ABCF->ab-angle a

Percentage Accurate: 19.2% → 61.3%

Time: 19.4s

Alternatives: 15

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: 19.2% 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: 61.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \sqrt{2 \cdot F}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := 2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\\ t_4 := t\_2 - {B\_m}^{2}\\ t_5 := \frac{\sqrt{t\_3 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_4}\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+184}:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot t\_1\right) \cdot \frac{\sqrt{2 \cdot C}}{-\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;t\_5 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, 2 \cdot C\right)}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 F)))
        (t_2 (* (* 4.0 A) C))
        (t_3 (* 2.0 (* (- (pow B_m 2.0) t_2) F)))
        (t_4 (- t_2 (pow B_m 2.0)))
        (t_5
         (/
          (sqrt (* t_3 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_4)))
   (if (<= t_5 -2e+184)
     (*
      (* (sqrt t_0) t_1)
      (/ (sqrt (* 2.0 C)) (- (fma A (* C -4.0) (* B_m B_m)))))
     (if (<= t_5 -5e-191)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (- t_0))
       (if (<= t_5 INFINITY)
         (/ (sqrt (* t_3 (fma (/ (* B_m B_m) A) -0.5 (* 2.0 C)))) t_4)
         (* (sqrt (/ 1.0 B_m)) (- t_1)))))))

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 = sqrt((2.0 * F));
	double t_2 = (4.0 * A) * C;
	double t_3 = 2.0 * ((pow(B_m, 2.0) - t_2) * F);
	double t_4 = t_2 - pow(B_m, 2.0);
	double t_5 = sqrt((t_3 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_4;
	double tmp;
	if (t_5 <= -2e+184) {
		tmp = (sqrt(t_0) * t_1) * (sqrt((2.0 * C)) / -fma(A, (C * -4.0), (B_m * B_m)));
	} else if (t_5 <= -5e-191) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / -t_0;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * fma(((B_m * B_m) / A), -0.5, (2.0 * C)))) / t_4;
	} else {
		tmp = sqrt((1.0 / B_m)) * -t_1;
	}
	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 = sqrt(Float64(2.0 * F))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F))
	t_4 = Float64(t_2 - (B_m ^ 2.0))
	t_5 = Float64(sqrt(Float64(t_3 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_4)
	tmp = 0.0
	if (t_5 <= -2e+184)
		tmp = Float64(Float64(sqrt(t_0) * t_1) * Float64(sqrt(Float64(2.0 * C)) / Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m)))));
	elseif (t_5 <= -5e-191)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / Float64(-t_0));
	elseif (t_5 <= Inf)
		tmp = Float64(sqrt(Float64(t_3 * fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(2.0 * C)))) / t_4);
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-t_1));
	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 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$3 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+184], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -5e-191], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[Sqrt[N[(t$95$3 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-t$95$1)), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000003e184

   1. Initial program 4.5%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 16.3

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

   5. Applied rewrites 16.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 26.2%

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

   9. Applied rewrites 26.1%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-fma.f64 N/A

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

       14. associate-*r* N/A

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

       15. lift-*.f64 N/A

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

       16. *-commutative N/A

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

       17. sqrt-prod N/A

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

       18. pow1/2 N/A

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

       19. lift-sqrt.f64 N/A

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

   11. Applied rewrites 38.2%

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

   if -2.00000000000000003e184 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e-191

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if -5.0000000000000001e-191 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 19.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 29.9

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

   5. Applied rewrites 29.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 17.4

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

   5. Applied rewrites 17.4%

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

   7. Applied rewrites 26.7%

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

   9. Applied rewrites 26.7%

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

   11. Applied rewrites 26.8%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+184}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot F}\right) \cdot \frac{\sqrt{2 \cdot C}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := t\_0 \cdot \left(2 \cdot F\right)\\ t_2 := \sqrt{2 \cdot F}\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+184}:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot t\_2\right) \cdot \frac{\sqrt{2 \cdot C}}{-\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-t\_2\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (* t_0 (* 2.0 F)))
        (t_2 (sqrt (* 2.0 F)))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B_m 2.0)))))
   (if (<= t_4 -2e+184)
     (*
      (* (sqrt t_0) t_2)
      (/ (sqrt (* 2.0 C)) (- (fma A (* C -4.0) (* B_m B_m)))))
     (if (<= t_4 -5e-191)
       (/
        (sqrt (* t_1 (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (- t_0))
       (if (<= t_4 INFINITY)
         (* (sqrt (* t_1 (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)))) (/ -1.0 t_0))
         (* (sqrt (/ 1.0 B_m)) (- t_2)))))))

