ABCF->ab-angle b

Percentage Accurate: 21.2% → 54.4%

Time: 23.1s

Alternatives: 14

Speedup: 5.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 21.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 54.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot t\_3}}{-t\_3}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{F \cdot \left(\frac{{B\_m}^{2}}{-{A}^{2}} - 4\right)}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma C (* A -4.0) (pow B_m 2.0))))
   (if (<= t_2 -1e-183)
     (/
      (* (sqrt (* F (+ A (- C (hypot B_m (- A C)))))) (sqrt (* 2.0 t_3)))
      (- t_3))
     (if (<= t_2 2e+216)
       (/
        (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
        (- t_0))
       (if (<= t_2 INFINITY)
         (* 0.5 (sqrt (/ (* F (- (/ (pow B_m 2.0) (- (pow A 2.0))) 4.0)) C)))
         (/ (sqrt (* 2.0 (* F (- C (hypot C B_m))))) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -1e-183) {
		tmp = (sqrt((F * (A + (C - hypot(B_m, (A - C)))))) * sqrt((2.0 * t_3))) / -t_3;
	} else if (t_2 <= 2e+216) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 0.5 * sqrt(((F * ((pow(B_m, 2.0) / -pow(A, 2.0)) - 4.0)) / C));
	} else {
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -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(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (t_2 <= -1e-183)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))) * sqrt(Float64(2.0 * t_3))) / Float64(-t_3));
	elseif (t_2 <= 2e+216)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	elseif (t_2 <= Inf)
		tmp = Float64(0.5 * sqrt(Float64(Float64(F * Float64(Float64((B_m ^ 2.0) / Float64(-(A ^ 2.0))) - 4.0)) / C)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - hypot(C, B_m))))) / 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 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-183], N[(N[(N[Sqrt[N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[t$95$2, 2e+216], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(0.5 * N[Sqrt[N[(N[(F * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / (-N[Power[A, 2.0], $MachinePrecision])), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000001e-183

   1. Initial program 55.7%

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

   3. Simplified 49.6%

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

      1. pow1/2 49.6%

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

      2. associate-*r* 62.3%

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

      3. unpow-prod-down 68.0%

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

      4. associate-+r- 67.8%

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

      5. hypot-undefine 58.6%

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

      6. unpow2 58.6%

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

      7. unpow2 58.6%

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

      8. +-commutative 58.6%

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

      9. unpow2 58.6%

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

      10. unpow2 58.6%

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

      11. hypot-define 67.8%

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

      12. pow1/2 67.8%

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

   6. Applied egg-rr 67.8%

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

      1. unpow1/2 67.8%

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

      2. associate-+r- 68.0%

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

      3. hypot-undefine 58.6%

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

      4. unpow2 58.6%

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

      5. unpow2 58.6%

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

      6. +-commutative 58.6%

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

      7. unpow2 58.6%

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

      8. unpow2 58.6%

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

      9. hypot-undefine 68.0%

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

   8. Simplified 68.0%

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

   if -1.00000000000000001e-183 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 2e216

   1. Initial program 33.7%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in C around inf 32.9%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)}\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 32.9%

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

   7. Simplified 32.9%

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

   if 2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.5%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in B around 0 17.8%

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

   6. Taylor expanded in A around -inf 3.5%

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

   7. Taylor expanded in C around inf 52.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 1.3%

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

      1. mul-1-neg 1.3%

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

      2. +-commutative 1.3%

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

      3. unpow2 1.3%

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

      4. unpow2 1.3%

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

      5. hypot-define 21.6%

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

   5. Simplified 21.6%

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

      1. neg-sub0 21.6%

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

      2. associate-*l/ 21.7%

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

      3. pow1/2 21.7%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 21.7%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 21.8%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 21.8%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 21.8%

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

      2. distribute-neg-frac2 21.8%

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

      3. unpow1/2 21.7%

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

   9. Simplified 21.7%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{F \cdot \left(\frac{{B}^{2}}{-{A}^{2}} - 4\right)}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot t\_0\\ t_2 := -t\_0\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{F \cdot \left(\frac{{B\_m}^{2}}{-{A}^{2}} - 4\right)}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

