ABCF->ab-angle b

Percentage Accurate: 18.5% → 46.3%

Time: 25.9s

Alternatives: 14

Speedup: 5.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.5% 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: 46.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ t_1 := A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\\ t_2 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_2\right) \cdot \left(2 \cdot t\_1\right)}}{-t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(2 \cdot t\_0\right) \cdot t\_1\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\_m\right) - A}}}\\ \end{array} \end{array} \]

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 C (* A -4.0) (pow B_m 2.0)))
        (t_1 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))
        (t_2 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e-90)
     (/ (sqrt (* (* F t_2) (* 2.0 t_1))) (- t_2))
     (if (<= (pow B_m 2.0) 2e+89)
       (*
        (sqrt
         (/
          (* F (+ A (- C (hypot B_m (- A C)))))
          (fma -4.0 (* A C) (pow B_m 2.0))))
        (- (sqrt 2.0)))
       (if (<= (pow B_m 2.0) 2e+150)
         (/ (sqrt (* F (* (* 2.0 t_0) t_1))) (- t_0))
         (/
          -1.0
          (*
           (/ B_m (sqrt 2.0))
           (sqrt (/ (/ -1.0 F) (- (hypot A B_m) A))))))))))

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(C, (A * -4.0), pow(B_m, 2.0));
	double t_1 = A + (A + (-0.5 * (pow(B_m, 2.0) / C)));
	double t_2 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-90) {
		tmp = sqrt(((F * t_2) * (2.0 * t_1))) / -t_2;
	} else if (pow(B_m, 2.0) <= 2e+89) {
		tmp = sqrt(((F * (A + (C - hypot(B_m, (A - C))))) / fma(-4.0, (A * C), pow(B_m, 2.0)))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 2e+150) {
		tmp = sqrt((F * ((2.0 * t_0) * t_1))) / -t_0;
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((-1.0 / F) / (hypot(A, B_m) - A))));
	}
	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(C, Float64(A * -4.0), (B_m ^ 2.0))
	t_1 = Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))
	t_2 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-90)
		tmp = Float64(sqrt(Float64(Float64(F * t_2) * Float64(2.0 * t_1))) / Float64(-t_2));
	elseif ((B_m ^ 2.0) <= 2e+89)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C))))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 2e+150)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(2.0 * t_0) * t_1))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(-1.0 / F) / Float64(hypot(A, B_m) - A)))));
	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[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-90], N[(N[Sqrt[N[(N[(F * t$95$2), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+89], N[(N[Sqrt[N[(N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+150], N[(N[Sqrt[N[(F * N[(N[(2.0 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / F), $MachinePrecision] / N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999995e-91

   1. Initial program 13.9%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in C around inf 26.8%

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

      1. mul-1-neg 26.8%

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

   7. Simplified 26.8%

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

   if 9.99999999999999995e-91 < (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999999e89

   1. Initial program 38.1%

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

   3. Taylor expanded in F around 0 42.4%

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

      1. mul-1-neg 42.4%

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

      2. cancel-sign-sub-inv 42.4%

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

      3. metadata-eval 42.4%

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

      4. +-commutative 42.4%

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

   5. Simplified 50.8%

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

   if 1.99999999999999999e89 < (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999996e150

   1. Initial program 21.2%

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

   3. Simplified 22.7%

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

   5. Taylor expanded in C around inf 30.4%

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

      1. mul-1-neg 30.4%

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

   7. Simplified 30.4%

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

   if 1.99999999999999996e150 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.0%

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

   3. Taylor expanded in C around 0 7.4%

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

      1. mul-1-neg 7.4%

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

      2. +-commutative 7.4%

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

   5. Simplified 7.4%

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

      1. add-cbrt-cube 6.6%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 6.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 24.4%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 25.5%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]

   9. Simplified 25.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]
   10. Step-by-step derivation

       1. associate-*l/ 25.5%

          \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}}{B}} \]

