ABCF->ab-angle a

Percentage Accurate: 19.0% → 64.1%

Time: 25.8s

Alternatives: 10

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 64.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)\\ t_1 := C \cdot \left(4 \cdot A\right)\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\\ t_3 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\frac{C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{t\_0}} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C, 1 - \frac{A}{C}, A + C\right)} \cdot \sqrt{t\_2}}{t\_1 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\_m\right)}\right) \cdot \frac{e^{\log 2 \cdot 0.5}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (pow B_m 2.0)))
        (t_1 (* C (* 4.0 A)))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_1) F)))
        (t_3
         (/
          (sqrt (* t_2 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_3 -1e-196)
     (*
      (sqrt 2.0)
      (* (sqrt (/ (+ C (+ A (hypot B_m (- A C)))) t_0)) (- (sqrt F))))
     (if (<= t_3 0.0)
       (*
        (sqrt (* F (/ (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)) t_0)))
        (- (sqrt 2.0)))
       (if (<= t_3 INFINITY)
         (/
          (* (sqrt (fma C (- 1.0 (/ A C)) (+ A C))) (sqrt t_2))
          (- t_1 (* B_m B_m)))
         (*
          (* (sqrt F) (sqrt (+ C (hypot C B_m))))
          (/ (exp (* (log 2.0) 0.5)) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), pow(B_m, 2.0));
	double t_1 = C * (4.0 * A);
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_1) * F);
	double t_3 = sqrt((t_2 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -1e-196) {
		tmp = sqrt(2.0) * (sqrt(((C + (A + hypot(B_m, (A - C)))) / t_0)) * -sqrt(F));
	} else if (t_3 <= 0.0) {
		tmp = sqrt((F * (((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)) / t_0))) * -sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt(fma(C, (1.0 - (A / C)), (A + C))) * sqrt(t_2)) / (t_1 - (B_m * B_m));
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(C, B_m)))) * (exp((log(2.0) * 0.5)) / -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(-4.0, Float64(A * C), (B_m ^ 2.0))
	t_1 = Float64(C * Float64(4.0 * A))
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F))
	t_3 = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -1e-196)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) / t_0)) * Float64(-sqrt(F))));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)) / t_0))) * Float64(-sqrt(2.0)));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(fma(C, Float64(1.0 - Float64(A / C)), Float64(A + C))) * sqrt(t_2)) / Float64(t_1 - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(C, B_m)))) * Float64(exp(Float64(log(2.0) * 0.5)) / 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[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$2 * 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]}, If[LessEqual[t$95$3, -1e-196], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(F * N[(N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(C * N[(1.0 - N[(A / C), $MachinePrecision]), $MachinePrecision] + N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $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))) < -1e-196