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 (* F t_0))
        (t_2 (- t_0))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B_m 2.0)))))
   (if (<= t_4 -1e-183)
     (/ (sqrt (* t_1 (* 2.0 (+ A (- C (hypot B_m (- A C))))))) t_2)
     (if (<= t_4 2e+216)
       (/ (sqrt (* t_1 (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C))))))) t_2)
       (if (<= t_4 INFINITY)
         (* 0.5 (sqrt (/ (* F (- (/ (pow B_m 2.0) (- (pow A 2.0))) 4.0)) C)))
         (/ (sqrt (* 2.0 (* F (- C (hypot C B_m))))) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = F * t_0;
	double t_2 = -t_0;
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B_m, 2.0));
	double tmp;
	if (t_4 <= -1e-183) {
		tmp = sqrt((t_1 * (2.0 * (A + (C - hypot(B_m, (A - C))))))) / t_2;
	} else if (t_4 <= 2e+216) {
		tmp = sqrt((t_1 * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = 0.5 * sqrt(((F * ((pow(B_m, 2.0) / -pow(A, 2.0)) - 4.0)) / C));
	} else {
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -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(F * t_0)
	t_2 = Float64(-t_0)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_4 <= -1e-183)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * Float64(A + Float64(C - hypot(B_m, Float64(A - C))))))) / t_2);
	elseif (t_4 <= 2e+216)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(0.5 * sqrt(Float64(Float64(F * Float64(Float64((B_m ^ 2.0) / Float64(-(A ^ 2.0))) - 4.0)) / C)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - hypot(C, B_m))))) / 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 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1e-183], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 2e+216], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(0.5 * N[Sqrt[N[(N[(F * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / (-N[Power[A, 2.0], $MachinePrecision])), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000001e-183

   1. Initial program 55.7%

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

   3. Simplified 63.4%

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

   if -1.00000000000000001e-183 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 2e216

   1. Initial program 33.7%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in C around inf 32.9%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)}\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 32.9%

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

   7. Simplified 32.9%

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

   if 2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.5%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in B around 0 17.8%

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

   6. Taylor expanded in A around -inf 3.5%

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

   7. Taylor expanded in C around inf 52.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 1.3%

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

      1. mul-1-neg 1.3%

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

      2. +-commutative 1.3%

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

      3. unpow2 1.3%

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

      4. unpow2 1.3%

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

      5. hypot-define 21.6%

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

   5. Simplified 21.6%

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

      1. neg-sub0 21.6%

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

      2. associate-*l/ 21.7%

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

      3. pow1/2 21.7%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 21.7%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 21.8%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 21.8%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 21.8%

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

      2. distribute-neg-frac2 21.8%

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

      3. unpow1/2 21.7%

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

   9. Simplified 21.7%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{F \cdot \left(\frac{{B}^{2}}{-{A}^{2}} - 4\right)}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 34.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -1.3 \cdot 10^{-290}:\\ \;\;\;\;0.5 \cdot \sqrt{F \cdot \frac{\mathsf{fma}\left(-1, {\left(\frac{B\_m}{A}\right)}^{2}, -4\right)}{C}}\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -1.3e-290)
   (* 0.5 (sqrt (* F (/ (fma -1.0 (pow (/ B_m A) 2.0) -4.0) C))))
   (if (<= C 1.4e+76)
     (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))
     (/
      (sqrt (* (* 4.0 A) (* -4.0 (* A (* C F)))))
      (- (fma B_m B_m (* A (* C -4.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -1.3e-290) {
		tmp = 0.5 * sqrt((F * (fma(-1.0, pow((B_m / A), 2.0), -4.0) / C)));
	} else if (C <= 1.4e+76) {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	} else {
		tmp = sqrt(((4.0 * A) * (-4.0 * (A * (C * F))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	}
	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 <= -1.3e-290)
		tmp = Float64(0.5 * sqrt(Float64(F * Float64(fma(-1.0, (Float64(B_m / A) ^ 2.0), -4.0) / C))));
	elseif (C <= 1.4e+76)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(-4.0 * Float64(A * Float64(C * F))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -1.3e-290], N[(0.5 * N[Sqrt[N[(F * N[(N[(-1.0 * N[Power[N[(B$95$m / A), $MachinePrecision], 2.0], $MachinePrecision] + -4.0), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 1.4e+76], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.3e-290