       2. pow1/3 24.4%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{B} \]

       3. pow-pow 27.7%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

       4. metadata-eval 27.7%

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

       5. pow1/2 27.7%

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

       6. sqrt-prod 27.7%

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

       7. distribute-frac-neg2 27.7%

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

       8. clear-num 27.7%

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

       9. associate-*r* 27.7%

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

   11. Applied egg-rr 27.7%

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

   12. Taylor expanded in F around 0 7.4%

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

       1. mul-1-neg 7.4%

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

       2. distribute-rgt-neg-in 7.4%

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

       3. associate-/r* 8.1%

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

       4. unpow2 8.1%

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

       5. unpow2 8.1%

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

       6. hypot-define 28.7%

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

   14. Simplified 28.7%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\right) - A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ t_2 := \left(4 \cdot A\right) \cdot C - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(\sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot t\_2\right)\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot t\_1}}{-t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (fma C (* A -4.0) (pow B_m 2.0)))
        (t_2 (- (* (* 4.0 A) C) (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))
            (* 2.0 (* F t_2))))
          t_2)))
   (if (<= t_3 -2e-206)
     (/
      (* (sqrt (* F (+ A (- C (hypot B_m (- A C)))))) (sqrt (* 2.0 t_1)))
      (- t_1))
     (if (<= t_3 INFINITY)
       (/
        (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
        (- t_0))
       (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double t_2 = ((4.0 * A) * C) - pow(B_m, 2.0);
	double t_3 = sqrt(((sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * t_2)))) / t_2;
	double tmp;
	if (t_3 <= -2e-206) {
		tmp = (sqrt((F * (A + (C - hypot(B_m, (A - C)))))) * sqrt((2.0 * t_1))) / -t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	t_2 = Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))) - Float64(A + C)) * Float64(2.0 * Float64(F * t_2)))) / t_2)
	tmp = 0.0
	if (t_3 <= -2e-206)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))) * sqrt(Float64(2.0 * t_1))) / Float64(-t_1));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-206], N[(N[(N[Sqrt[N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000006e-206

   1. Initial program 36.0%

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

   3. Simplified 36.2%

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

      1. pow1/2 36.3%

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

      2. associate-*r* 43.9%

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

      3. unpow-prod-down 58.5%

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

      4. associate-+r- 57.2%

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

      5. hypot-undefine 44.8%

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

      6. unpow2 44.8%

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

      7. unpow2 44.8%

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

      8. +-commutative 44.8%

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

      9. unpow2 44.8%

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

      10. unpow2 44.8%

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

      11. hypot-define 57.2%

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

      12. pow1/2 57.2%

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

   6. Applied egg-rr 57.2%

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

      1. unpow1/2 57.2%

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

      2. associate-+r- 58.5%

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

      3. hypot-undefine 44.8%

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

      4. unpow2 44.8%

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

      5. unpow2 44.8%

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

      6. +-commutative 44.8%

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

      7. unpow2 44.8%

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

      8. unpow2 44.8%

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

      9. hypot-undefine 58.5%

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

   8. Simplified 58.5%

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

   if -2.00000000000000006e-206 < (/.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 14.5%

