0 * (A * C))
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-318:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 50000000000.0:
		tmp = -1.0 / (t_1 / math.sqrt(((F * (2.0 * t_1)) * (2.0 * A))))
	elif math.pow(B_m, 2.0) <= 2e+139:
		tmp = t_0 * -math.sqrt((F * (-0.5 * (math.pow(B_m, 2.0) / C))))
	else:
		tmp = t_0 * -math.sqrt((F * (A - math.hypot(B_m, A))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	t_1 = Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = Float64(-1.0 / Float64(t_1 / sqrt(Float64(Float64(F * Float64(2.0 * t_1)) * Float64(2.0 * A)))));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	t_1 = (B_m ^ 2.0) + (-4.0 * (A * C));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = -1.0 / (t_1 / sqrt(((F * (2.0 * t_1)) * (2.0 * A))));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = t_0 * -sqrt((F * (-0.5 * ((B_m ^ 2.0) / C))));
	else
		tmp = t_0 * -sqrt((F * (A - hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-318], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 50000000000.0], N[(-1.0 / N[(t$95$1 / N[Sqrt[N[(N[(F * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+139], N[(t$95$0 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$0 * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000024e-318

   1. Initial program 13.6%

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

   3. Taylor expanded in F around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. *-commutative 12.7%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 19.0%

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

      8. cancel-sign-sub-inv 19.0%

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

   5. Simplified 19.0%

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

   6. Taylor expanded in A around -inf 26.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 2.0000024e-318 < (pow.f64 B #s(literal 2 binary64)) < 5e10

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf 24.4%

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

      1. clear-num 24.3%

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

      2. inv-pow 24.3%

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

      3. associate-*l* 24.5%

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

      4. associate-*r* 24.5%

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

      5. associate-*l* 24.5%

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

   5. Applied egg-rr 24.5%

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

      1. unpow-1 24.5%

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

      2. cancel-sign-sub-inv 24.5%

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

      3. metadata-eval 24.5%

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

      4. cancel-sign-sub-inv 24.5%

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

      5. metadata-eval 24.5%

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

   7. Simplified 24.5%

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

   if 5e10 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e139

   1. Initial program 9.6%

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

   3. Taylor expanded in A around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. unpow2 2.3%

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

      3. unpow2 2.3%

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

      4. hypot-define 3.2%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in C around inf 13.4%

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

   if 2.00000000000000007e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.7%

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

   3. Taylor expanded in C around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. hypot-define 25.1%

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

   5. Simplified 25.1%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 50000000000:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} + -4 \cdot \left(A \cdot C\right)}{\sqrt{\left(F \cdot \left(2 \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(2 \cdot A\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 50000000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-318)
   (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))
   (if (<= (pow B_m 2.0) 50000000000.0)
     (/
      (sqrt (* (* 2.0 A) (* 2.0 (* -4.0 (* A (* C F))))))
      (- (* (* 4.0 A) C) (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 2e+139)
       (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F C)))))
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m 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) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-318) {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 50000000000.0) {
		tmp = sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+139) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / C)));
	} else {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-318) {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 50000000000.0) {
		tmp = Math.sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 2e+139) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-0.5 * (F / C)));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (A - Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-318:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 50000000000.0:
		tmp = math.sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 2e+139:
		tmp = math.sqrt(2.0) * -math.sqrt((-0.5 * (F / C)))
	else:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (A - math.hypot(B_m, A))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * A) * Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / C)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / C)));
	else
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-318], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 50000000000.0], N[(N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+139], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], 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]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000024e-318

   1. Initial program 13.6%

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

   3. Taylor expanded in F around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. *-commutative 12.7%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 19.0%

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

      8. cancel-sign-sub-inv 19.0%

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

   5. Simplified 19.0%

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

   6. Taylor expanded in A around -inf 26.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 2.0000024e-318 < (pow.f64 B #s(literal 2 binary64)) < 5e10

