ABCF->ab-angle b

Percentage Accurate: 18.9% → 48.2%

Time: 14.6s

Alternatives: 11

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 48.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A + A\right) \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;-\sqrt{\left(\frac{\left(C + A\right) - \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)} \cdot F\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 2.15e-51)
     (/
      (sqrt (* t_0 (* (fma (/ (* B_m B_m) C) -0.5 (+ A A)) (* 2.0 F))))
      (- t_0))
     (if (<= B_m 1.75e+112)
       (-
        (sqrt
         (*
          (*
           (/ (- (+ C A) (hypot B_m (- A C))) (fma (* C A) -4.0 (* B_m B_m)))
           F)
          2.0)))
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* (- A (hypot B_m A)) 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((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 2.15e-51) {
		tmp = sqrt((t_0 * (fma(((B_m * B_m) / C), -0.5, (A + A)) * (2.0 * F)))) / -t_0;
	} else if (B_m <= 1.75e+112) {
		tmp = -sqrt((((((C + A) - hypot(B_m, (A - C))) / fma((C * A), -4.0, (B_m * B_m))) * F) * 2.0));
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt(((A - hypot(B_m, A)) * 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(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 2.15e-51)
		tmp = Float64(sqrt(Float64(t_0 * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, Float64(A + A)) * Float64(2.0 * F)))) / Float64(-t_0));
	elseif (B_m <= 1.75e+112)
		tmp = Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(C + A) - hypot(B_m, Float64(A - C))) / fma(Float64(C * A), -4.0, Float64(B_m * B_m))) * F) * 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(Float64(A - hypot(B_m, A)) * 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[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.15e-51], N[(N[Sqrt[N[(t$95$0 * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + N[(A + A), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 1.75e+112], (-N[Sqrt[N[(N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.1499999999999999e-51

   1. Initial program 19.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. lower-neg.f64 16.6

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

   5. Applied rewrites 16.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 16.6%

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

   8. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 16.8

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

   10. Applied rewrites 16.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 16.9

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

   12. Applied rewrites 16.9%

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

   if 2.1499999999999999e-51 < B < 1.74999999999999998e112

   1. Initial program 30.9%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 51.5%

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

   7. Applied rewrites 51.8%

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

   if 1.74999999999999998e112 < B

   1. Initial program 6.1%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 44.1

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

   5. Applied rewrites 44.1%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A + A\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;-\sqrt{\left(\frac{\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot F\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 43.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\left(F \cdot 2\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\frac{C + A}{B\_m} - 1}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e+68)
     (/ (sqrt (* (+ A A) (* (* F 2.0) t_0))) (- t_0))
     (* (- (sqrt 2.0)) (sqrt (* F (/ (- (/ (+ C A) B_m) 1.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+68) {
		tmp = sqrt(((A + A) * ((F * 2.0) * t_0))) / -t_0;
	} else {
		tmp = -sqrt(2.0) * sqrt((F * ((((C + A) / B_m) - 1.0) / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+68)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(Float64(F * 2.0) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(F * Float64(Float64(Float64(Float64(C + A) / B_m) - 1.0) / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+68], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e68

   1. Initial program 23.0%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. lower-neg.f64 19.4

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

   5. Applied rewrites 19.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 19.4%

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

   if 5.0000000000000004e68 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.9%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 30.0%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 28.8%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\frac{C + A}{B} - 1}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\frac{C + A}{B\_m} - 1}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e-102)
   (/
    (sqrt (* (* -4.0 (* A C)) (* (* F 2.0) (+ A A))))
    (- (fma (* -4.0 C) A (* B_m B_m))))
   (* (- (sqrt 2.0)) (sqrt (* F (/ (- (/ (+ C A) B_m) 1.0) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 4e-102) {
		tmp = sqrt(((-4.0 * (A * C)) * ((F * 2.0) * (A + A)))) / -fma((-4.0 * C), A, (B_m * B_m));
	} else {
		tmp = -sqrt(2.0) * sqrt((F * ((((C + A) / B_m) - 1.0) / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-102)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(A * C)) * Float64(Float64(F * 2.0) * Float64(A + A)))) / Float64(-fma(Float64(-4.0 * C), A, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(F * Float64(Float64(Float64(Float64(C + A) / B_m) - 1.0) / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-102], N[(N[Sqrt[N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999973e-102