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

   3. Simplified 25.0%

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

   5. Taylor expanded in C around inf 35.6%

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

      1. mul-1-neg 35.6%

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

   7. Simplified 35.6%

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

   if +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 C around 0 2.0%

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

      1. mul-1-neg 2.0%

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

      2. +-commutative 2.0%

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

      3. unpow2 2.0%

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

      4. unpow2 2.0%

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

      5. hypot-define 20.4%

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

   5. Simplified 20.4%

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

      1. neg-sub0 20.4%

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

      2. associate-*l/ 20.4%

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

      3. pow1/2 20.4%

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

      4. pow1/2 20.4%

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

      5. hypot-undefine 2.0%

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

      6. unpow2 2.0%

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

      7. unpow2 2.0%

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

      8. pow-prod-down 2.0%

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

      9. unpow2 2.0%

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

      10. unpow2 2.0%

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

      11. hypot-undefine 20.5%

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

   7. Applied egg-rr 20.5%

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

      1. neg-sub0 20.5%

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

      2. distribute-neg-frac2 20.5%

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

      3. unpow1/2 20.5%

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

   9. Simplified 20.5%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\_m\right) - A}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-46)
     (/
      (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
      (- t_0))
     (/
      -1.0
      (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ -1.0 F) (- (hypot A B_m) A))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-46) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((-1.0 / F) / (hypot(A, B_m) - A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-46)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(-1.0 / F) / Float64(hypot(A, B_m) - A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-46], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / F), $MachinePrecision] / N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000005e-46

   1. Initial program 14.7%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in C around inf 25.7%

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

      1. mul-1-neg 25.7%

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

   7. Simplified 25.7%

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

   if 2.00000000000000005e-46 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.9%

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

   3. Taylor expanded in C around 0 12.2%

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

      1. mul-1-neg 12.2%

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

      2. +-commutative 12.2%

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

   5. Simplified 12.2%

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

      1. add-cbrt-cube 10.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 9.7%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 22.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 23.6%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]

   9. Simplified 23.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]
   10. Step-by-step derivation

       1. associate-*l/ 23.5%

          \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}}{B}} \]

       2. pow1/3 22.5%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{B} \]

       3. pow-pow 27.0%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

       4. metadata-eval 27.0%

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

       5. pow1/2 27.0%

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

       6. sqrt-prod 27.0%

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

       7. distribute-frac-neg2 27.0%

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

       8. clear-num 27.0%

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

       9. associate-*r* 27.0%

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

   11. Applied egg-rr 27.0%

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

   12. Taylor expanded in F around 0 12.1%

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

       1. mul-1-neg 12.1%

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

       2. distribute-rgt-neg-in 12.1%

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

       3. associate-/r* 12.6%

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

       4. unpow2 12.6%

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

       5. unpow2 12.6%

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

       6. hypot-define 27.4%

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

   14. Simplified 27.4%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\right) - A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\_m\right) - A}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e-90)
     (/ (sqrt (* (* 4.0 A) (* F t_0))) (- t_0))
     (/
      -1.0
      (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ -1.0 F) (- (hypot A B_m) A))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-90) {
		tmp = sqrt(((4.0 * A) * (F * t_0))) / -t_0;
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((-1.0 / F) / (hypot(A, B_m) - A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-90)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * t_0))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(-1.0 / F) / Float64(hypot(A, B_m) - A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-90], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / F), $MachinePrecision] / N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999995e-91

   1. Initial program 13.9%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in A around -inf 26.0%

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

   if 9.99999999999999995e-91 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.4%

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

   3. Taylor expanded in C around 0 11.6%

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

      1. mul-1-neg 11.6%

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

      2. +-commutative 11.6%

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

   5. Simplified 11.6%

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

      1. add-cbrt-cube 9.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 9.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 21.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 22.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]

   9. Simplified 22.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]
   10. Step-by-step derivation

       1. associate-*l/ 22.3%

          \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}}{B}} \]

       2. pow1/3 21.3%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{B} \]

       3. pow-pow 25.6%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

       4. metadata-eval 25.6%

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

       5. pow1/2 25.6%

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

       6. sqrt-prod 25.6%

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

       7. distribute-frac-neg2 25.6%

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

       8. clear-num 25.6%

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

       9. associate-*r* 25.6%

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

   11. Applied egg-rr 25.6%

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

   12. Taylor expanded in F around 0 11.5%

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

       1. mul-1-neg 11.5%

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

       2. distribute-rgt-neg-in 11.5%

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

       3. associate-/r* 12.0%

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

       4. unpow2 12.0%

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

       5. unpow2 12.0%

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

       6. hypot-define 25.8%

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

   14. Simplified 25.8%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\right) - A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\_m\right) - A}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-90)
   (/
    (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
    (- (fma C (* A -4.0) (pow B_m 2.0))))
   (/ -1.0 (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ -1.0 F) (- (hypot A B_m) A)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-90) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((-1.0 / F) / (hypot(A, B_m) - A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-90)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(-fma(C, Float64(A * -4.0), (B_m ^ 2.0))));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(-1.0 / F) / Float64(hypot(A, B_m) - A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-90], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / F), $MachinePrecision] / N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999995e-91