   1. Initial program 47.3%

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

   3. Taylor expanded in F around 0 48.9%

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

   5. Simplified 64.5%

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

      1. *-commutative 64.5%

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

      2. sqrt-prod 77.8%

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

      3. associate-+l+ 79.5%

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

   7. Applied egg-rr 79.5%

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

   if -1e-196 < (/.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))) < 0.0

   1. Initial program 3.6%

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

   3. Taylor expanded in F around 0 19.9%

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

   5. Simplified 18.3%

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

   6. Taylor expanded in A around -inf 26.8%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 27.2%

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

   3. Taylor expanded in C around inf 36.1%

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

      1. associate-*r/ 36.1%

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

      2. mul-1-neg 36.1%

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

   5. Simplified 36.1%

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

      1. unpow2 36.1%

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

   7. Applied egg-rr 36.1%

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

      1. pow1/2 36.9%

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

      2. *-commutative 36.9%

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

      3. unpow-prod-down 51.3%

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

      4. pow1/2 50.4%

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

      5. +-commutative 50.4%

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

      6. +-commutative 50.4%

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

      7. fma-define 51.2%

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

      8. pow1/2 51.2%

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

      9. *-commutative 51.2%

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

      10. *-commutative 51.2%

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

      11. *-commutative 51.2%

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

   9. Applied egg-rr 51.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 1.9%

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

      1. mul-1-neg 1.9%

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

      2. *-commutative 1.9%

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

      3. *-commutative 1.9%

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

      4. +-commutative 1.9%

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

      5. unpow2 1.9%

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

      6. unpow2 1.9%

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

      7. hypot-define 20.5%

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

   5. Simplified 20.5%

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

      1. sqrt-prod 31.2%

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

   7. Applied egg-rr 31.2%

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

      1. pow1/2 31.2%

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

      2. pow-to-exp 31.2%

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

   9. Applied egg-rr 31.2%

      \[\leadsto -\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\color{blue}{e^{\log 2 \cdot 0.5}}}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}} \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\frac{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C, 1 - \frac{A}{C}, A + C\right)} \cdot \sqrt{2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)}}{C \cdot \left(4 \cdot A\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \cdot \frac{e^{\log 2 \cdot 0.5}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 4.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 7 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 5.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_1\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\_m\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 4.2e-172)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (* B_m B_m)))
     (if (<= B_m 7e-55)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (if (<= B_m 5.4e+43)
         (/
          (* (sqrt (* 2.0 (* F t_1))) (sqrt (+ A (+ C (hypot (- A C) B_m)))))
          (- t_1))
         (*
          (* (sqrt F) (sqrt (+ C (hypot C B_m))))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 4.2e-172) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 7e-55) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (B_m <= 5.4e+43) {
		tmp = (sqrt((2.0 * (F * t_1))) * sqrt((A + (C + hypot((A - C), B_m))))) / -t_1;
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(C, B_m)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 4.2e-172)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif (B_m <= 7e-55)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 5.4e+43)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_1))) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / Float64(-t_1));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(C, B_m)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.2e-172], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7e-55], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 5.4e+43], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.1999999999999999e-172

   1. Initial program 16.4%

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

   3. Taylor expanded in C around inf 8.5%

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

      1. associate-*r/ 8.5%

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

      2. mul-1-neg 8.5%

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

   5. Simplified 8.5%

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

      1. unpow2 8.5%

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

   7. Applied egg-rr 8.5%

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

   8. Taylor expanded in A around 0 14.6%

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

   if 4.1999999999999999e-172 < B < 7.00000000000000051e-55

   1. Initial program 14.0%

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

   3. Taylor expanded in F around 0 15.2%

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

   5. Simplified 22.6%

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

   6. Taylor expanded in C around inf 25.5%

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

   if 7.00000000000000051e-55 < B < 5.4000000000000004e43

   1. Initial program 44.8%

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

   3. Simplified 53.9%

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

      1. associate-*r* 53.9%

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

      2. associate-+r+ 53.1%

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

      3. hypot-undefine 44.8%

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

      4. unpow2 44.8%

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

      5. unpow2 44.8%

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

      6. +-commutative 44.8%

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

      7. sqrt-prod 48.7%

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

      8. *-commutative 48.7%

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

      9. associate-+l+ 48.7%

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

   6. Applied egg-rr 69.5%

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

   if 5.4000000000000004e43 < B

   1. Initial program 12.5%

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

   3. Taylor expanded in A around 0 19.2%

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

      1. mul-1-neg 19.2%

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

      2. *-commutative 19.2%

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

      3. *-commutative 19.2%

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

      4. +-commutative 19.2%

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

      5. unpow2 19.2%

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

      6. unpow2 19.2%

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

      7. hypot-define 51.3%

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

   5. Simplified 51.3%

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

      1. sqrt-prod 73.9%

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

   7. Applied egg-rr 73.9%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(4 \cdot A\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;B\_m \leq 3 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\_m\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= B_m 3e-173)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (* B_m B_m)))
     (if (<= B_m 5.6e+15)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (* (* (sqrt F) (sqrt (+ C (hypot C B_m)))) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (B_m <= 3e-173) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 5.6e+15) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(C, B_m)))) * (sqrt(2.0) / -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 t_0 = C * (4.0 * A);
	double tmp;
	if (B_m <= 3e-173) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 5.6e+15) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((C + Math.hypot(C, B_m)))) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	tmp = 0
	if B_m <= 3e-173:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m))
	elif B_m <= 5.6e+15:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	else:
		tmp = (math.sqrt(F) * math.sqrt((C + math.hypot(C, B_m)))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if (B_m <= 3e-173)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif (B_m <= 5.6e+15)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(C, B_m)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	tmp = 0.0;
	if (B_m <= 3e-173)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	elseif (B_m <= 5.6e+15)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	else
		tmp = (sqrt(F) * sqrt((C + hypot(C, B_m)))) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3e-173], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5.6e+15], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.0000000000000001e-173