   1. Initial program 22.2%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in B around 0 16.6%

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

   6. Taylor expanded in A around -inf 4.2%

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

   7. Taylor expanded in C around inf 17.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{F \cdot \left(-1 \cdot \frac{{B}^{2}}{{A}^{2}} - 4\right)}{C}}} \]
   8. Step-by-step derivation

      1. associate-/l* 17.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{F \cdot \frac{-1 \cdot \frac{{B}^{2}}{{A}^{2}} - 4}{C}}} \]

      2. fmm-def 17.5%

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

      3. metadata-eval 17.5%

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

      4. unpow2 17.5%

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

      5. unpow2 17.5%

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

      6. times-frac 18.7%

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

      7. unpow2 18.7%

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

   9. Simplified 18.7%

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

   if -1.3e-290 < C < 1.3999999999999999e76

   1. Initial program 28.3%

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

   3. Taylor expanded in C around 0 11.5%

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

      1. mul-1-neg 11.5%

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

      2. +-commutative 11.5%

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

      3. unpow2 11.5%

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

      4. unpow2 11.5%

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

      5. hypot-define 21.4%

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

   5. Simplified 21.4%

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

      1. neg-sub0 21.4%

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

      2. associate-*l/ 21.5%

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

      3. pow1/2 21.5%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B} \]

      4. pow1/2 21.5%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 21.5%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 21.5%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 21.5%

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

      2. distribute-neg-frac2 21.5%

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

      3. unpow1/2 21.5%

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

      4. associate-*r* 21.5%

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

   9. Simplified 21.5%

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

   if 1.3999999999999999e76 < C

   1. Initial program 8.9%

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

   3. Simplified 12.6%

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

   5. Taylor expanded in B around 0 13.3%

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

   6. Taylor expanded in A around -inf 28.3%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.3 \cdot 10^{-290}:\\ \;\;\;\;0.5 \cdot \sqrt{F \cdot \frac{\mathsf{fma}\left(-1, {\left(\frac{B}{A}\right)}^{2}, -4\right)}{C}}\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.15 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.15e-94)
   (/
    (sqrt (* -8.0 (* A (* C (* F (+ A A))))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.15e-94) {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.15e-94)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / 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_] := If[LessEqual[B$95$m, 1.15e-94], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.15e-94

   1. Initial program 23.2%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in B around 0 18.1%

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

   6. Taylor expanded in C around inf 17.0%

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

      1. mul-1-neg 17.0%

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

   8. Simplified 17.0%

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

   if 1.15e-94 < B

   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 C around 0 17.1%

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

      1. mul-1-neg 17.1%

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

      2. +-commutative 17.1%

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

      3. unpow2 17.1%

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

      4. unpow2 17.1%

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

      5. hypot-define 38.2%

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

   5. Simplified 38.2%

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

      1. neg-sub0 38.2%

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

      2. associate-*l/ 38.2%

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

      3. pow1/2 38.2%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B} \]

      4. pow1/2 38.2%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 38.3%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 38.3%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 38.3%

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

      2. distribute-neg-frac2 38.3%

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

      3. unpow1/2 38.3%

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

      4. associate-*r* 38.3%

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

   9. Simplified 38.3%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.15 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

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 8e-141)
   (/
    (sqrt (* (* 4.0 A) (* -4.0 (* A (* C F)))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- 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 <= 8e-141) {
		tmp = sqrt(((4.0 * A) * (-4.0 * (A * (C * F))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -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 <= 8e-141)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(-4.0 * Float64(A * Float64(C * F))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / 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_] := If[LessEqual[B$95$m, 8e-141], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.0000000000000003e-141

   1. Initial program 22.9%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in B around 0 18.3%

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

   6. Taylor expanded in A around -inf 16.3%

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

   if 8.0000000000000003e-141 < B

   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 0 16.8%

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

      1. mul-1-neg 16.8%

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

      2. +-commutative 16.8%

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

      3. unpow2 16.8%

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

      4. unpow2 16.8%

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

      5. hypot-define 37.5%

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

   5. Simplified 37.5%

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

      1. neg-sub0 37.5%

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

      2. associate-*l/ 37.5%

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

      3. pow1/2 37.5%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B} \]

      4. pow1/2 37.5%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 37.6%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 37.6%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 37.6%

