ABCF->ab-angle a

Percentage Accurate: 18.9% → 52.3%

Time: 18.5s

Alternatives: 10

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: 18.9% 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: 52.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := -t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-109}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{t\_0 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B\_m \cdot B\_m\right)}{A}\right)\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(2 \cdot C\right)} \cdot \sqrt{F}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B\_m + C}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C)))) (t_1 (- t_0)))
   (if (<= (pow B_m 2.0) 1e-109)
     (*
      (sqrt 2.0)
      (/ (sqrt (* t_0 (* F (fma 2.0 C (/ (* -0.5 (* B_m B_m)) A))))) t_1))
     (if (<= (pow B_m 2.0) 2e+70)
       (*
        (sqrt 2.0)
        (/
         (* (sqrt (* (fma B_m B_m (* C (* -4.0 A))) (* 2.0 C))) (sqrt F))
         t_1))
       (* (/ (sqrt 2.0) -1.0) (/ (* (sqrt F) (sqrt (+ B_m C))) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = -t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-109) {
		tmp = sqrt(2.0) * (sqrt((t_0 * (F * fma(2.0, C, ((-0.5 * (B_m * B_m)) / A))))) / t_1);
	} else if (pow(B_m, 2.0) <= 2e+70) {
		tmp = sqrt(2.0) * ((sqrt((fma(B_m, B_m, (C * (-4.0 * A))) * (2.0 * C))) * sqrt(F)) / t_1);
	} else {
		tmp = (sqrt(2.0) / -1.0) * ((sqrt(F) * sqrt((B_m + C))) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(-t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-109)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(t_0 * Float64(F * fma(2.0, C, Float64(Float64(-0.5 * Float64(B_m * B_m)) / A))))) / t_1));
	elseif ((B_m ^ 2.0) <= 2e+70)
		tmp = Float64(sqrt(2.0) * Float64(Float64(sqrt(Float64(fma(B_m, B_m, Float64(C * Float64(-4.0 * A))) * Float64(2.0 * C))) * sqrt(F)) / t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) / -1.0) * Float64(Float64(sqrt(F) * sqrt(Float64(B_m + C))) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-109], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(t$95$0 * N[(F * N[(2.0 * C + N[(N[(-0.5 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+70], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(B$95$m * B$95$m + N[(C * N[(-4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.9%

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

   4. Applied rewrites 21.5%

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

   5. Taylor expanded in C around inf

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      4. metadata-eval N/A

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

      5. lower-*.f64 30.4

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

   7. Applied rewrites 30.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lower-neg.f64 30.4

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

   9. Applied rewrites 30.4%

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

   10. Taylor expanded in A around -inf

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 31.3

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

   12. Applied rewrites 31.3%

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

   if 9.9999999999999999e-110 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000015e70

   1. Initial program 31.9%

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in C around inf

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      4. metadata-eval N/A

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

      5. lower-*.f64 22.9

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

   7. Applied rewrites 22.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lower-neg.f64 22.9

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

   9. Applied rewrites 22.9%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. sqrt-prod N/A

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

       7. pow1/2 N/A

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

       8. lower-*.f64 N/A

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

   11. Applied rewrites 29.4%

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

   if 2.00000000000000015e70 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.6%

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

   4. Applied rewrites 13.6%

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

   5. Taylor expanded in A around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 12.3

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 23.0%

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

   12. Applied rewrites 35.1%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-109}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(2 \cdot C\right)} \cdot \sqrt{F}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B + C}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{-8 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B\_m + C}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-106)
   (*
    (sqrt 2.0)
    (/ (sqrt (* -8.0 (* A (* F (* C C))))) (- (fma B_m B_m (* -4.0 (* A C))))))
   (if (<= (pow B_m 2.0) 2e+53)
     (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))) (- B_m))
     (* (/ (sqrt 2.0) -1.0) (/ (* (sqrt F) (sqrt (+ B_m C))) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-106) {
		tmp = sqrt(2.0) * (sqrt((-8.0 * (A * (F * (C * C))))) / -fma(B_m, B_m, (-4.0 * (A * C))));
	} else if (pow(B_m, 2.0) <= 2e+53) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C))))))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / -1.0) * ((sqrt(F) * sqrt((B_m + C))) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-106)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(-8.0 * Float64(A * Float64(F * Float64(C * C))))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))));
	elseif ((B_m ^ 2.0) <= 2e+53)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / -1.0) * Float64(Float64(sqrt(F) * sqrt(Float64(B_m + C))) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-106], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(-8.0 * N[(A * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+53], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999941e-107

