ABCF->ab-angle a

Percentage Accurate: 18.6% → 57.7%

Time: 24.6s

Alternatives: 12

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.6% 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: 57.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -t\_0\\ \mathbf{if}\;B\_m \leq 6 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(C \cdot 4\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 1.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (- t_0)))
   (if (<= B_m 6e-178)
     (/ (sqrt (* (* t_0 F) (* C 4.0))) t_1)
     (if (<= B_m 3.8e-80)
       (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
       (if (<= B_m 1.15e+54)
         (/
          (*
           (sqrt (* 2.0 (* F (fma B_m B_m (* -4.0 (* A C))))))
           (sqrt (+ A (+ C (hypot (- A C) B_m)))))
          t_1)
         (*
          (* (sqrt (+ C (hypot B_m C))) (sqrt F))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -t_0;
	double tmp;
	if (B_m <= 6e-178) {
		tmp = sqrt(((t_0 * F) * (C * 4.0))) / t_1;
	} else if (B_m <= 3.8e-80) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else if (B_m <= 1.15e+54) {
		tmp = (sqrt((2.0 * (F * fma(B_m, B_m, (-4.0 * (A * C)))))) * sqrt((A + (C + hypot((A - C), B_m))))) / t_1;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-t_0)
	tmp = 0.0
	if (B_m <= 6e-178)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(C * 4.0))) / t_1);
	elseif (B_m <= 3.8e-80)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 1.15e+54)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))))) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[B$95$m, 6e-178], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 3.8e-80], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 1.15e+54], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 5.9999999999999997e-178