   1. Initial program 16.4%

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

   3. Taylor expanded in C around inf 8.5%

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

      1. associate-*r/ 8.5%

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

      2. mul-1-neg 8.5%

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

   5. Simplified 8.5%

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

      1. unpow2 8.5%

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

   7. Applied egg-rr 8.5%

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

   8. Taylor expanded in A around 0 14.6%

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

   if 3.0000000000000001e-173 < B < 5.6e15

   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 F around 0 28.4%

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

   5. Simplified 36.0%

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

   6. Taylor expanded in C around inf 23.0%

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

   if 5.6e15 < 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 A around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. *-commutative 22.7%

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

      3. *-commutative 22.7%

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

      4. +-commutative 22.7%

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

      5. unpow2 22.7%

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

      6. unpow2 22.7%

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

      7. hypot-define 50.1%

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

   5. Simplified 50.1%

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

      1. sqrt-prod 69.5%

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

   7. Applied egg-rr 69.5%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(4 \cdot A\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m \cdot \left(\frac{C}{B\_m} + 1\right)}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= B_m 4.5e-172)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (* B_m B_m)))
     (if (<= B_m 8.5e+15)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (*
        (/ (sqrt 2.0) B_m)
        (* (sqrt F) (- (sqrt (* B_m (+ (/ C B_m) 1.0))))))))))

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 t_0 = C * (4.0 * A);
	double tmp;
	if (B_m <= 4.5e-172) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 8.5e+15) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m * ((C / B_m) + 1.0))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if (b_m <= 4.5d-172) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m * b_m))
    else if (b_m <= 8.5d+15) then
        tmp = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt(f) * -sqrt((b_m * ((c / b_m) + 1.0d0))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (B_m <= 4.5e-172) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 8.5e+15) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((B_m * ((C / B_m) + 1.0))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	tmp = 0
	if B_m <= 4.5e-172:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m))
	elif B_m <= 8.5e+15:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((B_m * ((C / B_m) + 1.0))))
	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)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if (B_m <= 4.5e-172)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif (B_m <= 8.5e+15)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(B_m * Float64(Float64(C / B_m) + 1.0))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	tmp = 0.0;
	if (B_m <= 4.5e-172)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	elseif (B_m <= 8.5e+15)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m * ((C / B_m) + 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.5e-172], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8.5e+15], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m * N[(N[(C / B$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.50000000000000004e-172

   1. Initial program 16.4%

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

   3. Taylor expanded in C around inf 8.5%

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

      1. associate-*r/ 8.5%

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

      2. mul-1-neg 8.5%

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

   5. Simplified 8.5%

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

      1. unpow2 8.5%

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

   7. Applied egg-rr 8.5%

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

   8. Taylor expanded in A around 0 14.6%

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

   if 4.50000000000000004e-172 < B < 8.5e15

   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 F around 0 28.4%

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

   5. Simplified 36.0%

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

   6. Taylor expanded in C around inf 23.0%

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

   if 8.5e15 < 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 A around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. *-commutative 22.7%

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

      3. *-commutative 22.7%

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

      4. +-commutative 22.7%

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

      5. unpow2 22.7%

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

      6. unpow2 22.7%

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

      7. hypot-define 50.1%

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

   5. Simplified 50.1%

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

      1. sqrt-prod 69.5%

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

   7. Applied egg-rr 69.5%

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

   8. Taylor expanded in B around inf 62.9%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(4 \cdot A\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B \cdot \left(\frac{C}{B} + 1\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;B\_m \leq 1.95 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= B_m 1.95e-173)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (* B_m B_m)))
     (if (<= B_m 4.8e+15)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ B_m C)))))))))

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 = C * (4.0 * A);
	double tmp;
	if (B_m <= 1.95e-173) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 4.8e+15) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if (b_m <= 1.95d-173) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m * b_m))
    else if (b_m <= 4.8d+15) then
        tmp = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt(f) * -sqrt((b_m + c)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (B_m <= 1.95e-173) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 4.8e+15) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((B_m + C)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	tmp = 0
	if B_m <= 1.95e-173:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m))
	elif B_m <= 4.8e+15:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((B_m + C)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if (B_m <= 1.95e-173)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif (B_m <= 4.8e+15)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(B_m + C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	tmp = 0.0;
	if (B_m <= 1.95e-173)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	elseif (B_m <= 4.8e+15)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.95e-173], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.8e+15], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.94999999999999994e-173