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 * (2.0 * F);
	double t_2 = sqrt((2.0 * F));
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B_m, 2.0));
	double tmp;
	if (t_4 <= -2e+184) {
		tmp = (sqrt(t_0) * t_2) * (sqrt((2.0 * C)) / -fma(A, (C * -4.0), (B_m * B_m)));
	} else if (t_4 <= -5e-191) {
		tmp = sqrt((t_1 * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / -t_0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(2.0, C, (((B_m * B_m) * -0.5) / A)))) * (-1.0 / t_0);
	} else {
		tmp = sqrt((1.0 / B_m)) * -t_2;
	}
	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 * Float64(2.0 * F))
	t_2 = sqrt(Float64(2.0 * F))
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_4 <= -2e+184)
		tmp = Float64(Float64(sqrt(t_0) * t_2) * Float64(sqrt(Float64(2.0 * C)) / Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m)))));
	elseif (t_4 <= -5e-191)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / Float64(-t_0));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A)))) * Float64(-1.0 / t_0));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-t_2));
	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 = N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+184], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-191], N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-t$95$2)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000003e184

   1. Initial program 4.5%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 16.3

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

   5. Applied rewrites 16.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 26.2%

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

   9. Applied rewrites 26.1%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-fma.f64 N/A

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

       14. associate-*r* N/A

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

       15. lift-*.f64 N/A

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

       16. *-commutative N/A

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

       17. sqrt-prod N/A

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

       18. pow1/2 N/A

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

       19. lift-sqrt.f64 N/A

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

   11. Applied rewrites 38.2%

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

   if -2.00000000000000003e184 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e-191

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if -5.0000000000000001e-191 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 19.5%

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

   4. Applied rewrites 19.4%

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

   5. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 29.8

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

   7. Applied rewrites 29.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 17.4

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

   5. Applied rewrites 17.4%

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

   7. Applied rewrites 26.7%

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

   9. Applied rewrites 26.7%

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

   11. Applied rewrites 26.8%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+184}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot F}\right) \cdot \frac{\sqrt{2 \cdot C}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \sqrt{2 \cdot F}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+84}:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot t\_1\right) \cdot \frac{\sqrt{2 \cdot C}}{-\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 F)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -2e+84)
     (*
      (* (sqrt t_0) t_1)
      (/ (sqrt (* 2.0 C)) (- (fma A (* C -4.0) (* B_m B_m)))))
     (if (<= t_3 -5e-191)
       (*
        (sqrt
         (/ (* F (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))) t_0))
        (- (sqrt 2.0)))
       (if (<= t_3 INFINITY)
         (*
          (sqrt (* (* t_0 (* 2.0 F)) (fma 2.0 C (/ (* (* B_m B_m) -0.5) A))))
          (/ -1.0 t_0))
         (* (sqrt (/ 1.0 B_m)) (- t_1)))))))

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 = sqrt((2.0 * F));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -2e+84) {
		tmp = (sqrt(t_0) * t_1) * (sqrt((2.0 * C)) / -fma(A, (C * -4.0), (B_m * B_m)));
	} else if (t_3 <= -5e-191) {
		tmp = sqrt(((F * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / t_0)) * -sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((t_0 * (2.0 * F)) * fma(2.0, C, (((B_m * B_m) * -0.5) / A)))) * (-1.0 / t_0);
	} else {
		tmp = sqrt((1.0 / B_m)) * -t_1;
	}
	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 = sqrt(Float64(2.0 * F))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -2e+84)
		tmp = Float64(Float64(sqrt(t_0) * t_1) * Float64(sqrt(Float64(2.0 * C)) / Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m)))));
	elseif (t_3 <= -5e-191)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / t_0)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A)))) * Float64(-1.0 / t_0));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-t_1));
	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 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+84], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-191], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-t$95$1)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000012e84