   1. Initial program 15.8%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in A around -inf 17.1%

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

      1. *-commutative 17.1%

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

   7. Simplified 17.1%

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

   if 5.9999999999999997e-178 < B < 3.79999999999999967e-80

   1. Initial program 15.2%

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

   3. Taylor expanded in C around inf 9.3%

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

      1. associate-*r/ 9.3%

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

      2. mul-1-neg 9.3%

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

   5. Simplified 9.3%

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

   6. Taylor expanded in F around 0 10.0%

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

      1. mul-1-neg 10.0%

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

      2. mul-1-neg 10.0%

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

      3. distribute-frac-neg 10.0%

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

      4. cancel-sign-sub-inv 10.0%

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

      5. metadata-eval 10.0%

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

   8. Simplified 10.0%

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

   9. Taylor expanded in A around inf 27.5%

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

   if 3.79999999999999967e-80 < B < 1.14999999999999997e54

   1. Initial program 38.4%

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

   3. Simplified 39.2%

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

      1. associate-*r* 39.2%

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

      2. associate-+r+ 38.6%

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

      3. hypot-undefine 38.4%

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

      4. unpow2 38.4%

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

      5. unpow2 38.4%

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

      6. +-commutative 38.4%

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

      7. sqrt-prod 42.1%

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

      8. *-commutative 42.1%

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

      9. associate-*r* 42.1%

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

      10. associate-+l+ 42.1%

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

   6. Applied egg-rr 46.9%

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

   if 1.14999999999999997e54 < B

   1. Initial program 12.3%

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

   3. Taylor expanded in A around 0 22.5%

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

      1. mul-1-neg 22.5%

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

      2. unpow2 22.5%

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

      3. unpow2 22.5%

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

      4. hypot-define 52.5%

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

   5. Simplified 52.5%

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

      1. pow1/2 52.5%

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

      2. *-commutative 52.5%

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

      3. unpow-prod-down 69.4%

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

      4. pow1/2 69.4%

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

      5. pow1/2 69.4%

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

   7. Applied egg-rr 69.4%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 1.85 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(C \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 8.6 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 6.6 \cdot 10^{+38}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 1.85e-178)
     (/ (sqrt (* (* t_0 F) (* C 4.0))) (- t_0))
     (if (<= B_m 8.6e-69)
       (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
       (if (<= B_m 6.6e+38)
         (-
          (sqrt
           (*
            2.0
            (*
             F
             (/
              (+ (+ A C) (hypot B_m (- A C)))
              (fma C (* A -4.0) (pow B_m 2.0)))))))
         (*
          (* (sqrt (+ C (hypot B_m C))) (sqrt F))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 1.85e-178) {
		tmp = sqrt(((t_0 * F) * (C * 4.0))) / -t_0;
	} else if (B_m <= 8.6e-69) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else if (B_m <= 6.6e+38) {
		tmp = -sqrt((2.0 * (F * (((A + C) + hypot(B_m, (A - C))) / fma(C, (A * -4.0), pow(B_m, 2.0))))));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 1.85e-178)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(C * 4.0))) / Float64(-t_0));
	elseif (B_m <= 8.6e-69)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 6.6e+38)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(C, Float64(A * -4.0), (B_m ^ 2.0)))))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.85e-178], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 8.6e-69], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 6.6e+38], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.85000000000000002e-178

   1. Initial program 15.8%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in A around -inf 17.1%

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

      1. *-commutative 17.1%

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

   7. Simplified 17.1%

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

   if 1.85000000000000002e-178 < B < 8.59999999999999999e-69

   1. Initial program 18.4%

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

   3. Taylor expanded in C around inf 8.1%

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

      1. associate-*r/ 8.1%

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

      2. mul-1-neg 8.1%

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

   5. Simplified 8.1%

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

   6. Taylor expanded in F around 0 8.8%

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

      1. mul-1-neg 8.8%

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

      2. mul-1-neg 8.8%

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

      3. distribute-frac-neg 8.8%

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

      4. cancel-sign-sub-inv 8.8%

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

      5. metadata-eval 8.8%

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

   8. Simplified 8.8%

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

   9. Taylor expanded in A around inf 28.6%

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

   if 8.59999999999999999e-69 < B < 6.5999999999999998e38

   1. Initial program 39.7%

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

   3. Taylor expanded in F around 0 44.7%

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

   5. Simplified 51.8%

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

      1. *-commutative 51.8%

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

      2. pow1/2 51.8%

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

      3. pow1/2 51.8%

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

      4. pow-prod-down 52.2%

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

      5. associate-+r+ 52.2%

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

      6. +-commutative 52.2%

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

      7. fma-undefine 52.2%

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

      8. *-commutative 52.2%

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

      9. associate-*l* 52.2%

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

      10. fma-define 52.2%

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

   7. Applied egg-rr 52.2%

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

      1. unpow1/2 52.2%

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

   9. Simplified 52.2%

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

   if 6.5999999999999998e38 < B

   1. Initial program 13.2%

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

   3. Taylor expanded in A around 0 23.1%

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

      1. mul-1-neg 23.1%

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

      2. unpow2 23.1%

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

      3. unpow2 23.1%

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

      4. hypot-define 51.9%

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

   5. Simplified 51.9%

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

      1. pow1/2 51.9%

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

      2. *-commutative 51.9%

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

      3. unpow-prod-down 68.1%

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

      4. pow1/2 68.1%

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

      5. pow1/2 68.1%

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

   7. Applied egg-rr 68.1%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.85 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{+38}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 1.55 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(C \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 1.55 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 1.55e-178)
     (/ (sqrt (* (* t_0 F) (* C 4.0))) (- t_0))
     (if (<= B_m 1.55e-65)
       (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
       (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 1.55e-178) {
		tmp = sqrt(((t_0 * F) * (C * 4.0))) / -t_0;
	} else if (B_m <= 1.55e-65) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 1.55e-178)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(C * 4.0))) / Float64(-t_0));
	elseif (B_m <= 1.55e-65)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.55e-178], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 1.55e-65], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.55e-178