   1. Initial program 20.9%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. lower-neg.f64 24.1

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

   5. Applied rewrites 24.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 24.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 24.2%

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

   10. Taylor expanded in A around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 23.5

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

   12. Applied rewrites 23.5%

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

   if 3.99999999999999973e-102 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 24.5%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\frac{C + A}{B} - 1}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \frac{-0.5}{C}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \frac{\frac{C + A}{B\_m} - 1}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 2e+57)
     (* t_0 (sqrt (* F (/ -0.5 C))))
     (* t_0 (sqrt (* F (/ (- (/ (+ C A) B_m) 1.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e+57) {
		tmp = t_0 * sqrt((F * (-0.5 / C)));
	} else {
		tmp = t_0 * sqrt((F * ((((C + A) / B_m) - 1.0) / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if ((b_m ** 2.0d0) <= 2d+57) then
        tmp = t_0 * sqrt((f * ((-0.5d0) / c)))
    else
        tmp = t_0 * sqrt((f * ((((c + a) / b_m) - 1.0d0) / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e+57) {
		tmp = t_0 * Math.sqrt((F * (-0.5 / C)));
	} else {
		tmp = t_0 * Math.sqrt((F * ((((C + A) / B_m) - 1.0) / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e+57:
		tmp = t_0 * math.sqrt((F * (-0.5 / C)))
	else:
		tmp = t_0 * math.sqrt((F * ((((C + A) / B_m) - 1.0) / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+57)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(-0.5 / C))));
	else
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(Float64(Float64(Float64(C + A) / B_m) - 1.0) / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e+57)
		tmp = t_0 * sqrt((F * (-0.5 / C)));
	else
		tmp = t_0 * sqrt((F * ((((C + A) / B_m) - 1.0) / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+57], N[(t$95$0 * N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(F * N[(N[(N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e57

   1. Initial program 22.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 16.7%

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

   if 2.0000000000000001e57 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 30.4%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 28.5%

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\frac{C + A}{B} - 1}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 46.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A - \mathsf{hypot}\left(B\_m, A\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A + A\right) \cdot \left(2 \cdot F\right)\right)}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{t\_0}{B\_m \cdot B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{t\_0 \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- A (hypot B_m A))) (t_1 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 2.15e-51)
     (/
      (sqrt (* t_1 (* (fma (/ (* B_m B_m) C) -0.5 (+ A A)) (* 2.0 F))))
      (- t_1))
     (if (<= B_m 1.75e+112)
       (* (- (sqrt 2.0)) (sqrt (* F (/ t_0 (* B_m B_m)))))
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* t_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 = A - hypot(B_m, A);
	double t_1 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 2.15e-51) {
		tmp = sqrt((t_1 * (fma(((B_m * B_m) / C), -0.5, (A + A)) * (2.0 * F)))) / -t_1;
	} else if (B_m <= 1.75e+112) {
		tmp = -sqrt(2.0) * sqrt((F * (t_0 / (B_m * B_m))));
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt((t_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 = Float64(A - hypot(B_m, A))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 2.15e-51)
		tmp = Float64(sqrt(Float64(t_1 * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, Float64(A + A)) * Float64(2.0 * F)))) / Float64(-t_1));
	elseif (B_m <= 1.75e+112)
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(F * Float64(t_0 / Float64(B_m * B_m)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(t_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[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.15e-51], N[(N[Sqrt[N[(t$95$1 * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + N[(A + A), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[B$95$m, 1.75e+112], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(F * N[(t$95$0 / N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.1499999999999999e-51

