ABCF->ab-angle a

Percentage Accurate: 18.4% → 58.9%

Time: 17.1s

Alternatives: 16

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.4% 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: 58.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -\sqrt{F}\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, 2 \cdot C\right)}}{-t\_2}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(t\_0 \cdot 2\right)} \cdot t\_1}{t\_2}\\ \mathbf{elif}\;B\_m \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot t\_0} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot t\_1\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (sqrt F)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= B_m 4.3e-128)
     (/
      (sqrt (* (* 2.0 (* t_2 F)) (fma (/ (* B_m B_m) A) -0.5 (* 2.0 C))))
      (- t_2))
     (if (<= B_m 1.05e-85)
       (/ (* (sqrt (* (* 2.0 C) (* t_0 2.0))) t_1) t_2)
       (if (<= B_m 7.2e+59)
         (*
          (sqrt (* (* 2.0 F) t_0))
          (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) (- t_0)))
         (* (* (sqrt (+ (hypot C B_m) C)) t_1) (/ (sqrt 2.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -sqrt(F);
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 4.3e-128) {
		tmp = sqrt(((2.0 * (t_2 * F)) * fma(((B_m * B_m) / A), -0.5, (2.0 * C)))) / -t_2;
	} else if (B_m <= 1.05e-85) {
		tmp = (sqrt(((2.0 * C) * (t_0 * 2.0))) * t_1) / t_2;
	} else if (B_m <= 7.2e+59) {
		tmp = sqrt(((2.0 * F) * t_0)) * (sqrt(((hypot((A - C), B_m) + A) + C)) / -t_0);
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * t_1) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-sqrt(F))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 4.3e-128)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(2.0 * C)))) / Float64(-t_2));
	elseif (B_m <= 1.05e-85)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * C) * Float64(t_0 * 2.0))) * t_1) / t_2);
	elseif (B_m <= 7.2e+59)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / Float64(-t_0)));
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * t_1) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.3e-128], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[B$95$m, 1.05e-85], N[(N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B$95$m, 7.2e+59], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.29999999999999994e-128

   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 A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 17.6

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

   5. Applied rewrites 17.6%

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

   if 4.29999999999999994e-128 < B < 1.05e-85

   1. Initial program 1.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 8.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 37.6

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

   7. Applied rewrites 37.6%

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

   if 1.05e-85 < B < 7.1999999999999997e59

   1. Initial program 32.5%

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

   4. Applied rewrites 60.2%

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

   if 7.1999999999999997e59 < B

   1. Initial program 14.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 78.3%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.3 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right)} \cdot \left(-\sqrt{F}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -\sqrt{F}\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 1.85 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{-t\_2}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(t\_0 \cdot 2\right)} \cdot t\_1}{t\_2}\\ \mathbf{elif}\;B\_m \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot t\_0} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot t\_1\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (sqrt F)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= B_m 1.85e-132)
     (/ (sqrt (* (* 2.0 (* t_2 F)) (* 2.0 C))) (- t_2))
     (if (<= B_m 1.05e-85)
       (/ (* (sqrt (* (* 2.0 C) (* t_0 2.0))) t_1) t_2)
       (if (<= B_m 7.2e+59)
         (*
          (sqrt (* (* 2.0 F) t_0))
          (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) (- t_0)))
         (* (* (sqrt (+ (hypot C B_m) C)) t_1) (/ (sqrt 2.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -sqrt(F);
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 1.85e-132) {
		tmp = sqrt(((2.0 * (t_2 * F)) * (2.0 * C))) / -t_2;
	} else if (B_m <= 1.05e-85) {
		tmp = (sqrt(((2.0 * C) * (t_0 * 2.0))) * t_1) / t_2;
	} else if (B_m <= 7.2e+59) {
		tmp = sqrt(((2.0 * F) * t_0)) * (sqrt(((hypot((A - C), B_m) + A) + C)) / -t_0);
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * t_1) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-sqrt(F))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 1.85e-132)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(2.0 * C))) / Float64(-t_2));
	elseif (B_m <= 1.05e-85)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * C) * Float64(t_0 * 2.0))) * t_1) / t_2);
	elseif (B_m <= 7.2e+59)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / Float64(-t_0)));
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * t_1) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.85e-132], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[B$95$m, 1.05e-85], N[(N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B$95$m, 7.2e+59], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.8500000000000001e-132