   1. Initial program 10.9%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 17.0

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

   5. Applied rewrites 17.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 26.3%

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

   9. Applied rewrites 26.2%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-fma.f64 N/A

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

       14. associate-*r* N/A

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

       15. lift-*.f64 N/A

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

       16. *-commutative N/A

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

       17. sqrt-prod N/A

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

       18. pow1/2 N/A

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

       19. lift-sqrt.f64 N/A

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

   11. Applied rewrites 37.5%

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

   if -2.00000000000000012e84 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e-191

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 90.6%

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

   if -5.0000000000000001e-191 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 19.5%

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

   4. Applied rewrites 19.4%

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

   5. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 29.8

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

   7. Applied rewrites 29.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 17.4

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

   5. Applied rewrites 17.4%

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

   7. Applied rewrites 26.7%

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

   9. Applied rewrites 26.7%

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

   11. Applied rewrites 26.8%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+84}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot F}\right) \cdot \frac{\sqrt{2 \cdot C}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot \left(-\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot F\right)}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-110}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \sqrt{F}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e-276)
   (*
    (/ (sqrt (* 2.0 C)) (fma A (* C -4.0) (* B_m B_m)))
    (- (sqrt (* (* A -8.0) (* C F)))))
   (if (<= (pow B_m 2.0) 5e-110)
     (*
      (/ -1.0 (fma B_m B_m (* -4.0 (* A C))))
      (sqrt (* -16.0 (* F (* A (* C C))))))
     (if (<= (pow B_m 2.0) 5e+251)
       (/
        (* (sqrt (* 2.0 (+ C (sqrt (fma B_m B_m (* C C)))))) (sqrt F))
        (- B_m))
       (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))))

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) <= 4e-276) {
		tmp = (sqrt((2.0 * C)) / fma(A, (C * -4.0), (B_m * B_m))) * -sqrt(((A * -8.0) * (C * F)));
	} else if (pow(B_m, 2.0) <= 5e-110) {
		tmp = (-1.0 / fma(B_m, B_m, (-4.0 * (A * C)))) * sqrt((-16.0 * (F * (A * (C * C)))));
	} else if (pow(B_m, 2.0) <= 5e+251) {
		tmp = (sqrt((2.0 * (C + sqrt(fma(B_m, B_m, (C * C)))))) * sqrt(F)) / -B_m;
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	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) <= 4e-276)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) / fma(A, Float64(C * -4.0), Float64(B_m * B_m))) * Float64(-sqrt(Float64(Float64(A * -8.0) * Float64(C * F)))));
	elseif ((B_m ^ 2.0) <= 5e-110)
		tmp = Float64(Float64(-1.0 / fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(-16.0 * Float64(F * Float64(A * Float64(C * C))))));
	elseif ((B_m ^ 2.0) <= 5e+251)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) * sqrt(F)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	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], 4e-276], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-110], N[(N[(-1.0 / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-16.0 * N[(F * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+251], N[(N[(N[Sqrt[N[(2.0 * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4e-276

   1. Initial program 24.7%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 31.6

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

   5. Applied rewrites 31.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 29.4%

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

   9. Applied rewrites 29.4%

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

   10. Taylor expanded in A around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 26.8

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

   12. Applied rewrites 26.8%

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

   if 4e-276 < (pow.f64 B #s(literal 2 binary64)) < 5e-110

   1. Initial program 14.1%

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

   4. Applied rewrites 14.2%

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

   5. Taylor expanded in B around -inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 3.7

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      6. lower-*.f64 24.4

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   10. Applied rewrites 24.4%

       \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   if 5e-110 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000005e251

   1. Initial program 25.6%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 18.7%

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

   7. Applied rewrites 21.5%

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

   if 5.0000000000000005e251 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.4%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 24.9