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf 24.4%

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

   4. Taylor expanded in B around 0 19.0%

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

   if 5e10 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e139

   1. Initial program 9.6%

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

   3. Taylor expanded in F around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 28.1%

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

      8. cancel-sign-sub-inv 28.1%

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

   5. Simplified 28.1%

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

   6. Taylor expanded in C around 0 18.1%

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

      1. mul-1-neg 18.1%

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

      2. +-commutative 18.1%

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

      3. unpow2 18.1%

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

      4. unpow2 18.1%

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

      5. hypot-define 23.5%

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

   8. Simplified 23.5%

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

   9. Taylor expanded in A around -inf 21.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \]

   if 2.00000000000000007e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.7%

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

   3. Taylor expanded in C around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. hypot-define 25.1%

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

   5. Simplified 25.1%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 50000000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 50000000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A}{B\_m} + -1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

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 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 2e-318)
     (* (sqrt (* F (/ -0.5 C))) t_0)
     (if (<= (pow B_m 2.0) 50000000000.0)
       (/
        (sqrt (* (* 2.0 A) (* 2.0 (* -4.0 (* A (* C F))))))
        (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (if (<= (pow B_m 2.0) 2e+139)
         (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F C)))))
         (* (sqrt (* F (/ (+ (/ A B_m) -1.0) B_m))) t_0))))))

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 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-318) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else if (pow(B_m, 2.0) <= 50000000000.0) {
		tmp = sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+139) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / C)));
	} else {
		tmp = sqrt((F * (((A / B_m) + -1.0) / B_m))) * t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if ((b_m ** 2.0d0) <= 2d-318) then
        tmp = sqrt((f * ((-0.5d0) / c))) * t_0
    else if ((b_m ** 2.0d0) <= 50000000000.0d0) then
        tmp = sqrt(((2.0d0 * a) * (2.0d0 * ((-4.0d0) * (a * (c * f)))))) / (((4.0d0 * a) * c) - (b_m ** 2.0d0))
    else if ((b_m ** 2.0d0) <= 2d+139) then
        tmp = sqrt(2.0d0) * -sqrt(((-0.5d0) * (f / c)))
    else
        tmp = sqrt((f * (((a / b_m) + (-1.0d0)) / b_m))) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-318) {
		tmp = Math.sqrt((F * (-0.5 / C))) * t_0;
	} else if (Math.pow(B_m, 2.0) <= 50000000000.0) {
		tmp = Math.sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 2e+139) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-0.5 * (F / C)));
	} else {
		tmp = Math.sqrt((F * (((A / B_m) + -1.0) / B_m))) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-318:
		tmp = math.sqrt((F * (-0.5 / C))) * t_0
	elif math.pow(B_m, 2.0) <= 50000000000.0:
		tmp = math.sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 2e+139:
		tmp = math.sqrt(2.0) * -math.sqrt((-0.5 * (F / C)))
	else:
		tmp = math.sqrt((F * (((A / B_m) + -1.0) / B_m))) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * A) * Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A / B_m) + -1.0) / B_m))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-318)
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	elseif ((B_m ^ 2.0) <= 50000000000.0)
		tmp = sqrt(((2.0 * A) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 2e+139)
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / C)));
	else
		tmp = sqrt((F * (((A / B_m) + -1.0) / B_m))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-318], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 50000000000.0], N[(N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+139], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(A / B$95$m), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000024e-318

   1. Initial program 13.6%

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

   3. Taylor expanded in F around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. *-commutative 12.7%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 19.0%

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

      8. cancel-sign-sub-inv 19.0%

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

   5. Simplified 19.0%

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

   6. Taylor expanded in A around -inf 26.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 2.0000024e-318 < (pow.f64 B #s(literal 2 binary64)) < 5e10