   1. Initial program 17.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 17.7

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

   5. Applied rewrites 17.7%

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

   if 1.8500000000000001e-132 < B < 1.05e-85

   1. Initial program 1.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 10.1%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 44.5

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

   7. Applied rewrites 44.5%

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

   if 1.05e-85 < B < 7.1999999999999997e59

   1. Initial program 32.5%

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

   4. Applied rewrites 60.2%

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

   if 7.1999999999999997e59 < B

   1. Initial program 14.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 78.3%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.85 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right)} \cdot \left(-\sqrt{F}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := -\sqrt{F}\\ \mathbf{if}\;B\_m \leq 1.85 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{C \cdot \mathsf{fma}\left(-16, \left(A \cdot C\right) \cdot F, 4 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot F\right)\right)}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(t\_0 \cdot 2\right)} \cdot t\_2}{t\_1}\\ \mathbf{elif}\;B\_m \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot t\_0} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot t\_2\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2 (- (sqrt F))))
   (if (<= B_m 1.85e-132)
     (/
      (sqrt (* C (fma -16.0 (* (* A C) F) (* 4.0 (* (* B_m B_m) F)))))
      (- t_1))
     (if (<= B_m 1.05e-85)
       (/ (* (sqrt (* (* 2.0 C) (* t_0 2.0))) t_2) t_1)
       (if (<= B_m 7.2e+59)
         (*
          (sqrt (* (* 2.0 F) t_0))
          (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) (- t_0)))
         (* (* (sqrt (+ (hypot C B_m) C)) t_2) (/ (sqrt 2.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(F);
	double tmp;
	if (B_m <= 1.85e-132) {
		tmp = sqrt((C * fma(-16.0, ((A * C) * F), (4.0 * ((B_m * B_m) * F))))) / -t_1;
	} else if (B_m <= 1.05e-85) {
		tmp = (sqrt(((2.0 * C) * (t_0 * 2.0))) * t_2) / t_1;
	} else if (B_m <= 7.2e+59) {
		tmp = sqrt(((2.0 * F) * t_0)) * (sqrt(((hypot((A - C), B_m) + A) + C)) / -t_0);
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * t_2) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(-sqrt(F))
	tmp = 0.0
	if (B_m <= 1.85e-132)
		tmp = Float64(sqrt(Float64(C * fma(-16.0, Float64(Float64(A * C) * F), Float64(4.0 * Float64(Float64(B_m * B_m) * F))))) / Float64(-t_1));
	elseif (B_m <= 1.05e-85)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * C) * Float64(t_0 * 2.0))) * t_2) / t_1);
	elseif (B_m <= 7.2e+59)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / Float64(-t_0)));
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * t_2) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[F], $MachinePrecision])}, If[LessEqual[B$95$m, 1.85e-132], N[(N[Sqrt[N[(C * N[(-16.0 * N[(N[(A * C), $MachinePrecision] * F), $MachinePrecision] + N[(4.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[B$95$m, 1.05e-85], N[(N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 7.2e+59], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.8500000000000001e-132

   1. Initial program 17.4%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{{C}^{2} \cdot \left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 11.9%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in C around 0

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

   8. Applied rewrites 17.7%

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

   if 1.8500000000000001e-132 < B < 1.05e-85

   1. Initial program 1.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 10.1%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 44.5