   1. Initial program 15.8%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in A around -inf 17.1%

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

      1. *-commutative 17.1%

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

   7. Simplified 17.1%

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

   if 1.55e-178 < B < 1.55000000000000008e-65

   1. Initial program 18.4%

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

   3. Taylor expanded in C around inf 8.1%

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

      1. associate-*r/ 8.1%

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

      2. mul-1-neg 8.1%

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

   5. Simplified 8.1%

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

   6. Taylor expanded in F around 0 8.8%

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

      1. mul-1-neg 8.8%

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

      2. mul-1-neg 8.8%

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

      3. distribute-frac-neg 8.8%

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

      4. cancel-sign-sub-inv 8.8%

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

      5. metadata-eval 8.8%

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

   8. Simplified 8.8%

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

   9. Taylor expanded in A around inf 28.6%

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

   if 1.55000000000000008e-65 < B

   1. Initial program 18.5%

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

   3. Taylor expanded in A around 0 25.0%

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

      1. mul-1-neg 25.0%

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

      2. unpow2 25.0%

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

      3. unpow2 25.0%

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

      4. hypot-define 48.1%

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

   5. Simplified 48.1%

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

      1. pow1/2 48.1%

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

      2. *-commutative 48.1%

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

      3. unpow-prod-down 61.0%

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

      4. pow1/2 61.0%

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

      5. pow1/2 61.0%

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

   7. Applied egg-rr 61.0%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.55 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(C \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 6.4 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 1.1e-178)
     (/ (sqrt (* (* t_0 F) (* C 4.0))) (- t_0))
     (if (<= B_m 6.4e-26)
       (* (sqrt (* -0.5 (/ F A))) (- (sqrt 2.0)))
       (* (sqrt (/ 2.0 B_m)) (- (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 1.1e-178) {
		tmp = sqrt(((t_0 * F) * (C * 4.0))) / -t_0;
	} else if (B_m <= 6.4e-26) {
		tmp = sqrt((-0.5 * (F / A))) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 / B_m)) * -sqrt(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(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 1.1e-178)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(C * 4.0))) / Float64(-t_0));
	elseif (B_m <= 6.4e-26)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.1e-178], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 6.4e-26], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.1000000000000001e-178

   1. Initial program 15.8%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in A around -inf 17.1%

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

      1. *-commutative 17.1%

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

   7. Simplified 17.1%

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

   if 1.1000000000000001e-178 < B < 6.4000000000000002e-26

   1. Initial program 18.9%

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

   3. Taylor expanded in C around inf 6.6%

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

      1. associate-*r/ 6.6%

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

      2. mul-1-neg 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in F around 0 7.8%

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

      1. mul-1-neg 7.8%

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

      2. mul-1-neg 7.8%

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

      3. distribute-frac-neg 7.8%

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

      4. cancel-sign-sub-inv 7.8%

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

      5. metadata-eval 7.8%

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

   8. Simplified 7.8%

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

   9. Taylor expanded in A around inf 22.9%

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

   if 6.4000000000000002e-26 < B

   1. Initial program 18.4%

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

   3. Taylor expanded in B around inf 43.8%

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

      1. mul-1-neg 43.8%

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

      2. *-commutative 43.8%

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

   5. Simplified 43.8%

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

      1. *-commutative 43.8%

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

      2. pow1/2 43.8%

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

      3. pow1/2 43.8%

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

      4. pow-prod-down 43.9%

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

   7. Applied egg-rr 43.9%

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

      1. unpow1/2 43.9%

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

   9. Simplified 43.9%

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

       1. *-un-lft-identity 43.9%

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

       2. associate-*l/ 43.9%

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

   11. Applied egg-rr 43.9%

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

       1. *-lft-identity 43.9%

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

   13. Simplified 43.9%

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

       1. pow1/2 43.9%

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

       2. associate-/l* 43.9%

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

       3. unpow-prod-down 60.2%

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

       4. pow1/2 60.2%

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

   15. Applied egg-rr 60.2%

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

       1. unpow1/2 60.2%

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

   17. Simplified 60.2%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7e+69)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
   (/ (sqrt (* F 2.0)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7e+69) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 7d+69) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / a)))
    else
        tmp = sqrt((f * 2.0d0)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7e+69) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else {
		tmp = Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7e+69:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	else:
		tmp = math.sqrt((F * 2.0)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7e+69)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 7e+69)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	else
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7e+69], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.99999999999999974e69