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

   5. Applied rewrites 24.9%

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

   7. Applied rewrites 37.1%

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

   9. Applied rewrites 37.2%

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

   11. Applied rewrites 37.2%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot F\right)}\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-110}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \sqrt{F}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \sqrt{F}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e-24)
     (/ (sqrt (* (* 2.0 C) (* (* 2.0 F) t_0))) (- t_0))
     (if (<= (pow B_m 2.0) 5e+251)
       (/
        (* (sqrt (* 2.0 (+ C (sqrt (fma B_m B_m (* C C)))))) (sqrt F))
        (- B_m))
       (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))))

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(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-24) {
		tmp = sqrt(((2.0 * C) * ((2.0 * F) * t_0))) / -t_0;
	} else if (pow(B_m, 2.0) <= 5e+251) {
		tmp = (sqrt((2.0 * (C + sqrt(fma(B_m, B_m, (C * C)))))) * sqrt(F)) / -B_m;
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	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(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-24)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 5e+251)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) * sqrt(F)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-24], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+251], N[(N[(N[Sqrt[N[(2.0 * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e-24

   1. Initial program 22.1%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 27.4

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

   5. Applied rewrites 27.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 27.4%

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

   if 4.9999999999999998e-24 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000005e251

   1. Initial program 25.3%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 23.5%

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

   if 5.0000000000000005e251 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.4%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 24.9

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

   5. Applied rewrites 24.9%

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

   7. Applied rewrites 37.1%

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

   9. Applied rewrites 37.2%

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

   11. Applied rewrites 37.2%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \sqrt{F}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-110}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \sqrt{F}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-110)
   (*
    (/ -1.0 (fma B_m B_m (* -4.0 (* A C))))
    (sqrt (* -16.0 (* A (* F (* C C))))))
   (if (<= (pow B_m 2.0) 5e+251)
     (/ (* (sqrt (* 2.0 (+ C (sqrt (fma B_m B_m (* C C)))))) (sqrt F)) (- B_m))
     (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-110) {
		tmp = (-1.0 / fma(B_m, B_m, (-4.0 * (A * C)))) * sqrt((-16.0 * (A * (F * (C * C)))));
	} else if (pow(B_m, 2.0) <= 5e+251) {
		tmp = (sqrt((2.0 * (C + sqrt(fma(B_m, B_m, (C * C)))))) * sqrt(F)) / -B_m;
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-110)
		tmp = Float64(Float64(-1.0 / fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(-16.0 * Float64(A * Float64(F * Float64(C * C))))));
	elseif ((B_m ^ 2.0) <= 5e+251)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) * sqrt(F)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-110], N[(N[(-1.0 / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-16.0 * N[(A * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+251], N[(N[(N[Sqrt[N[(2.0 * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e-110

   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 20.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      5. lower-*.f64 22.7

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   7. Applied rewrites 22.7%

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   if 5e-110 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000005e251

   1. Initial program 25.6%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 18.7%

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

   7. Applied rewrites 21.5%

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

   if 5.0000000000000005e251 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.4%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 24.9

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

   5. Applied rewrites 24.9%

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

   7. Applied rewrites 37.1%

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

   9. Applied rewrites 37.2%

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

   11. Applied rewrites 37.2%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-110}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \sqrt{F}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 18500000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 18500000.0)
   (*
    (/ -1.0 (fma B_m B_m (* -4.0 (* A C))))
    (sqrt (* -16.0 (* F (* A (* C C))))))
   (/ (- (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) {
	double tmp;
	if (B_m <= 18500000.0) {
		tmp = (-1.0 / fma(B_m, B_m, (-4.0 * (A * C)))) * sqrt((-16.0 * (F * (A * (C * C)))));
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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 <= 18500000.0)
		tmp = Float64(Float64(-1.0 / fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(-16.0 * Float64(F * Float64(A * Float64(C * C))))));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 18500000.0], N[(N[(-1.0 / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-16.0 * N[(F * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.85e7