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

   7. Applied rewrites 44.5%

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

   if 1.05e-85 < B < 7.1999999999999997e59

   1. Initial program 32.5%

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

   4. Applied rewrites 60.2%

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

   if 7.1999999999999997e59 < B

   1. Initial program 14.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 78.3%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.85 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{C \cdot \mathsf{fma}\left(-16, \left(A \cdot C\right) \cdot F, 4 \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right)} \cdot \left(-\sqrt{F}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;\frac{\sqrt{C \cdot \mathsf{fma}\left(-16, \left(A \cdot C\right) \cdot F, 4 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot F\right)\right)}}{-\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.7e+60)
   (/
    (sqrt (* C (fma -16.0 (* (* A C) F) (* 4.0 (* (* B_m B_m) F)))))
    (- (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (* (* (sqrt (+ (hypot C B_m) C)) (- (sqrt F))) (/ (sqrt 2.0) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.7e+60) {
		tmp = sqrt((C * fma(-16.0, ((A * C) * F), (4.0 * ((B_m * B_m) * F))))) / -(pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * -sqrt(F)) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.7e+60)
		tmp = Float64(sqrt(Float64(C * fma(-16.0, Float64(Float64(A * C) * F), Float64(4.0 * Float64(Float64(B_m * B_m) * F))))) / Float64(-Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))));
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * Float64(-sqrt(F))) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.7e+60], N[(N[Sqrt[N[(C * N[(-16.0 * N[(N[(A * C), $MachinePrecision] * F), $MachinePrecision] + N[(4.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.7e60

   1. Initial program 18.8%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{{C}^{2} \cdot \left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 13.0%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in C around 0

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

   8. Applied rewrites 19.4%

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

   if 1.7e60 < B

   1. Initial program 14.2%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 79.5%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;\frac{\sqrt{C \cdot \mathsf{fma}\left(-16, \left(A \cdot C\right) \cdot F, 4 \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{F}\\ \mathbf{if}\;B\_m \leq 3 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot t\_0\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt F))))
   (if (<= B_m 3e-278)
     (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
     (if (<= B_m 2.7e-155)
       (/
        (sqrt (* (* (* (* C C) F) A) -16.0))
        (- (- (pow B_m 2.0) (* (* 4.0 A) C))))
       (if (<= B_m 8e+50)
         (*
          (* t_0 (sqrt (/ (* C 2.0) (fma -4.0 (* C A) (* B_m B_m)))))
          (sqrt 2.0))
         (* (* (sqrt (+ (hypot C B_m) C)) t_0) (/ (sqrt 2.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(F);
	double tmp;
	if (B_m <= 3e-278) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else if (B_m <= 2.7e-155) {
		tmp = sqrt(((((C * C) * F) * A) * -16.0)) / -(pow(B_m, 2.0) - ((4.0 * A) * C));
	} else if (B_m <= 8e+50) {
		tmp = (t_0 * sqrt(((C * 2.0) / fma(-4.0, (C * A), (B_m * B_m))))) * sqrt(2.0);
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * t_0) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(F))
	tmp = 0.0
	if (B_m <= 3e-278)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 2.7e-155)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * F) * A) * -16.0)) / Float64(-Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))));
	elseif (B_m <= 8e+50)
		tmp = Float64(Float64(t_0 * sqrt(Float64(Float64(C * 2.0) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))) * sqrt(2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * t_0) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[F], $MachinePrecision])}, If[LessEqual[B$95$m, 3e-278], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.7e-155], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 8e+50], N[(N[(t$95$0 * N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3e-278

   1. Initial program 15.9%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{A}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.8%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2} \]