   1. Initial program 13.9%

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

   3. Simplified 20.9%

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

   5. Taylor expanded in C around inf 25.1%

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

      1. associate-*r* 25.1%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

      2. mul-1-neg 25.1%

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

   7. Simplified 25.1%

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

   if 9.99999999999999995e-91 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.4%

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

   3. Taylor expanded in C around 0 11.6%

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

      1. mul-1-neg 11.6%

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

      2. +-commutative 11.6%

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

   5. Simplified 11.6%

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

      1. add-cbrt-cube 9.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 9.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 21.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 22.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]

   9. Simplified 22.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]
   10. Step-by-step derivation

       1. associate-*l/ 22.3%

          \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}}{B}} \]

       2. pow1/3 21.3%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{B} \]

       3. pow-pow 25.6%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

       4. metadata-eval 25.6%

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

       5. pow1/2 25.6%

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

       6. sqrt-prod 25.6%

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

       7. distribute-frac-neg2 25.6%

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

       8. clear-num 25.6%

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

       9. associate-*r* 25.6%

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

   11. Applied egg-rr 25.6%

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

   12. Taylor expanded in F around 0 11.5%

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

       1. mul-1-neg 11.5%

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

       2. distribute-rgt-neg-in 11.5%

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

       3. associate-/r* 12.0%

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

       4. unpow2 12.0%

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

       5. unpow2 12.0%

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

       6. hypot-define 25.8%

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

   14. Simplified 25.8%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\right) - A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\_m\right) - A}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-90)
   (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
   (/ -1.0 (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ -1.0 F) (- (hypot A B_m) A)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-90) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((-1.0 / F) / (hypot(A, B_m) - A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-90) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * Math.sqrt(((-1.0 / F) / (Math.hypot(A, B_m) - A))));
	}
	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 math.pow(B_m, 2.0) <= 1e-90:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	else:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * math.sqrt(((-1.0 / F) / (math.hypot(A, B_m) - A))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-90)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(-1.0 / F) / Float64(hypot(A, B_m) - A)))));
	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 ^ 2.0) <= 1e-90)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	else
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((-1.0 / F) / (hypot(A, B_m) - A))));
	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[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-90], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / F), $MachinePrecision] / N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999995e-91

   1. Initial program 13.9%

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

   3. Simplified 20.9%

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

   5. Taylor expanded in C around inf 18.1%

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

      1. associate-*r* 18.1%

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

   7. Simplified 18.1%

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

   8. Taylor expanded in C around inf 25.0%

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

      1. associate-*r* 25.1%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

      2. mul-1-neg 25.1%

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

   10. Simplified 25.0%

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

   if 9.99999999999999995e-91 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.4%

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

   3. Taylor expanded in C around 0 11.6%

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

      1. mul-1-neg 11.6%

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

      2. +-commutative 11.6%

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

   5. Simplified 11.6%

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

      1. add-cbrt-cube 9.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 9.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 21.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 22.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]

   9. Simplified 22.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}} \]
   10. Step-by-step derivation

       1. associate-*l/ 22.3%

          \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot \sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}}{B}} \]

       2. pow1/3 21.3%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{B} \]