   1. Initial program 17.2%

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

   4. Applied rewrites 16.8%

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

   5. Taylor expanded in B around -inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 4.3

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

   7. Applied rewrites 4.3%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      6. lower-*.f64 14.1

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   10. Applied rewrites 14.1%

       \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   if 1.85e7 < B

   1. Initial program 12.6%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 48.4

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

   5. Applied rewrites 48.4%

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

   7. Applied rewrites 48.7%

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

   9. Applied rewrites 48.6%

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

   11. Applied rewrites 67.0%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 18500000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6e-53)
   (*
    (/ -1.0 (fma B_m B_m (* -4.0 (* A C))))
    (sqrt (* -16.0 (* A (* F (* C C))))))
   (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6e-53) {
		tmp = (-1.0 / fma(B_m, B_m, (-4.0 * (A * C)))) * sqrt((-16.0 * (A * (F * (C * C)))));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	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 <= 6e-53)
		tmp = Float64(Float64(-1.0 / fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(-16.0 * Float64(A * Float64(F * Float64(C * C))))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6e-53], N[(N[(-1.0 / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-16.0 * N[(A * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.0000000000000004e-53

   1. Initial program 15.6%

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

   4. Applied rewrites 15.2%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

      5. lower-*.f64 14.4

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   7. Applied rewrites 14.4%

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   if 6.0000000000000004e-53 < B

   1. Initial program 17.4%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 41.9

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

   5. Applied rewrites 41.9%

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

   7. Applied rewrites 58.4%

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

   9. Applied rewrites 58.4%

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

   11. Applied rewrites 58.5%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.9% accurate, 11.2× speedup

Math

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

FPCore

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

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((1.0 / B_m)) * -sqrt((2.0 * F));
}

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((1.0d0 / b_m)) * -sqrt((2.0d0 * f))
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((1.0 / B_m)) * -Math.sqrt((2.0 * F));
}

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((1.0 / B_m)) * -math.sqrt((2.0 * F))

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(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))))
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((1.0 / B_m)) * -sqrt((2.0 * F));
end

Wolfram

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

Derivation

1. Initial program 16.2%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.9

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

5. Applied rewrites 13.9%

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

7. Applied rewrites 18.1%

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

9. Applied rewrites 18.1%

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

11. Applied rewrites 18.1%

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

Alternative 10: 36.9% 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 16.2%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.9

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

5. Applied rewrites 13.9%

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

7. Applied rewrites 14.0%

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

9. Applied rewrites 14.0%

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

11. Applied rewrites 18.1%

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

12. Final simplification 18.1%

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

Alternative 11: 36.9% accurate, 12.6× speedup

Math

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

FPCore

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

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 / B_m)) * -sqrt(F);
}

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 / b_m)) * -sqrt(f)
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 / B_m)) * -Math.sqrt(F);
}

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 / B_m)) * -math.sqrt(F)

Julia

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

Wolfram

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

Derivation

1. Initial program 16.2%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.9

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

5. Applied rewrites 13.9%

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

7. Applied rewrites 18.1%

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

9. Applied rewrites 18.1%

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

10. Final simplification 18.1%

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

Alternative 12: 28.6% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.2%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.9

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

5. Applied rewrites 13.9%

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

7. Applied rewrites 14.0%

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

9. Applied rewrites 14.0%

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

11. Applied rewrites 26.3%

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

Alternative 13: 28.4% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.2%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.9

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

5. Applied rewrites 13.9%

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

7. Applied rewrites 14.0%

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

Alternative 14: 28.4% 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 16.2%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.9

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

5. Applied rewrites 13.9%

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

7. Applied rewrites 14.0%

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

9. Applied rewrites 14.0%

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

Alternative 15: 2.4% accurate, 18.2× speedup

Math

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

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

Derivation

1. Initial program 16.2%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.9

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

5. Applied rewrites 13.9%

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

7. Applied rewrites 18.1%

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

9. Applied rewrites 18.1%

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

11. Applied rewrites 1.9%

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

Reproduce

herbie shell --seed 2024221 
(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))))