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf 24.4%

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

   4. Taylor expanded in B around 0 19.0%

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

   if 5e10 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e139

   1. Initial program 9.6%

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

   3. Taylor expanded in F around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. associate-/l* 13.3%

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

      4. associate--l+ 13.8%

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

      5. unpow2 13.8%

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

      6. unpow2 13.8%

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

      7. hypot-undefine 28.1%

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

      8. cancel-sign-sub-inv 28.1%

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

   5. Simplified 28.1%

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

   6. Taylor expanded in C around 0 18.1%

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

      1. mul-1-neg 18.1%

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

      2. +-commutative 18.1%

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

      3. unpow2 18.1%

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

      4. unpow2 18.1%

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

      5. hypot-define 23.5%

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

   8. Simplified 23.5%

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

   9. Taylor expanded in A around -inf 21.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \]

   if 2.00000000000000007e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.7%

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

   3. Taylor expanded in F around 0 13.3%

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

      1. mul-1-neg 13.3%

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

      2. *-commutative 13.3%

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

      3. associate-/l* 19.7%

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

      4. associate--l+ 19.7%

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

      5. unpow2 19.7%

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

      6. unpow2 19.7%

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

      7. hypot-undefine 24.7%

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

      8. cancel-sign-sub-inv 24.7%

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

   5. Simplified 24.7%

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

   6. Taylor expanded in C around 0 19.8%

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

      1. mul-1-neg 19.8%

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

      2. +-commutative 19.8%

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

      3. unpow2 19.8%

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

      4. unpow2 19.8%

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

      5. hypot-define 23.8%

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

   8. Simplified 23.8%

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

   9. Taylor expanded in B around inf 21.0%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\frac{A}{B} - 1}{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 50000000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A}{B} + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A}{B\_m} + -1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

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 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 2e+139)
     (* (sqrt (* F (/ -0.5 C))) t_0)
     (* (sqrt (* F (/ (+ (/ A B_m) -1.0) B_m))) t_0))))

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 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e+139) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * (((A / B_m) + -1.0) / B_m))) * t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if ((b_m ** 2.0d0) <= 2d+139) then
        tmp = sqrt((f * ((-0.5d0) / c))) * t_0
    else
        tmp = sqrt((f * (((a / b_m) + (-1.0d0)) / b_m))) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e+139) {
		tmp = Math.sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = Math.sqrt((F * (((A / B_m) + -1.0) / B_m))) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e+139:
		tmp = math.sqrt((F * (-0.5 / C))) * t_0
	else:
		tmp = math.sqrt((F * (((A / B_m) + -1.0) / B_m))) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+139)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A / B_m) + -1.0) / B_m))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e+139)
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	else
		tmp = sqrt((F * (((A / B_m) + -1.0) / B_m))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+139], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(A / B$95$m), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e139

   1. Initial program 17.3%

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

   3. Taylor expanded in F around 0 14.4%

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

      1. mul-1-neg 14.4%

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

      2. *-commutative 14.4%

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

      3. associate-/l* 17.1%

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

      4. associate--l+ 17.5%

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

      5. unpow2 17.5%

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

      6. unpow2 17.5%

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

      7. hypot-undefine 23.6%

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

      8. cancel-sign-sub-inv 23.6%

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

   5. Simplified 23.6%

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

   6. Taylor expanded in A around -inf 17.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 2.00000000000000007e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.7%

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

   3. Taylor expanded in F around 0 13.3%

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

      1. mul-1-neg 13.3%

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

      2. *-commutative 13.3%

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

      3. associate-/l* 19.7%

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

      4. associate--l+ 19.7%

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

      5. unpow2 19.7%

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

      6. unpow2 19.7%

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

      7. hypot-undefine 24.7%

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

      8. cancel-sign-sub-inv 24.7%

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

   5. Simplified 24.7%

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

   6. Taylor expanded in C around 0 19.8%

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

      1. mul-1-neg 19.8%

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

      2. +-commutative 19.8%

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

      3. unpow2 19.8%

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

      4. unpow2 19.8%

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

      5. hypot-define 23.8%

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

   8. Simplified 23.8%

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

   9. Taylor expanded in B around inf 21.0%

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

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A}{B} + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.5e+69)
   (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F C)))))
   (* (sqrt 2.0) (- (sqrt (- (/ 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 <= 4.5e+69) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / C)));
	} else {
		tmp = sqrt(2.0) * -sqrt(-(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 <= 4.5d+69) then
        tmp = sqrt(2.0d0) * -sqrt(((-0.5d0) * (f / c)))
    else
        tmp = sqrt(2.0d0) * -sqrt(-(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 <= 4.5e+69) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-0.5 * (F / C)));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt(-(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 <= 4.5e+69:
		tmp = math.sqrt(2.0) * -math.sqrt((-0.5 * (F / C)))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt(-(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 <= 4.5e+69)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / C)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-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 <= 4.5e+69)
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / C)));
	else
		tmp = sqrt(2.0) * -sqrt(-(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, 4.5e+69], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[(-N[(F / B$95$m), $MachinePrecision])], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.4999999999999999e69