   1. Initial program 16.4%

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

   3. Taylor expanded in C around inf 8.5%

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

      1. associate-*r/ 8.5%

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

      2. mul-1-neg 8.5%

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

   5. Simplified 8.5%

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

      1. unpow2 8.5%

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

   7. Applied egg-rr 8.5%

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

   8. Taylor expanded in A around 0 14.6%

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

   if 1.94999999999999994e-173 < B < 4.8e15

   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 F around 0 28.4%

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

   5. Simplified 36.0%

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

   6. Taylor expanded in C around inf 23.0%

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

   if 4.8e15 < 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 A around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. *-commutative 22.7%

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

      3. *-commutative 22.7%

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

      4. +-commutative 22.7%

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

      5. unpow2 22.7%

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

      6. unpow2 22.7%

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

      7. hypot-define 50.1%

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

   5. Simplified 50.1%

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

      1. sqrt-prod 69.5%

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

   7. Applied egg-rr 69.5%

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

   8. Taylor expanded in C around 0 62.9%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.95 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(4 \cdot A\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;B\_m \leq 1.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 1.02 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= B_m 1.2e-171)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (* B_m B_m)))
     (if (<= B_m 1.02e+117)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (B_m <= 1.2e-171) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 1.02e+117) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if (b_m <= 1.2d-171) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m * b_m))
    else if (b_m <= 1.02d+117) then
        tmp = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    else
        tmp = sqrt(f) * -sqrt((2.0d0 / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (B_m <= 1.2e-171) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 1.02e+117) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(F) * -Math.sqrt((2.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	tmp = 0
	if B_m <= 1.2e-171:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m))
	elif B_m <= 1.02e+117:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt(F) * -math.sqrt((2.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if (B_m <= 1.2e-171)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif (B_m <= 1.02e+117)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	tmp = 0.0;
	if (B_m <= 1.2e-171)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	elseif (B_m <= 1.02e+117)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	else
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.2e-171], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.02e+117], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.19999999999999993e-171

   1. Initial program 16.4%

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

   3. Taylor expanded in C around inf 8.5%

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

      1. associate-*r/ 8.5%

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

      2. mul-1-neg 8.5%

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

   5. Simplified 8.5%

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

      1. unpow2 8.5%

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

   7. Applied egg-rr 8.5%

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

   8. Taylor expanded in A around 0 14.6%

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

   if 1.19999999999999993e-171 < B < 1.02e117

   1. Initial program 29.5%

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

   3. Taylor expanded in F around 0 38.0%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in C around inf 23.2%

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

   if 1.02e117 < B

   1. Initial program 6.8%

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

   3. Taylor expanded in B around inf 48.8%

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

      1. mul-1-neg 48.8%

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

      2. *-commutative 48.8%

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

   5. Simplified 48.8%

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

      1. *-commutative 48.8%

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

      2. pow1/2 48.8%

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

      3. pow1/2 48.8%

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

      4. pow-prod-down 49.0%

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

   7. Applied egg-rr 49.0%

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

      1. unpow1/2 49.0%

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

   9. Simplified 49.0%

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

       1. associate-*l/ 49.2%

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

   11. Applied egg-rr 49.2%

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

       1. associate-/l* 49.2%

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

   13. Simplified 49.2%

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

       1. pow1/2 49.2%

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

       2. *-commutative 49.2%

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

       3. unpow-prod-down 76.8%

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

       4. pow1/2 76.8%

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

       5. pow1/2 76.8%

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

   15. Applied egg-rr 76.8%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(4 \cdot A\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 10^{+118}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1e+118)
   (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
   (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1e+118) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1d+118) then
        tmp = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    else
        tmp = sqrt(f) * -sqrt((2.0d0 / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1e+118) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(F) * -Math.sqrt((2.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1e+118:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt(F) * -math.sqrt((2.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1e+118)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1e+118)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	else
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1e+118], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9.99999999999999967e117