   1. Initial program 19.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. lower-neg.f64 16.6

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

   5. Applied rewrites 16.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 16.6%

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

   8. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 16.8

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

   10. Applied rewrites 16.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 16.9

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

   12. Applied rewrites 16.9%

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

   if 2.1499999999999999e-51 < B < 1.74999999999999998e112

   1. Initial program 30.9%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in C around 0

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

   8. Applied rewrites 42.8%

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

   if 1.74999999999999998e112 < B

   1. Initial program 6.1%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 44.1

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

   5. Applied rewrites 44.1%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A + A\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{A - \mathsf{hypot}\left(B, A\right)}{B \cdot B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A + A\right) \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 2.15e-51)
     (/
      (sqrt (* t_0 (* (fma (/ (* B_m B_m) C) -0.5 (+ A A)) (* 2.0 F))))
      (- t_0))
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* (- A (hypot B_m A)) 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((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 2.15e-51) {
		tmp = sqrt((t_0 * (fma(((B_m * B_m) / C), -0.5, (A + A)) * (2.0 * F)))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt(((A - hypot(B_m, A)) * 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(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 2.15e-51)
		tmp = Float64(sqrt(Float64(t_0 * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, Float64(A + A)) * Float64(2.0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(Float64(A - hypot(B_m, A)) * 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[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.15e-51], N[(N[Sqrt[N[(t$95$0 * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + N[(A + A), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.1499999999999999e-51

   1. Initial program 19.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. lower-neg.f64 16.6

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

   5. Applied rewrites 16.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 16.6%

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

   8. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 16.8

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

   10. Applied rewrites 16.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 16.9

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

   12. Applied rewrites 16.9%

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

   if 2.1499999999999999e-51 < B

   1. Initial program 19.2%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 39.6

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

   5. Applied rewrites 39.6%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A + A\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \frac{-0.5}{C}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \frac{-1}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 2e+57)
     (* t_0 (sqrt (* F (/ -0.5 C))))
     (* t_0 (sqrt (* F (/ -1.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e+57) {
		tmp = t_0 * sqrt((F * (-0.5 / C)));
	} else {
		tmp = t_0 * sqrt((F * (-1.0 / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if ((b_m ** 2.0d0) <= 2d+57) then
        tmp = t_0 * sqrt((f * ((-0.5d0) / c)))
    else
        tmp = t_0 * sqrt((f * ((-1.0d0) / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e+57) {
		tmp = t_0 * Math.sqrt((F * (-0.5 / C)));
	} else {
		tmp = t_0 * Math.sqrt((F * (-1.0 / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e+57:
		tmp = t_0 * math.sqrt((F * (-0.5 / C)))
	else:
		tmp = t_0 * math.sqrt((F * (-1.0 / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+57)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(-0.5 / C))));
	else
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(-1.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e+57)
		tmp = t_0 * sqrt((F * (-0.5 / C)));
	else
		tmp = t_0 * sqrt((F * (-1.0 / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+57], N[(t$95$0 * N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(F * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e57

   1. Initial program 22.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 16.7%

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

   if 2.0000000000000001e57 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 30.4%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 26.6%

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

Alternative 8: 27.1% accurate, 11.2× speedup

Math

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

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)) (sqrt (* F (/ -1.0 B_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.6%

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

3. Taylor expanded in F around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 30.3%

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

6. Taylor expanded in B around inf

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

8. Applied rewrites 14.2%

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

Alternative 9: 1.7% accurate, 12.9× speedup

Math

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

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 (/ (fma F B_m (* B_m F)) (* B_m B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((fma(F, B_m, (B_m * F)) / (B_m * 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 sqrt(Float64(fma(F, B_m, Float64(B_m * F)) / Float64(B_m * 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[(F * B$95$m + N[(B$95$m * F), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 19.6%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 2.0

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

5. Applied rewrites 2.0%

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

7. Applied rewrites 2.0%

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

9. Applied rewrites 1.9%

   \[\leadsto \sqrt{\frac{\mathsf{fma}\left(F, B, B \cdot F\right)}{B \cdot B}} \]
10. Add Preprocessing

Alternative 10: 1.6% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.6%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 2.0

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

5. Applied rewrites 2.0%

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

7. Applied rewrites 2.0%

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

Alternative 11: 1.6% accurate, 18.2× speedup

Math

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

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 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 19.6%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 2.0

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

5. Applied rewrites 2.0%

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

7. Applied rewrites 2.0%

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

9. Applied rewrites 2.0%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))