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

      2. distribute-neg-frac2 37.6%

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

      3. unpow1/2 37.6%

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

      4. associate-*r* 37.6%

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

   9. Simplified 37.6%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-261}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

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 (<= F -3.9e-261)
   (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))
   (/ (sqrt (* 2.0 (* F (- C (hypot C B_m))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -3.9e-261) {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	} else {
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -B_m;
	}
	return tmp;
}

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 (F <= -3.9e-261) {
		tmp = Math.sqrt(((2.0 * F) * (A - Math.hypot(B_m, A)))) / -B_m;
	} else {
		tmp = Math.sqrt((2.0 * (F * (C - Math.hypot(C, B_m))))) / -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 F <= -3.9e-261:
		tmp = math.sqrt(((2.0 * F) * (A - math.hypot(B_m, A)))) / -B_m
	else:
		tmp = math.sqrt((2.0 * (F * (C - math.hypot(C, B_m))))) / -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 (F <= -3.9e-261)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - hypot(C, B_m))))) / 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 (F <= -3.9e-261)
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	else
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -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[F, -3.9e-261], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.90000000000000017e-261

   1. Initial program 21.5%

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

   3. Taylor expanded in C around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. +-commutative 9.5%

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

      3. unpow2 9.5%

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

      4. unpow2 9.5%

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

      5. hypot-define 17.5%

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

   5. Simplified 17.5%

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

      1. neg-sub0 17.5%

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

      2. associate-*l/ 17.5%

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

      3. pow1/2 17.5%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B} \]

      4. pow1/2 17.5%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 17.5%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 17.5%

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

      2. distribute-neg-frac2 17.5%

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

      3. unpow1/2 17.5%

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

      4. associate-*r* 17.5%

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

   9. Simplified 17.5%

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

   if -3.90000000000000017e-261 < F

   1. Initial program 21.5%

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

   3. Taylor expanded in A around 0 6.7%

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

      1. mul-1-neg 6.7%

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

      2. +-commutative 6.7%

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

      3. unpow2 6.7%

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

      4. unpow2 6.7%

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

      5. hypot-define 21.1%

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

   5. Simplified 21.1%

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

      1. neg-sub0 21.1%

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

      2. associate-*l/ 21.1%

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

      3. pow1/2 21.1%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 21.2%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 21.3%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 21.3%

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

      2. distribute-neg-frac2 21.3%

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

      3. unpow1/2 21.1%

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

   9. Simplified 21.1%

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

Alternative 7: 35.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\frac{{B\_m}^{2} \cdot \left(-F\right)}{C}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

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 2e-117)
   (/ (sqrt (/ (* (pow B_m 2.0) (- F)) C)) (- B_m))
   (/ (sqrt (* 2.0 (* F (- C (hypot C B_m))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2e-117) {
		tmp = sqrt(((pow(B_m, 2.0) * -F) / C)) / -B_m;
	} else {
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -B_m;
	}
	return tmp;
}

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 <= 2e-117) {
		tmp = Math.sqrt(((Math.pow(B_m, 2.0) * -F) / C)) / -B_m;
	} else {
		tmp = Math.sqrt((2.0 * (F * (C - Math.hypot(C, B_m))))) / -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 <= 2e-117:
		tmp = math.sqrt(((math.pow(B_m, 2.0) * -F) / C)) / -B_m
	else:
		tmp = math.sqrt((2.0 * (F * (C - math.hypot(C, B_m))))) / -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 <= 2e-117)
		tmp = Float64(sqrt(Float64(Float64((B_m ^ 2.0) * Float64(-F)) / C)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - hypot(C, B_m))))) / 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 (B_m <= 2e-117)
		tmp = sqrt((((B_m ^ 2.0) * -F) / C)) / -B_m;
	else
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -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, 2e-117], N[(N[Sqrt[N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * (-F)), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.00000000000000006e-117

   1. Initial program 22.8%

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

   3. Taylor expanded in A around 0 3.8%

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

      1. mul-1-neg 3.8%

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

      2. +-commutative 3.8%

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

      3. unpow2 3.8%

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

      4. unpow2 3.8%

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

      5. hypot-define 8.0%

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

   5. Simplified 8.0%

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

      1. neg-sub0 8.0%

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

      2. associate-*l/ 8.0%

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

      3. pow1/2 8.0%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 8.1%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 8.1%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 8.1%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 8.1%