   1. Initial program 15.3%

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

   3. Taylor expanded in F around 0 14.4%

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

      1. mul-1-neg 14.4%

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

      2. *-commutative 14.4%

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

      3. associate-/l* 18.1%

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

      4. associate--l+ 18.4%

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

      5. unpow2 18.4%

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

      6. unpow2 18.4%

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

      7. hypot-undefine 23.6%

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

      8. cancel-sign-sub-inv 23.6%

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

   5. Simplified 23.6%

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

   6. Taylor expanded in C around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. +-commutative 17.3%

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

      3. unpow2 17.3%

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

      4. unpow2 17.3%

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

      5. hypot-define 20.6%

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

   8. Simplified 20.6%

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

   9. Taylor expanded in A around -inf 15.1%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \]

   if 4.4999999999999999e69 < B

   1. Initial program 7.5%

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

   3. Taylor expanded in F around 0 11.9%

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

      1. mul-1-neg 11.9%

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

      2. *-commutative 11.9%

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

      3. associate-/l* 18.3%

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

      4. associate--l+ 18.3%

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

      5. unpow2 18.3%

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

      6. unpow2 18.3%

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

      7. hypot-undefine 25.9%

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

      8. cancel-sign-sub-inv 25.9%

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

   5. Simplified 25.9%

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

   6. Taylor expanded in C around 0 18.5%

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

      1. mul-1-neg 18.5%

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

      2. +-commutative 18.5%

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

      3. unpow2 18.5%

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

      4. unpow2 18.5%

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

      5. hypot-define 23.9%

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

   8. Simplified 23.9%

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

   9. Taylor expanded in A around 0 45.1%

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

       1. associate-*r/ 45.1%

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

       2. mul-1-neg 45.1%

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

   11. Simplified 45.1%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.3e+69)
   (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0)))
   (* (sqrt 2.0) (- (sqrt (- (/ 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 <= 3.3e+69) {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * -sqrt(-(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 <= 3.3d+69) then
        tmp = sqrt((f * ((-0.5d0) / c))) * -sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * -sqrt(-(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 <= 3.3e+69) {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt(-(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 <= 3.3e+69:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt(-(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 <= 3.3e+69)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-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 <= 3.3e+69)
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	else
		tmp = sqrt(2.0) * -sqrt(-(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, 3.3e+69], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[(-N[(F / B$95$m), $MachinePrecision])], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.2999999999999999e69