   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 19.9%

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

   5. Simplified 28.6%

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

   6. Taylor expanded in A around -inf 17.9%

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

   if 6.99999999999999974e69 < B

   1. Initial program 11.8%

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

   3. Taylor expanded in B around inf 46.4%

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

      1. mul-1-neg 46.4%

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

      2. *-commutative 46.4%

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

   5. Simplified 46.4%

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

      1. *-commutative 46.4%

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

      2. pow1/2 46.4%

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

      3. pow1/2 46.4%

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

      4. pow-prod-down 46.5%

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

   7. Applied egg-rr 46.5%

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

      1. unpow1/2 46.5%

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

   9. Simplified 46.5%

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

       1. sqrt-prod 46.4%

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

       2. sqrt-undiv 68.2%

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

       3. *-commutative 68.2%

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

       4. associate-*r/ 68.2%

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

       5. pow1/2 68.2%

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

       6. pow1/2 68.2%

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

       7. pow-prod-down 68.4%

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

   11. Applied egg-rr 68.4%

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

       1. unpow1/2 68.4%

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

   13. Simplified 68.4%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

3. Taylor expanded in B around inf 16.2%

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

   1. mul-1-neg 16.2%

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

   2. *-commutative 16.2%

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

5. Simplified 16.2%

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

   1. *-commutative 16.2%

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

   2. pow1/2 16.3%

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

   3. pow1/2 16.3%

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

   4. pow-prod-down 16.4%

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

7. Applied egg-rr 16.4%

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

   1. unpow1/2 16.3%

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

9. Simplified 16.3%

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

    1. *-un-lft-identity 16.3%

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

    2. associate-*l/ 16.3%

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

11. Applied egg-rr 16.3%

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

    1. *-lft-identity 16.3%

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

13. Simplified 16.3%

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

    1. pow1/2 16.4%

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

    2. associate-/l* 16.4%

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

    3. unpow-prod-down 21.3%

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

    4. pow1/2 21.3%

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

15. Applied egg-rr 21.3%

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

    1. unpow1/2 21.3%

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

17. Simplified 21.3%

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

18. Final simplification 21.3%

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

Alternative 7: 28.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.4 \cdot 10^{+93}:\\ \;\;\;\;-{\left(\frac{F \cdot 2}{B\_m}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{2}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 1.4e+93)
   (- (pow (/ (* F 2.0) B_m) 0.5))
   (* (sqrt (* C 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) {
	double tmp;
	if (C <= 1.4e+93) {
		tmp = -pow(((F * 2.0) / B_m), 0.5);
	} else {
		tmp = sqrt((C * F)) * (2.0 / -B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 1.4d+93) then
        tmp = -(((f * 2.0d0) / b_m) ** 0.5d0)
    else
        tmp = sqrt((c * f)) * (2.0d0 / -b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 1.4e+93) {
		tmp = -Math.pow(((F * 2.0) / B_m), 0.5);
	} else {
		tmp = Math.sqrt((C * F)) * (2.0 / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 1.4e+93:
		tmp = -math.pow(((F * 2.0) / B_m), 0.5)
	else:
		tmp = math.sqrt((C * F)) * (2.0 / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 1.4e+93)
		tmp = Float64(-(Float64(Float64(F * 2.0) / B_m) ^ 0.5));
	else
		tmp = Float64(sqrt(Float64(C * F)) * Float64(2.0 / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 1.4e+93)
		tmp = -(((F * 2.0) / B_m) ^ 0.5);
	else
		tmp = sqrt((C * F)) * (2.0 / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 1.4e+93], (-N[Power[N[(N[(F * 2.0), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision]), N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(2.0 / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.39999999999999994e93