   if 3e-278 < B < 2.69999999999999981e-155

   1. Initial program 27.8%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.1

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

   5. Applied rewrites 28.1%

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

   if 2.69999999999999981e-155 < B < 8.0000000000000006e50

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 9.1%

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

   10. Applied rewrites 25.3%

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

   if 8.0000000000000006e50 < B

   1. Initial program 15.4%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 78.6%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\left(\left(-\sqrt{F}\right) \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{F}\\ \mathbf{if}\;B\_m \leq 8.4 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 2.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0, -4, \left(B\_m \cdot B\_m\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B\_m \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot t\_0\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt F))))
   (if (<= B_m 8.4e-280)
     (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
     (if (<= B_m 2.35e-155)
       (/
        (sqrt
         (*
          (fma
           (/ (* (fma 0.0 -4.0 (* (* B_m B_m) 2.0)) F) C)
           2.0
           (* (* F A) -16.0))
          (* C C)))
        (- (fma B_m B_m (* -4.0 (* A C)))))
       (if (<= B_m 8e+50)
         (*
          (* t_0 (sqrt (/ (* C 2.0) (fma -4.0 (* C A) (* B_m B_m)))))
          (sqrt 2.0))
         (* (* (sqrt (+ (hypot C B_m) C)) t_0) (/ (sqrt 2.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(F);
	double tmp;
	if (B_m <= 8.4e-280) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else if (B_m <= 2.35e-155) {
		tmp = sqrt((fma(((fma(0.0, -4.0, ((B_m * B_m) * 2.0)) * F) / C), 2.0, ((F * A) * -16.0)) * (C * C))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else if (B_m <= 8e+50) {
		tmp = (t_0 * sqrt(((C * 2.0) / fma(-4.0, (C * A), (B_m * B_m))))) * sqrt(2.0);
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * t_0) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(F))
	tmp = 0.0
	if (B_m <= 8.4e-280)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 2.35e-155)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(fma(0.0, -4.0, Float64(Float64(B_m * B_m) * 2.0)) * F) / C), 2.0, Float64(Float64(F * A) * -16.0)) * Float64(C * C))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	elseif (B_m <= 8e+50)
		tmp = Float64(Float64(t_0 * sqrt(Float64(Float64(C * 2.0) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))) * sqrt(2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * t_0) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[F], $MachinePrecision])}, If[LessEqual[B$95$m, 8.4e-280], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.35e-155], N[(N[Sqrt[N[(N[(N[(N[(N[(0.0 * -4.0 + N[(N[(B$95$m * B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision] * 2.0 + N[(N[(F * A), $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision] * N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 8e+50], N[(N[(t$95$0 * N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 8.40000000000000003e-280

   1. Initial program 15.9%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{A}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.8%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2} \]

   if 8.40000000000000003e-280 < B < 2.3499999999999999e-155

   1. Initial program 27.8%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{{C}^{2} \cdot \left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 23.3%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)} \]

      6. lower-*.f64 23.3

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)} \]

   8. Applied rewrites 23.3%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

   if 2.3499999999999999e-155 < B < 8.0000000000000006e50

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 9.1%

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

   10. Applied rewrites 25.3%

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

   if 8.0000000000000006e50 < B

   1. Initial program 15.4%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 78.6%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.4 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\left(\left(-\sqrt{F}\right) \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{F}\\ \mathbf{if}\;B\_m \leq 8.4 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 2.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0, -4, \left(B\_m \cdot B\_m\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B\_m \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;B\_m \leq 1.8 \cdot 10^{+206}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \frac{\sqrt{F \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot t\_0\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt F))))
   (if (<= B_m 8.4e-280)
     (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
     (if (<= B_m 2.35e-155)
       (/
        (sqrt
         (*
          (fma
           (/ (* (fma 0.0 -4.0 (* (* B_m B_m) 2.0)) F) C)
           2.0
           (* (* F A) -16.0))
          (* C C)))
        (- (fma B_m B_m (* -4.0 (* A C)))))
       (if (<= B_m 8e+50)
         (*
          (* t_0 (sqrt (/ (* C 2.0) (fma -4.0 (* C A) (* B_m B_m)))))
          (sqrt 2.0))
         (if (<= B_m 1.8e+206)
           (* (sqrt (+ (hypot C B_m) C)) (/ (sqrt (* F 2.0)) (- B_m)))
           (* (* (sqrt (+ B_m C)) t_0) (/ (sqrt 2.0) B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(F);
	double tmp;
	if (B_m <= 8.4e-280) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else if (B_m <= 2.35e-155) {
		tmp = sqrt((fma(((fma(0.0, -4.0, ((B_m * B_m) * 2.0)) * F) / C), 2.0, ((F * A) * -16.0)) * (C * C))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else if (B_m <= 8e+50) {
		tmp = (t_0 * sqrt(((C * 2.0) / fma(-4.0, (C * A), (B_m * B_m))))) * sqrt(2.0);
	} else if (B_m <= 1.8e+206) {
		tmp = sqrt((hypot(C, B_m) + C)) * (sqrt((F * 2.0)) / -B_m);
	} else {
		tmp = (sqrt((B_m + C)) * t_0) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(F))
	tmp = 0.0
	if (B_m <= 8.4e-280)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 2.35e-155)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(fma(0.0, -4.0, Float64(Float64(B_m * B_m) * 2.0)) * F) / C), 2.0, Float64(Float64(F * A) * -16.0)) * Float64(C * C))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	elseif (B_m <= 8e+50)
		tmp = Float64(Float64(t_0 * sqrt(Float64(Float64(C * 2.0) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))) * sqrt(2.0));
	elseif (B_m <= 1.8e+206)
		tmp = Float64(sqrt(Float64(hypot(C, B_m) + C)) * Float64(sqrt(Float64(F * 2.0)) / Float64(-B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * t_0) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[F], $MachinePrecision])}, If[LessEqual[B$95$m, 8.4e-280], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.35e-155], N[(N[Sqrt[N[(N[(N[(N[(N[(0.0 * -4.0 + N[(N[(B$95$m * B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision] * 2.0 + N[(N[(F * A), $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision] * N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 8e+50], N[(N[(t$95$0 * N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.8e+206], N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < 8.40000000000000003e-280