   1. Initial program 20.2%

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

   3. Taylor expanded in F around 0 21.8%

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

   5. Simplified 27.9%

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

   6. Taylor expanded in C around inf 18.0%

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

   if 9.99999999999999967e117 < B

   1. Initial program 6.8%

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

   3. Taylor expanded in B around inf 48.8%

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

      1. mul-1-neg 48.8%

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

      2. *-commutative 48.8%

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

   5. Simplified 48.8%

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

      1. *-commutative 48.8%

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

      2. pow1/2 48.8%

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

      3. pow1/2 48.8%

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

      4. pow-prod-down 49.0%

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

   7. Applied egg-rr 49.0%

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

      1. unpow1/2 49.0%

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

   9. Simplified 49.0%

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

       1. associate-*l/ 49.2%

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

   11. Applied egg-rr 49.2%

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

       1. associate-/l* 49.2%

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

   13. Simplified 49.2%

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

       1. pow1/2 49.2%

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

       2. *-commutative 49.2%

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

       3. unpow-prod-down 76.8%

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

       4. pow1/2 76.8%

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

       5. pow1/2 76.8%

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

   15. Applied egg-rr 76.8%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 10^{+118}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B\_m \cdot B\_m - t\_0\right)\right)\right) \cdot \left(\left(A + C\right) + C \cdot \left(1 - \frac{A}{C}\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= F -5e-310)
     (/
      (sqrt
       (* (* 2.0 (* F (- (* B_m B_m) t_0))) (+ (+ A C) (* C (- 1.0 (/ A C))))))
      (- t_0 (* B_m B_m)))
     (* (sqrt F) (- (sqrt (/ 2.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (F <= -5e-310) {
		tmp = sqrt(((2.0 * (F * ((B_m * B_m) - t_0))) * ((A + C) + (C * (1.0 - (A / C)))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if (f <= (-5d-310)) then
        tmp = sqrt(((2.0d0 * (f * ((b_m * b_m) - t_0))) * ((a + c) + (c * (1.0d0 - (a / c)))))) / (t_0 - (b_m * b_m))
    else
        tmp = sqrt(f) * -sqrt((2.0d0 / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (F <= -5e-310) {
		tmp = Math.sqrt(((2.0 * (F * ((B_m * B_m) - t_0))) * ((A + C) + (C * (1.0 - (A / C)))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = Math.sqrt(F) * -Math.sqrt((2.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	tmp = 0
	if F <= -5e-310:
		tmp = math.sqrt(((2.0 * (F * ((B_m * B_m) - t_0))) * ((A + C) + (C * (1.0 - (A / C)))))) / (t_0 - (B_m * B_m))
	else:
		tmp = math.sqrt(F) * -math.sqrt((2.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) - t_0))) * Float64(Float64(A + C) + Float64(C * Float64(1.0 - Float64(A / C)))))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = sqrt(((2.0 * (F * ((B_m * B_m) - t_0))) * ((A + C) + (C * (1.0 - (A / C)))))) / (t_0 - (B_m * B_m));
	else
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e-310], N[(N[Sqrt[N[(N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[(C * N[(1.0 - N[(A / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 16.7%