   1. Initial program 15.3%

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

   3. Taylor expanded in F around 0 14.4%

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

      1. mul-1-neg 14.4%

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

      2. *-commutative 14.4%

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

      3. associate-/l* 18.1%

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

      4. associate--l+ 18.4%

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

      5. unpow2 18.4%

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

      6. unpow2 18.4%

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

      7. hypot-undefine 23.6%

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

      8. cancel-sign-sub-inv 23.6%

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

   5. Simplified 23.6%

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

   6. Taylor expanded in A around -inf 15.1%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{C}}} \]

   if 3.2999999999999999e69 < B

   1. Initial program 7.5%

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

   3. Taylor expanded in F around 0 11.9%

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

      1. mul-1-neg 11.9%

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

      2. *-commutative 11.9%

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

      3. associate-/l* 18.3%

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

      4. associate--l+ 18.3%

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

      5. unpow2 18.3%

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

      6. unpow2 18.3%

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

      7. hypot-undefine 25.9%

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

      8. cancel-sign-sub-inv 25.9%

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

   5. Simplified 25.9%

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

   6. Taylor expanded in C around 0 18.5%

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

      1. mul-1-neg 18.5%

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

      2. +-commutative 18.5%

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

      3. unpow2 18.5%

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

      4. unpow2 18.5%

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

      5. hypot-define 23.9%

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

   8. Simplified 23.9%

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

   9. Taylor expanded in A around 0 45.1%

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

       1. associate-*r/ 45.1%

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

       2. mul-1-neg 45.1%

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

   11. Simplified 45.1%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.7 \cdot 10^{+226}:\\ \;\;\;\;-2 \cdot \left({\left(A \cdot F\right)}^{0.5} \cdot \frac{1}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -1.7e+226)
   (* -2.0 (* (pow (* A F) 0.5) (/ 1.0 B_m)))
   (* (sqrt 2.0) (- (sqrt (- (/ 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 (A <= -1.7e+226) {
		tmp = -2.0 * (pow((A * F), 0.5) * (1.0 / B_m));
	} else {
		tmp = sqrt(2.0) * -sqrt(-(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 (a <= (-1.7d+226)) then
        tmp = (-2.0d0) * (((a * f) ** 0.5d0) * (1.0d0 / b_m))
    else
        tmp = sqrt(2.0d0) * -sqrt(-(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 (A <= -1.7e+226) {
		tmp = -2.0 * (Math.pow((A * F), 0.5) * (1.0 / B_m));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt(-(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 A <= -1.7e+226:
		tmp = -2.0 * (math.pow((A * F), 0.5) * (1.0 / B_m))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt(-(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 (A <= -1.7e+226)
		tmp = Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) * Float64(1.0 / B_m)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-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 (A <= -1.7e+226)
		tmp = -2.0 * (((A * F) ^ 0.5) * (1.0 / B_m));
	else
		tmp = sqrt(2.0) * -sqrt(-(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[A, -1.7e+226], N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] * N[(1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[(-N[(F / B$95$m), $MachinePrecision])], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.69999999999999989e226

   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 A around -inf 21.2%

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

   4. Taylor expanded in B around inf 14.9%

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

      1. pow1/2 15.2%

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

   6. Applied egg-rr 15.2%

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

   if -1.69999999999999989e226 < A

   1. Initial program 14.7%

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

   3. Taylor expanded in F around 0 14.8%

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

      1. mul-1-neg 14.8%

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

      2. *-commutative 14.8%

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

      3. associate-/l* 19.2%

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

      4. associate--l+ 19.4%

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

      5. unpow2 19.4%

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

      6. unpow2 19.4%

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

      7. hypot-undefine 24.5%

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

      8. cancel-sign-sub-inv 24.5%

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

   5. Simplified 24.5%

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

   6. Taylor expanded in C around 0 18.5%

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

      1. mul-1-neg 18.5%

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

      2. +-commutative 18.5%

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

      3. unpow2 18.5%

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

      4. unpow2 18.5%

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

      5. hypot-define 21.4%

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

   8. Simplified 21.4%

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

   9. Taylor expanded in A around 0 11.3%

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

       1. associate-*r/ 11.3%

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

       2. mul-1-neg 11.3%

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

   11. Simplified 11.3%

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

4. Final simplification 11.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.7 \cdot 10^{+226}:\\ \;\;\;\;-2 \cdot \left({\left(A \cdot F\right)}^{0.5} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 9.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -2 \cdot \left({\left(A \cdot F\right)}^{0.5} \cdot \frac{1}{B\_m}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* -2.0 (* (pow (* A F) 0.5) (/ 1.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -2.0 * (pow((A * F), 0.5) * (1.0 / B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-2.0d0) * (((a * f) ** 0.5d0) * (1.0d0 / b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -2.0 * (Math.pow((A * F), 0.5) * (1.0 / B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -2.0 * (math.pow((A * F), 0.5) * (1.0 / B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) * Float64(1.0 / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -2.0 * (((A * F) ^ 0.5) * (1.0 / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] * N[(1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.9%

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

3. Taylor expanded in A around -inf 12.4%

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

4. Taylor expanded in B around inf 3.7%

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

   1. pow1/2 3.9%

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

6. Applied egg-rr 3.9%

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

7. Final simplification 3.9%

   \[\leadsto -2 \cdot \left({\left(A \cdot F\right)}^{0.5} \cdot \frac{1}{B}\right) \]
8. Add Preprocessing

Alternative 16: 9.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.9%

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

3. Taylor expanded in A around -inf 12.4%

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

4. Taylor expanded in B around inf 3.7%

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

   1. un-div-inv 3.7%

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

6. Applied egg-rr 3.7%

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

7. Final simplification 3.7%

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

Reproduce

herbie shell --seed 2024100 
(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))))