   1. Initial program 15.9%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{A}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.8%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2} \]

   if 8.40000000000000003e-280 < B < 2.3499999999999999e-155

   1. Initial program 27.8%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{{C}^{2} \cdot \left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 23.3%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)} \]

      6. lower-*.f64 23.3

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)} \]

   8. Applied rewrites 23.3%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

   if 2.3499999999999999e-155 < B < 8.0000000000000006e50

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 9.1%

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

   10. Applied rewrites 25.3%

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

   if 8.0000000000000006e50 < B < 1.80000000000000014e206

   1. Initial program 24.8%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 54.6

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 72.3%

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

   9. Applied rewrites 69.8%

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

   if 1.80000000000000014e206 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 78.7%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.4 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\left(\left(-\sqrt{F}\right) \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{+206}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \frac{\sqrt{F \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{F}\\ \mathbf{if}\;B\_m \leq 8.4 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 2.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0, -4, \left(B\_m \cdot B\_m\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B\_m \leq 1.55 \cdot 10^{+56}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot t\_0\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt F))))
   (if (<= B_m 8.4e-280)
     (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
     (if (<= B_m 2.35e-155)
       (/
        (sqrt
         (*
          (fma
           (/ (* (fma 0.0 -4.0 (* (* B_m B_m) 2.0)) F) C)
           2.0
           (* (* F A) -16.0))
          (* C C)))
        (- (fma B_m B_m (* -4.0 (* A C)))))
       (if (<= B_m 1.55e+56)
         (*
          (* t_0 (sqrt (/ (* C 2.0) (fma -4.0 (* C A) (* B_m B_m)))))
          (sqrt 2.0))
         (* (* (sqrt (+ B_m C)) t_0) (/ (sqrt 2.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(F);
	double tmp;
	if (B_m <= 8.4e-280) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else if (B_m <= 2.35e-155) {
		tmp = sqrt((fma(((fma(0.0, -4.0, ((B_m * B_m) * 2.0)) * F) / C), 2.0, ((F * A) * -16.0)) * (C * C))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else if (B_m <= 1.55e+56) {
		tmp = (t_0 * sqrt(((C * 2.0) / fma(-4.0, (C * A), (B_m * B_m))))) * sqrt(2.0);
	} else {
		tmp = (sqrt((B_m + C)) * t_0) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(F))
	tmp = 0.0
	if (B_m <= 8.4e-280)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 2.35e-155)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(fma(0.0, -4.0, Float64(Float64(B_m * B_m) * 2.0)) * F) / C), 2.0, Float64(Float64(F * A) * -16.0)) * Float64(C * C))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	elseif (B_m <= 1.55e+56)
		tmp = Float64(Float64(t_0 * sqrt(Float64(Float64(C * 2.0) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))) * sqrt(2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * t_0) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[F], $MachinePrecision])}, If[LessEqual[B$95$m, 8.4e-280], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.35e-155], N[(N[Sqrt[N[(N[(N[(N[(N[(0.0 * -4.0 + N[(N[(B$95$m * B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision] * 2.0 + N[(N[(F * A), $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision] * N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 1.55e+56], N[(N[(t$95$0 * N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 8.40000000000000003e-280

   1. Initial program 15.9%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{A}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.8%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2} \]

   if 8.40000000000000003e-280 < B < 2.3499999999999999e-155

   1. Initial program 27.8%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{{C}^{2} \cdot \left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 23.3%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)} \]