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

      2. distribute-neg-frac2 8.1%

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

      3. unpow1/2 8.0%

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

   9. Simplified 8.0%

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

   10. Taylor expanded in C around inf 13.2%

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

       1. associate-*r/ 13.2%

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

       2. associate-*r* 13.2%

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

       3. mul-1-neg 13.2%

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

   12. Simplified 13.2%

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

   if 2.00000000000000006e-117 < B

   1. Initial program 19.1%

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

   3. Taylor expanded in A around 0 17.9%

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

      1. mul-1-neg 17.9%

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

      2. +-commutative 17.9%

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

      3. unpow2 17.9%

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

      4. unpow2 17.9%

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

      5. hypot-define 37.2%

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

   5. Simplified 37.2%

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

      1. neg-sub0 37.2%

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

      2. associate-*l/ 37.2%

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

      3. pow1/2 37.2%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 37.2%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 37.3%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 37.3%

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

      2. distribute-neg-frac2 37.3%

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

      3. unpow1/2 37.3%

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

   9. Simplified 37.3%

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

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\frac{{B}^{2} \cdot \left(-F\right)}{C}}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\frac{{B\_m}^{2} \cdot \left(-F\right)}{C}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right) + C \cdot \left(2 \cdot F - \frac{C \cdot F}{B\_m}\right)}}{-B\_m}\\ \end{array} \end{array} \]

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-35)
   (/ (sqrt (/ (* (pow B_m 2.0) (- F)) C)) (- B_m))
   (/
    (sqrt (+ (* -2.0 (* B_m F)) (* C (- (* 2.0 F) (/ (* C F) B_m)))))
    (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7e-35) {
		tmp = sqrt(((pow(B_m, 2.0) * -F) / C)) / -B_m;
	} else {
		tmp = sqrt(((-2.0 * (B_m * F)) + (C * ((2.0 * F) - ((C * F) / B_m))))) / -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-35) then
        tmp = sqrt((((b_m ** 2.0d0) * -f) / c)) / -b_m
    else
        tmp = sqrt((((-2.0d0) * (b_m * f)) + (c * ((2.0d0 * f) - ((c * f) / b_m))))) / -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-35) {
		tmp = Math.sqrt(((Math.pow(B_m, 2.0) * -F) / C)) / -B_m;
	} else {
		tmp = Math.sqrt(((-2.0 * (B_m * F)) + (C * ((2.0 * F) - ((C * F) / B_m))))) / -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-35:
		tmp = math.sqrt(((math.pow(B_m, 2.0) * -F) / C)) / -B_m
	else:
		tmp = math.sqrt(((-2.0 * (B_m * F)) + (C * ((2.0 * F) - ((C * F) / B_m))))) / -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-35)
		tmp = Float64(sqrt(Float64(Float64((B_m ^ 2.0) * Float64(-F)) / C)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(Float64(-2.0 * Float64(B_m * F)) + Float64(C * Float64(Float64(2.0 * F) - Float64(Float64(C * F) / B_m))))) / 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 (B_m <= 7e-35)
		tmp = sqrt((((B_m ^ 2.0) * -F) / C)) / -B_m;
	else
		tmp = sqrt(((-2.0 * (B_m * F)) + (C * ((2.0 * F) - ((C * F) / B_m))))) / -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-35], N[(N[Sqrt[N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * (-F)), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision] + N[(C * N[(N[(2.0 * F), $MachinePrecision] - N[(N[(C * F), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.99999999999999992e-35

   1. Initial program 22.9%

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

   3. Taylor expanded in A around 0 5.7%

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

      1. mul-1-neg 5.7%

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

      2. +-commutative 5.7%

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

      3. unpow2 5.7%

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

      4. unpow2 5.7%

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

      5. hypot-define 10.2%

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

   5. Simplified 10.2%

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

      1. neg-sub0 10.2%

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

      2. associate-*l/ 10.2%

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

      3. pow1/2 10.2%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 10.2%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 10.2%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 10.2%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 10.2%