       3. pow-pow 25.6%

          \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

       4. metadata-eval 25.6%

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

       5. pow1/2 25.6%

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

       6. sqrt-prod 25.6%

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

       7. distribute-frac-neg2 25.6%

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

       8. clear-num 25.6%

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

       9. associate-*r* 25.6%

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

   11. Applied egg-rr 25.6%

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

   12. Taylor expanded in F around 0 11.5%

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

       1. mul-1-neg 11.5%

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

       2. distribute-rgt-neg-in 11.5%

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

       3. associate-/r* 12.0%

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

       4. unpow2 12.0%

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

       5. unpow2 12.0%

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

       6. hypot-define 25.8%

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

   14. Simplified 25.8%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{-1}{F}}{\mathsf{hypot}\left(A, B\right) - A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-90)
   (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
   (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-90) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-90) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A))))) / -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 math.pow(B_m, 2.0) <= 1e-90:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	else:
		tmp = math.sqrt((2.0 * (F * (A - math.hypot(B_m, A))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-90)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-90)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	else
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -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[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-90], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999995e-91

   1. Initial program 13.9%

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

   3. Simplified 20.9%

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

   5. Taylor expanded in C around inf 18.1%

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

      1. associate-*r* 18.1%

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

   7. Simplified 18.1%

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

   8. Taylor expanded in C around inf 25.0%

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

      1. associate-*r* 25.1%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

      2. mul-1-neg 25.1%

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

   10. Simplified 25.0%

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

   if 9.99999999999999995e-91 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.4%

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

   3. Taylor expanded in C around 0 11.6%

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

      1. mul-1-neg 11.6%

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

      2. +-commutative 11.6%

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

      3. unpow2 11.6%

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

      4. unpow2 11.6%

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

      5. hypot-define 25.5%

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

   5. Simplified 25.5%

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

      1. neg-sub0 25.5%

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

      2. associate-*l/ 25.6%

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

      3. pow1/2 25.6%

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

      4. pow1/2 25.6%

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

      5. hypot-undefine 11.7%

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

      6. unpow2 11.7%

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

      7. unpow2 11.7%

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

      8. pow-prod-down 11.7%

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

      9. unpow2 11.7%

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

      10. unpow2 11.7%

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

      11. hypot-undefine 25.6%

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

   7. Applied egg-rr 25.6%

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

      1. neg-sub0 25.6%

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

      2. distribute-neg-frac2 25.6%

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

      3. unpow1/2 25.6%

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

   9. Simplified 25.6%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A \cdot F - B\_m \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.5e-45)
   (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
   (/ (sqrt (* 2.0 (- (* A F) (* B_m F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.5e-45) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = sqrt((2.0 * ((A * F) - (B_m * F)))) / -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 <= 3.5d-45) then
        tmp = sqrt(((a * (-8.0d0)) * (c * (f * (a + a))))) / ((4.0d0 * a) * c)
    else
        tmp = sqrt((2.0d0 * ((a * f) - (b_m * f)))) / -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 <= 3.5e-45) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = Math.sqrt((2.0 * ((A * F) - (B_m * F)))) / -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 <= 3.5e-45:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	else:
		tmp = math.sqrt((2.0 * ((A * F) - (B_m * F)))) / -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 <= 3.5e-45)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(A * F) - Float64(B_m * F)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.5e-45)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	else
		tmp = sqrt((2.0 * ((A * F) - (B_m * F)))) / -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, 3.5e-45], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(A * F), $MachinePrecision] - N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.5e-45