   1. Initial program 21.4%

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

   4. Applied rewrites 22.0%

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

   5. Taylor expanded in C around inf

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      4. metadata-eval N/A

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

      5. lower-*.f64 30.7

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

   7. Applied rewrites 30.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lower-neg.f64 30.7

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

   9. Applied rewrites 30.7%

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

   10. Taylor expanded in C around inf

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

       1. rem-square-sqrt N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. rem-square-sqrt N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 19.9

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

   12. Applied rewrites 19.9%

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

   if 9.99999999999999941e-107 < (pow.f64 B #s(literal 2 binary64)) < 2e53

   1. Initial program 32.3%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 15.9

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

   5. Applied rewrites 15.9%

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in A around 0

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

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   10. Applied rewrites 19.2%

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

   if 2e53 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.4%

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in A around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 12.1

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

   7. Applied rewrites 12.1%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 22.6%

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

   12. Applied rewrites 34.4%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{-8 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B + C}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{t\_0 \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B\_m + C}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= (pow B_m 2.0) 5e+33)
     (* (sqrt 2.0) (/ (sqrt (* t_0 (* F (* 2.0 C)))) (- t_0)))
     (* (/ (sqrt 2.0) -1.0) (/ (* (sqrt F) (sqrt (+ B_m C))) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+33) {
		tmp = sqrt(2.0) * (sqrt((t_0 * (F * (2.0 * C)))) / -t_0);
	} else {
		tmp = (sqrt(2.0) / -1.0) * ((sqrt(F) * sqrt((B_m + C))) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+33)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(t_0 * Float64(F * Float64(2.0 * C)))) / Float64(-t_0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / -1.0) * Float64(Float64(sqrt(F) * sqrt(Float64(B_m + C))) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+33], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(t$95$0 * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999973e33

   1. Initial program 22.9%

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in C around inf

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      4. metadata-eval N/A

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

      5. lower-*.f64 28.9

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

   7. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lower-neg.f64 28.9

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

   9. Applied rewrites 28.9%

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

   if 4.99999999999999973e33 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.8%

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

   4. Applied rewrites 15.7%

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

   5. Taylor expanded in A around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 12.6

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 22.6%

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

   12. Applied rewrites 33.8%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B + C}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(t\_0 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B\_m + C}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= (pow B_m 2.0) 5e+33)
     (/ (sqrt (* 2.0 (* (* 2.0 C) (* t_0 F)))) (- t_0))
     (* (/ (sqrt 2.0) -1.0) (/ (* (sqrt F) (sqrt (+ B_m C))) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+33) {
		tmp = sqrt((2.0 * ((2.0 * C) * (t_0 * F)))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / -1.0) * ((sqrt(F) * sqrt((B_m + C))) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+33)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * Float64(t_0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / -1.0) * Float64(Float64(sqrt(F) * sqrt(Float64(B_m + C))) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+33], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999973e33

   1. Initial program 22.9%

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in C around inf

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      4. metadata-eval N/A

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

      5. lower-*.f64 28.9

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

   7. Applied rewrites 28.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. neg-mul-1 N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 27.3%

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

   if 4.99999999999999973e33 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.8%