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

   3. Taylor expanded in C around inf 22.0%

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

      1. associate-*r/ 22.0%

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

      2. mul-1-neg 22.0%

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

   5. Simplified 22.0%

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

      1. unpow2 22.0%

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

   7. Applied egg-rr 22.0%

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

      1. unpow2 22.0%

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

   9. Applied egg-rr 22.0%

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

   if -4.999999999999985e-310 < F

   1. Initial program 18.0%

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

   3. Taylor expanded in B around inf 17.8%

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

      1. mul-1-neg 17.8%

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

      2. *-commutative 17.8%

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

   5. Simplified 17.8%

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

      1. *-commutative 17.8%

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

      2. pow1/2 17.9%

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

      3. pow1/2 17.9%

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

      4. pow-prod-down 18.0%

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

   7. Applied egg-rr 18.0%

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

      1. unpow1/2 17.9%

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

   9. Simplified 17.9%

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

       1. associate-*l/ 17.9%

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

   11. Applied egg-rr 17.9%

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

       1. associate-/l* 17.9%

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

   13. Simplified 17.9%

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

       1. pow1/2 18.0%

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

       2. *-commutative 18.0%

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

       3. unpow-prod-down 24.0%

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

       4. pow1/2 24.0%

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

       5. pow1/2 24.0%

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

   15. Applied egg-rr 24.0%

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

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(A + C\right) + C \cdot \left(1 - \frac{A}{C}\right)\right)}}{C \cdot \left(4 \cdot A\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;F \leq 3.4 \cdot 10^{-256}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B\_m \cdot B\_m - t\_0\right)\right)\right) \cdot \left(\left(A + C\right) + C \cdot \left(1 - \frac{A}{C}\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= F 3.4e-256)
     (/
      (sqrt
       (* (* 2.0 (* F (- (* B_m B_m) t_0))) (+ (+ A C) (* C (- 1.0 (/ A C))))))
      (- t_0 (* B_m B_m)))
     (- (sqrt (* F (/ 2.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (F <= 3.4e-256) {
		tmp = sqrt(((2.0 * (F * ((B_m * B_m) - t_0))) * ((A + C) + (C * (1.0 - (A / C)))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = -sqrt((F * (2.0 / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if (f <= 3.4d-256) then
        tmp = sqrt(((2.0d0 * (f * ((b_m * b_m) - t_0))) * ((a + c) + (c * (1.0d0 - (a / c)))))) / (t_0 - (b_m * b_m))
    else
        tmp = -sqrt((f * (2.0d0 / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (F <= 3.4e-256) {
		tmp = Math.sqrt(((2.0 * (F * ((B_m * B_m) - t_0))) * ((A + C) + (C * (1.0 - (A / C)))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = -Math.sqrt((F * (2.0 / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	tmp = 0
	if F <= 3.4e-256:
		tmp = math.sqrt(((2.0 * (F * ((B_m * B_m) - t_0))) * ((A + C) + (C * (1.0 - (A / C)))))) / (t_0 - (B_m * B_m))
	else:
		tmp = -math.sqrt((F * (2.0 / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if (F <= 3.4e-256)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) - t_0))) * Float64(Float64(A + C) + Float64(C * Float64(1.0 - Float64(A / C)))))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(-sqrt(Float64(F * Float64(2.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	tmp = 0.0;
	if (F <= 3.4e-256)
		tmp = sqrt(((2.0 * (F * ((B_m * B_m) - t_0))) * ((A + C) + (C * (1.0 - (A / C)))))) / (t_0 - (B_m * B_m));
	else
		tmp = -sqrt((F * (2.0 / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 3.4e-256], N[(N[Sqrt[N[(N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[(C * N[(1.0 - N[(A / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if F < 3.4000000000000001e-256

   1. Initial program 13.7%

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

   3. Taylor expanded in C around inf 21.3%

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

      1. associate-*r/ 21.3%

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

      2. mul-1-neg 21.3%

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

   5. Simplified 21.3%

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

      1. unpow2 21.3%

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

   7. Applied egg-rr 21.3%

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

      1. unpow2 21.3%

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

   9. Applied egg-rr 21.3%

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

   if 3.4000000000000001e-256 < F

   1. Initial program 18.8%

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

   3. Taylor expanded in B around inf 18.3%

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

      1. mul-1-neg 18.3%

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

      2. *-commutative 18.3%

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

   5. Simplified 18.3%

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

      1. *-commutative 18.3%

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

      2. pow1/2 18.4%

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

      3. pow1/2 18.4%

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

      4. pow-prod-down 18.5%

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

   7. Applied egg-rr 18.5%

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

      1. unpow1/2 18.4%

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

   9. Simplified 18.4%

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

       1. associate-*l/ 18.5%

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

   11. Applied egg-rr 18.5%

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

       1. associate-/l* 18.5%

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

   13. Simplified 18.5%

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

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.4 \cdot 10^{-256}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(A + C\right) + C \cdot \left(1 - \frac{A}{C}\right)\right)}}{C \cdot \left(4 \cdot A\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around inf 15.6%

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

   1. mul-1-neg 15.6%

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

   2. *-commutative 15.6%

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

5. Simplified 15.6%

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

   1. *-commutative 15.6%

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

   2. pow1/2 15.7%

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

   3. pow1/2 15.7%

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

   4. pow-prod-down 15.8%

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

7. Applied egg-rr 15.8%

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

   1. unpow1/2 15.7%

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

9. Simplified 15.7%

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

    1. associate-*l/ 15.7%

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

11. Applied egg-rr 15.7%

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

    1. associate-/l* 15.7%

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

13. Simplified 15.7%

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

Reproduce

herbie shell --seed 2024130 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))