   1. Initial program 18.2%

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

   3. Taylor expanded in B around inf 17.7%

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

      1. mul-1-neg 17.7%

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

      2. *-commutative 17.7%

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

   5. Simplified 17.7%

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

      1. *-commutative 17.7%

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

      2. pow1/2 17.8%

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

      3. pow1/2 17.8%

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

      4. pow-prod-down 17.9%

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

   7. Applied egg-rr 17.9%

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

      1. unpow1/2 17.7%

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

   9. Simplified 17.7%

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

       1. pow1/2 17.9%

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

       2. associate-*l/ 17.9%

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

   11. Applied egg-rr 17.9%

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

   if 1.39999999999999994e93 < C

   1. Initial program 9.3%

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

   3. Taylor expanded in A around 0 6.5%

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

      1. mul-1-neg 6.5%

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

      2. unpow2 6.5%

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

      3. unpow2 6.5%

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

      4. hypot-define 22.7%

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

   5. Simplified 22.7%

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

      1. pow1/2 23.0%

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

      2. *-commutative 23.0%

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

      3. unpow-prod-down 25.4%

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

      4. pow1/2 25.4%

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

      5. pow1/2 25.4%

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

   7. Applied egg-rr 25.4%

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

      1. div-inv 25.4%

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

   9. Applied egg-rr 25.4%

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

   10. Taylor expanded in B around 0 11.1%

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

       1. unpow2 11.1%

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

       2. rem-square-sqrt 11.1%

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

   12. Simplified 11.1%

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

4. Final simplification 16.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.4 \cdot 10^{+93}:\\ \;\;\;\;-{\left(\frac{F \cdot 2}{B}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{2}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

3. Taylor expanded in B around inf 16.2%

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

   1. mul-1-neg 16.2%

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

   2. *-commutative 16.2%

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

5. Simplified 16.2%

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

   1. *-commutative 16.2%

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

   2. pow1/2 16.3%

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

   3. pow1/2 16.3%

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

   4. pow-prod-down 16.4%

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

7. Applied egg-rr 16.4%

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

   1. unpow1/2 16.3%

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

9. Simplified 16.3%

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

    1. pow1/2 16.4%

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

    2. associate-*l/ 16.4%

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

11. Applied egg-rr 16.4%

    \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]
12. Add Preprocessing

Alternative 9: 27.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

3. Taylor expanded in B around inf 16.2%

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

   1. mul-1-neg 16.2%

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

   2. *-commutative 16.2%

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

5. Simplified 16.2%

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

   1. *-commutative 16.2%

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

   2. pow1/2 16.3%

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

   3. pow1/2 16.3%

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

   4. pow-prod-down 16.4%

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

7. Applied egg-rr 16.4%

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

8. Final simplification 16.4%

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

Alternative 10: 27.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(Float64(F * 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[(N[(F * 2.0), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 16.9%

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

3. Taylor expanded in B around inf 16.2%

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

   1. mul-1-neg 16.2%

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

   2. *-commutative 16.2%

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

5. Simplified 16.2%

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

   1. *-commutative 16.2%

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

   2. pow1/2 16.3%

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

   3. pow1/2 16.3%

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

   4. pow-prod-down 16.4%

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

7. Applied egg-rr 16.4%

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

   1. unpow1/2 16.3%

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

9. Simplified 16.3%

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

    1. *-un-lft-identity 16.3%

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

    2. associate-*l/ 16.3%

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

11. Applied egg-rr 16.3%

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

    1. *-lft-identity 16.3%

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

13. Simplified 16.3%

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

Alternative 11: 27.6% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

3. Taylor expanded in B around inf 16.2%

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

   1. mul-1-neg 16.2%

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

   2. *-commutative 16.2%

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

5. Simplified 16.2%

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

   1. *-commutative 16.2%

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

   2. pow1/2 16.3%

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

   3. pow1/2 16.3%

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

   4. pow-prod-down 16.4%

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

7. Applied egg-rr 16.4%

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

   1. unpow1/2 16.3%

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

9. Simplified 16.3%

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

10. Final simplification 16.3%

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

Alternative 12: 27.6% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

3. Taylor expanded in B around inf 16.2%

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

   1. mul-1-neg 16.2%

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

   2. *-commutative 16.2%

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

5. Simplified 16.2%

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

   1. *-commutative 16.2%

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

   2. pow1/2 16.3%

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

   3. pow1/2 16.3%

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

   4. pow-prod-down 16.4%

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

7. Applied egg-rr 16.4%

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

   1. unpow1/2 16.3%

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

9. Simplified 16.3%

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

    1. *-un-lft-identity 16.3%

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

    2. associate-*l/ 16.3%

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

11. Applied egg-rr 16.3%

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

    1. *-lft-identity 16.3%

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

13. Simplified 16.3%

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

    1. *-un-lft-identity 16.3%

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

    2. associate-/l* 16.2%

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

15. Applied egg-rr 16.2%

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

    1. *-lft-identity 16.2%

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

17. Simplified 16.2%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))