      6. lower-*.f64 23.3

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)} \]

   8. Applied rewrites 23.3%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

   if 2.3499999999999999e-155 < B < 1.55000000000000002e56

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 9.1%

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

   10. Applied rewrites 25.3%

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

   if 1.55000000000000002e56 < B

   1. Initial program 15.4%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 69.0%

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

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.4 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(-\sqrt{F}\right) \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -\sqrt{F}\\ \mathbf{if}\;B\_m \leq 2.6 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 2.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot C\right) \cdot \mathsf{fma}\left(2 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot 2\right), \frac{F}{C}, -16 \cdot \left(F \cdot A\right)\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 1.55 \cdot 10^{+56}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\frac{C \cdot 2}{t\_0}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot t\_1\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))) (t_1 (- (sqrt F))))
   (if (<= B_m 2.6e-264)
     (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
     (if (<= B_m 2.35e-155)
       (/
        (sqrt
         (*
          (* C C)
          (fma (* 2.0 (* (* B_m B_m) 2.0)) (/ F C) (* -16.0 (* F A)))))
        (- t_0))
       (if (<= B_m 1.55e+56)
         (* (* t_1 (sqrt (/ (* C 2.0) t_0))) (sqrt 2.0))
         (* (* (sqrt (+ B_m C)) t_1) (/ (sqrt 2.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -sqrt(F);
	double tmp;
	if (B_m <= 2.6e-264) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else if (B_m <= 2.35e-155) {
		tmp = sqrt(((C * C) * fma((2.0 * ((B_m * B_m) * 2.0)), (F / C), (-16.0 * (F * A))))) / -t_0;
	} else if (B_m <= 1.55e+56) {
		tmp = (t_1 * sqrt(((C * 2.0) / t_0))) * sqrt(2.0);
	} else {
		tmp = (sqrt((B_m + C)) * t_1) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-sqrt(F))
	tmp = 0.0
	if (B_m <= 2.6e-264)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 2.35e-155)
		tmp = Float64(sqrt(Float64(Float64(C * C) * fma(Float64(2.0 * Float64(Float64(B_m * B_m) * 2.0)), Float64(F / C), Float64(-16.0 * Float64(F * A))))) / Float64(-t_0));
	elseif (B_m <= 1.55e+56)
		tmp = Float64(Float64(t_1 * sqrt(Float64(Float64(C * 2.0) / t_0))) * sqrt(2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * t_1) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, If[LessEqual[B$95$m, 2.6e-264], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.35e-155], N[(N[Sqrt[N[(N[(C * C), $MachinePrecision] * N[(N[(2.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(F / C), $MachinePrecision] + N[(-16.0 * N[(F * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 1.55e+56], N[(N[(t$95$1 * N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.6000000000000002e-264

   1. Initial program 15.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 18.8%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{A}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.7%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2} \]

   if 2.6000000000000002e-264 < B < 2.3499999999999999e-155

   1. Initial program 30.6%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{{C}^{2} \cdot \left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot \left(A + -1 \cdot A\right)\right) + 2 \cdot {B}^{2}\right)}{C}\right) \cdot {C}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 19.9%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0 \cdot A\right) \cdot A, -4, \left(B \cdot B\right) \cdot 2\right) \cdot F}{C}, 2, \left(F \cdot A\right) \cdot -16\right) \cdot \left(C \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 19.9%