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

      2. distribute-neg-frac2 10.2%

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

      3. unpow1/2 10.2%

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

   9. Simplified 10.2%

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

   10. Taylor expanded in C around inf 14.3%

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

       1. associate-*r/ 14.3%

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

       2. associate-*r* 14.3%

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

       3. mul-1-neg 14.3%

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

   12. Simplified 14.3%

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

   if 6.99999999999999992e-35 < B

   1. Initial program 17.7%

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

   3. Taylor expanded in A around 0 16.2%

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

      1. mul-1-neg 16.2%

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

      2. +-commutative 16.2%

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

      3. unpow2 16.2%

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

      4. unpow2 16.2%

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

      5. hypot-define 38.6%

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

   5. Simplified 38.6%

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

      1. neg-sub0 38.6%

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

      2. associate-*l/ 38.6%

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

      3. pow1/2 38.6%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 38.6%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 38.8%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 38.8%

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

      2. distribute-neg-frac2 38.8%

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

      3. unpow1/2 38.8%

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

   9. Simplified 38.8%

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

   10. Taylor expanded in C around 0 35.1%

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

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\frac{{B}^{2} \cdot \left(-F\right)}{C}}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right) + C \cdot \left(2 \cdot F - \frac{C \cdot F}{B}\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{{B\_m}^{2} \cdot \frac{F}{-C}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right) + C \cdot \left(2 \cdot F - \frac{C \cdot F}{B\_m}\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.4e-35)
   (/ (sqrt (* (pow B_m 2.0) (/ F (- C)))) (- B_m))
   (/
    (sqrt (+ (* -2.0 (* B_m F)) (* C (- (* 2.0 F) (/ (* C F) B_m)))))
    (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.4e-35) {
		tmp = sqrt((pow(B_m, 2.0) * (F / -C))) / -B_m;
	} else {
		tmp = sqrt(((-2.0 * (B_m * F)) + (C * ((2.0 * F) - ((C * F) / B_m))))) / -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 <= 1.4d-35) then
        tmp = sqrt(((b_m ** 2.0d0) * (f / -c))) / -b_m
    else
        tmp = sqrt((((-2.0d0) * (b_m * f)) + (c * ((2.0d0 * f) - ((c * f) / b_m))))) / -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 <= 1.4e-35) {
		tmp = Math.sqrt((Math.pow(B_m, 2.0) * (F / -C))) / -B_m;
	} else {
		tmp = Math.sqrt(((-2.0 * (B_m * F)) + (C * ((2.0 * F) - ((C * F) / B_m))))) / -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 <= 1.4e-35:
		tmp = math.sqrt((math.pow(B_m, 2.0) * (F / -C))) / -B_m
	else:
		tmp = math.sqrt(((-2.0 * (B_m * F)) + (C * ((2.0 * F) - ((C * F) / B_m))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.4e-35)
		tmp = Float64(sqrt(Float64((B_m ^ 2.0) * Float64(F / Float64(-C)))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(Float64(-2.0 * Float64(B_m * F)) + Float64(C * Float64(Float64(2.0 * F) - Float64(Float64(C * F) / B_m))))) / 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 (B_m <= 1.4e-35)
		tmp = sqrt(((B_m ^ 2.0) * (F / -C))) / -B_m;
	else
		tmp = sqrt(((-2.0 * (B_m * F)) + (C * ((2.0 * F) - ((C * F) / B_m))))) / -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, 1.4e-35], N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[(F / (-C)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision] + N[(C * N[(N[(2.0 * F), $MachinePrecision] - N[(N[(C * F), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.4e-35

   1. Initial program 22.9%

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

   3. Taylor expanded in A around 0 5.7%

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

      1. mul-1-neg 5.7%

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

      2. +-commutative 5.7%

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

      3. unpow2 5.7%

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

      4. unpow2 5.7%

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

      5. hypot-define 10.2%

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

   5. Simplified 10.2%

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

      1. neg-sub0 10.2%

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

      2. associate-*l/ 10.2%

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

      3. pow1/2 10.2%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 10.2%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 10.2%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 10.2%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 10.2%