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

   4. Applied rewrites 15.7%

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

   5. Taylor expanded in A around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 12.6

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 22.6%

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

   12. Applied rewrites 33.8%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B + C}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(-4 \cdot A\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t\_0 \cdot \left(2 \cdot C\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B\_m + C}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* -4.0 A)))))
   (if (<= (pow B_m 2.0) 5e+33)
     (/ (sqrt (* 2.0 (* F (* t_0 (* 2.0 C))))) (- t_0))
     (* (/ (sqrt 2.0) -1.0) (/ (* (sqrt F) (sqrt (+ B_m C))) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (-4.0 * A)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+33) {
		tmp = sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / -1.0) * ((sqrt(F) * sqrt((B_m + C))) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(-4.0 * A)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+33)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(t_0 * Float64(2.0 * C))))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / -1.0) * Float64(Float64(sqrt(F) * sqrt(Float64(B_m + C))) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(-4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+33], N[(N[Sqrt[N[(2.0 * N[(F * N[(t$95$0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999973e33

   1. Initial program 22.9%

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in C around inf

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C \cdot \left(2 + \color{blue}{0}\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      4. metadata-eval N/A

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

      5. lower-*.f64 28.9

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

   7. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lower-neg.f64 28.9

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

   9. Applied rewrites 28.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

   11. Applied rewrites 26.3%

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

   if 4.99999999999999973e33 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.8%

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

   4. Applied rewrites 15.7%

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

   5. Taylor expanded in A around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 12.6

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 22.6%

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

   12. Applied rewrites 33.8%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(2 \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B + C}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B\_m + C}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e+53)
   (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))) (- B_m))
   (* (/ (sqrt 2.0) -1.0) (/ (* (sqrt F) (sqrt (+ B_m C))) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e+53) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C))))))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / -1.0) * ((sqrt(F) * sqrt((B_m + C))) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+53)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / -1.0) * Float64(Float64(sqrt(F) * sqrt(Float64(B_m + C))) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+53], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e53

   1. Initial program 24.1%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 6.5

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

   5. Applied rewrites 6.5%

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

   7. Applied rewrites 6.5%

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

   8. Taylor expanded in A around 0

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

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   10. Applied rewrites 7.2%

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

   if 2e53 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.4%

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in A around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 12.1

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

   7. Applied rewrites 12.1%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 22.6%

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

   12. Applied rewrites 34.4%

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

4. Final simplification 17.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B + C}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.7% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return (math.sqrt(2.0) / -1.0) * ((math.sqrt(F) * math.sqrt((B_m + C))) / 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(2.0) / -1.0) * Float64(Float64(sqrt(F) * sqrt(Float64(B_m + C))) / B_m))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (sqrt(2.0) / -1.0) * ((sqrt(F) * sqrt((B_m + C))) / 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[(N[Sqrt[2.0], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

4. Applied rewrites 20.6%

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

5. Taylor expanded in A around 0

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

   1. associate-*l/ N/A

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

   2. *-lft-identity N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 9.1

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

7. Applied rewrites 9.1%

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

8. Taylor expanded in C around 0

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

10. Applied rewrites 12.4%

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

12. Applied rewrites 17.1%

    \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{B + C} \cdot \sqrt{F}}{B} \]

13. Final simplification 17.1%

    \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F} \cdot \sqrt{B + C}}{B} \]
14. Add Preprocessing

Alternative 8: 35.2% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in 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. distribute-rgt-neg-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-sqrt.f64 13.8

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

5. Applied rewrites 13.8%

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

7. Applied rewrites 13.8%

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

9. Applied rewrites 13.8%

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

11. Applied rewrites 17.3%

    \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
12. Add Preprocessing

Alternative 9: 35.2% accurate, 12.6× speedup

Math

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

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 20.0%

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

3. Taylor expanded in 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. distribute-rgt-neg-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-sqrt.f64 13.8

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

5. Applied rewrites 13.8%

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

7. Applied rewrites 13.8%

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

9. Applied rewrites 17.2%

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

Alternative 10: 27.1% 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 20.0%

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

3. Taylor expanded in 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. distribute-rgt-neg-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-sqrt.f64 13.8

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

5. Applied rewrites 13.8%

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

7. Applied rewrites 13.8%

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

9. Applied rewrites 13.8%

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

Reproduce

herbie shell --seed 2024234 
(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))))