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

   if 2.3499999999999999e-155 < B < 1.55000000000000002e56

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 9.1%

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

   10. Applied rewrites 25.3%

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

   if 1.55000000000000002e56 < B

   1. Initial program 15.4%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 69.0%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.6 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot C\right) \cdot \mathsf{fma}\left(2 \cdot \left(\left(B \cdot B\right) \cdot 2\right), \frac{F}{C}, -16 \cdot \left(F \cdot A\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(-\sqrt{F}\right) \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{F}\\ \mathbf{if}\;B\_m \leq 1.55 \cdot 10^{+56}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot t\_0\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt F))))
   (if (<= B_m 1.55e+56)
     (* (* t_0 (sqrt (/ (* C 2.0) (fma -4.0 (* C A) (* B_m B_m))))) (sqrt 2.0))
     (* (* (sqrt (+ B_m C)) t_0) (/ (sqrt 2.0) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(F);
	double tmp;
	if (B_m <= 1.55e+56) {
		tmp = (t_0 * sqrt(((C * 2.0) / fma(-4.0, (C * A), (B_m * B_m))))) * sqrt(2.0);
	} else {
		tmp = (sqrt((B_m + C)) * t_0) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(F))
	tmp = 0.0
	if (B_m <= 1.55e+56)
		tmp = Float64(Float64(t_0 * sqrt(Float64(Float64(C * 2.0) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))) * sqrt(2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * t_0) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[F], $MachinePrecision])}, If[LessEqual[B$95$m, 1.55e+56], N[(N[(t$95$0 * N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.55000000000000002e56

   1. Initial program 18.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 20.3%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 10.8%

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

   10. Applied rewrites 18.3%

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

   if 1.55000000000000002e56 < B

   1. Initial program 15.4%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 69.0%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.55 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(-\sqrt{F}\right) \cdot \sqrt{\frac{C \cdot 2}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9.5 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 9.5e+42)
   (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
   (* (* (sqrt (+ B_m C)) (- (sqrt F))) (/ (sqrt 2.0) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.5e+42) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else {
		tmp = (sqrt((B_m + C)) * -sqrt(F)) * (sqrt(2.0) / 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 <= 9.5d+42) then
        tmp = sqrt(((-0.5d0) * (f / a))) * -sqrt(2.0d0)
    else
        tmp = (sqrt((b_m + c)) * -sqrt(f)) * (sqrt(2.0d0) / 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 <= 9.5e+42) {
		tmp = Math.sqrt((-0.5 * (F / A))) * -Math.sqrt(2.0);
	} else {
		tmp = (Math.sqrt((B_m + C)) * -Math.sqrt(F)) * (Math.sqrt(2.0) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9.5e+42:
		tmp = math.sqrt((-0.5 * (F / A))) * -math.sqrt(2.0)
	else:
		tmp = (math.sqrt((B_m + C)) * -math.sqrt(F)) * (math.sqrt(2.0) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 9.5e+42)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))) * Float64(sqrt(2.0) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 9.5e+42)
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	else
		tmp = (sqrt((B_m + C)) * -sqrt(F)) * (sqrt(2.0) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9.5e+42], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9.50000000000000019e42

   1. Initial program 18.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{A}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.1%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2} \]

   if 9.50000000000000019e42 < B

   1. Initial program 16.5%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 77.8%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 68.5%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.5 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9.5 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 9.5e+42)
   (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
   (/ (sqrt (* F 2.0)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.5e+42) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 9.5d+42) then
        tmp = sqrt(((-0.5d0) * (f / a))) * -sqrt(2.0d0)
    else
        tmp = sqrt((f * 2.0d0)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.5e+42) {
		tmp = Math.sqrt((-0.5 * (F / A))) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9.5e+42:
		tmp = math.sqrt((-0.5 * (F / A))) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt((F * 2.0)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 9.5e+42)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 9.5e+42)
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	else
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9.5e+42], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9.50000000000000019e42

   1. Initial program 18.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{A}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.1%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2} \]

   if 9.50000000000000019e42 < B

   1. Initial program 16.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 47.8

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

   5. Applied rewrites 47.8%

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

   7. Applied rewrites 68.1%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.5 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.1% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 14.4

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

5. Applied rewrites 14.4%

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

7. Applied rewrites 18.9%

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

8. Final simplification 18.9%

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

Alternative 14: 35.1% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 14.4

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

5. Applied rewrites 14.4%

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

7. Applied rewrites 14.5%

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

9. Applied rewrites 18.9%

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

10. Final simplification 18.9%

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

Alternative 15: 27.7% accurate, 16.9× speedup

Math

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

FPCore

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

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) {
	return -sqrt(((F / B_m) * 2.0));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 14.4

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

5. Applied rewrites 14.4%

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

7. Applied rewrites 14.5%

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

Alternative 16: 27.7% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 14.4

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

5. Applied rewrites 14.4%

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

7. Applied rewrites 14.5%

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

9. Applied rewrites 14.4%

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

Reproduce

herbie shell --seed 2024339 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))