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

      2. distribute-neg-frac2 10.2%

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

      3. unpow1/2 10.2%

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

   9. Simplified 10.2%

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

   10. Taylor expanded in C around inf 14.3%

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

       1. mul-1-neg 14.3%

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

       2. associate-/l* 13.2%

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

       3. distribute-rgt-neg-in 13.2%

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

       4. distribute-neg-frac2 13.2%

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

   12. Simplified 13.2%

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

   if 1.4e-35 < B

   1. Initial program 17.7%

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

   3. Taylor expanded in A around 0 16.2%

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

      1. mul-1-neg 16.2%

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

      2. +-commutative 16.2%

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

      3. unpow2 16.2%

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

      4. unpow2 16.2%

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

      5. hypot-define 38.6%

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

   5. Simplified 38.6%

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

      1. neg-sub0 38.6%

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

      2. associate-*l/ 38.6%

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

      3. pow1/2 38.6%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 38.6%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 38.8%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 38.8%

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

      2. distribute-neg-frac2 38.8%

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

      3. unpow1/2 38.8%

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

   9. Simplified 38.8%

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

   10. Taylor expanded in C around 0 35.1%

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

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{{B}^{2} \cdot \frac{F}{-C}}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right) + C \cdot \left(2 \cdot F - \frac{C \cdot F}{B}\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.1% accurate, 5.9× speedup

Math

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

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 (* B_m 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 * (B_m * 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) * (b_m * 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 * (B_m * 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 * (B_m * 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(B_m * F))) / Float64(-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 * (B_m * 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[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 21.5%

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

3. Taylor expanded in A around 0 8.6%

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

   1. mul-1-neg 8.6%

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

   2. +-commutative 8.6%

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

   3. unpow2 8.6%

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

   4. unpow2 8.6%

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

   5. hypot-define 18.1%

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

5. Simplified 18.1%

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

   1. neg-sub0 18.1%

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

   2. associate-*l/ 18.1%

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

   3. pow1/2 18.1%

      \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

   4. pow1/2 18.1%

      \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

   5. pow-prod-down 18.2%

      \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

7. Applied egg-rr 18.2%

   \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
8. Step-by-step derivation

   1. neg-sub0 18.2%

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

   2. distribute-neg-frac2 18.2%

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

   3. unpow1/2 18.1%

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

9. Simplified 18.1%

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

10. Taylor expanded in C around 0 14.1%

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

Alternative 11: 10.0% accurate, 5.9× speedup

Math

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

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 (* A 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((A * 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((a * 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((A * 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((A * 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(A * 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((A * 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[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.5%

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

3. Taylor expanded in C around 0 9.3%

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

   1. mul-1-neg 9.3%

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

   2. +-commutative 9.3%

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

   3. unpow2 9.3%

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

   4. unpow2 9.3%

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

   5. hypot-define 17.9%

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

5. Simplified 17.9%

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

6. Taylor expanded in A around -inf 0.0%

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

   1. unpow2 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 4.0%

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

   4. rem-square-sqrt 4.0%

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

   5. metadata-eval 4.0%

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

8. Simplified 4.0%

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

Alternative 12: 2.0% accurate, 5.9× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (* 2.0 (/ 1.0 (/ B_m 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 * (1.0 / (B_m / 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 * (1.0d0 / (b_m / 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 * (1.0 / (B_m / 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 * (1.0 / (B_m / 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 sqrt(Float64(2.0 * Float64(1.0 / Float64(B_m / 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 * (1.0 / (B_m / 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[Sqrt[N[(2.0 * N[(1.0 / N[(B$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 21.5%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.2%

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

5. Simplified 2.2%

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

6. Taylor expanded in F around 0 2.2%

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

   1. sqrt-unprod 2.2%

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

8. Applied egg-rr 2.2%

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

   1. clear-num 2.7%

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

   2. inv-pow 2.7%

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

10. Applied egg-rr 2.7%

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

    1. unpow-1 2.7%

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

12. Simplified 2.7%

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

13. Final simplification 2.7%

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

Alternative 13: 2.0% accurate, 6.0× speedup

Math

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

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(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 21.5%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.2%

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

5. Simplified 2.2%

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

6. Taylor expanded in F around 0 2.2%

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

   1. sqrt-unprod 2.2%

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

   2. pow1/2 2.4%

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

8. Applied egg-rr 2.4%

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

9. Final simplification 2.4%

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

Alternative 14: 1.9% 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 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 21.5%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.2%

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

5. Simplified 2.2%

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

6. Taylor expanded in F around 0 2.2%

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

   1. sqrt-unprod 2.2%

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

8. Applied egg-rr 2.2%

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

9. Final simplification 2.2%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))