   1. Initial program 13.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in C around inf 12.1%

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

      1. associate-*r* 12.1%

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

   7. Simplified 12.1%

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

   8. Taylor expanded in C around inf 18.2%

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

      1. associate-*r* 17.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

      2. mul-1-neg 17.3%

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

   10. Simplified 18.2%

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

   if 3.5e-45 < B

   1. Initial program 20.5%

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

   3. Taylor expanded in C around 0 21.0%

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

      1. mul-1-neg 21.0%

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

      2. +-commutative 21.0%

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

      3. unpow2 21.0%

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

      4. unpow2 21.0%

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

      5. hypot-define 46.7%

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

   5. Simplified 46.7%

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

      1. neg-sub0 46.7%

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

      2. associate-*l/ 46.7%

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

      3. pow1/2 46.7%

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

      4. pow1/2 46.7%

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

      5. hypot-undefine 21.1%

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

      6. unpow2 21.1%

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

      7. unpow2 21.1%

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

      8. pow-prod-down 21.2%

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

      9. unpow2 21.2%

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

      10. unpow2 21.2%

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

      11. hypot-undefine 46.8%

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

   7. Applied egg-rr 46.8%

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

      1. neg-sub0 46.8%

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

      2. distribute-neg-frac2 46.8%

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

      3. unpow1/2 46.8%

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

   9. Simplified 46.8%

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

   10. Taylor expanded in A around 0 44.4%

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

       1. +-commutative 44.4%

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

       2. mul-1-neg 44.4%

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

       3. unsub-neg 44.4%

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

       4. *-commutative 44.4%

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

   12. Simplified 44.4%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A \cdot F - B \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.55 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A \cdot F - B\_m \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.55e-45)
   (/ (sqrt (* F (* -8.0 (* A (* C (+ A A)))))) (* (* 4.0 A) C))
   (/ (sqrt (* 2.0 (- (* A F) (* B_m F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.55e-45) {
		tmp = sqrt((F * (-8.0 * (A * (C * (A + A)))))) / ((4.0 * A) * C);
	} else {
		tmp = sqrt((2.0 * ((A * F) - (B_m * F)))) / -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 <= 2.55d-45) then
        tmp = sqrt((f * ((-8.0d0) * (a * (c * (a + a)))))) / ((4.0d0 * a) * c)
    else
        tmp = sqrt((2.0d0 * ((a * f) - (b_m * f)))) / -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 <= 2.55e-45) {
		tmp = Math.sqrt((F * (-8.0 * (A * (C * (A + A)))))) / ((4.0 * A) * C);
	} else {
		tmp = Math.sqrt((2.0 * ((A * F) - (B_m * F)))) / -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 <= 2.55e-45:
		tmp = math.sqrt((F * (-8.0 * (A * (C * (A + A)))))) / ((4.0 * A) * C)
	else:
		tmp = math.sqrt((2.0 * ((A * F) - (B_m * F)))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.55e-45)
		tmp = Float64(sqrt(Float64(F * Float64(-8.0 * Float64(A * Float64(C * Float64(A + A)))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(A * F) - Float64(B_m * F)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.55e-45)
		tmp = sqrt((F * (-8.0 * (A * (C * (A + A)))))) / ((4.0 * A) * C);
	else
		tmp = sqrt((2.0 * ((A * F) - (B_m * F)))) / -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, 2.55e-45], N[(N[Sqrt[N[(F * N[(-8.0 * N[(A * N[(C * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(A * F), $MachinePrecision] - N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.5499999999999999e-45

   1. Initial program 13.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in C around inf 12.1%

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

      1. associate-*r* 12.1%

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

   7. Simplified 12.1%

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

   8. Taylor expanded in C around inf 15.7%

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

      1. sub-neg 15.7%

         \[\leadsto \frac{\sqrt{F \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(A + \left(--1 \cdot A\right)\right)}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C} \]

      2. mul-1-neg 15.7%

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

      3. remove-double-neg 15.7%

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

   10. Simplified 15.7%

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

   if 2.5499999999999999e-45 < B

   1. Initial program 20.5%

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

   3. Taylor expanded in C around 0 21.0%

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

      1. mul-1-neg 21.0%

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

      2. +-commutative 21.0%

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

      3. unpow2 21.0%

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

      4. unpow2 21.0%

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

      5. hypot-define 46.7%

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

   5. Simplified 46.7%

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

      1. neg-sub0 46.7%

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

      2. associate-*l/ 46.7%

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

      3. pow1/2 46.7%

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

      4. pow1/2 46.7%

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

      5. hypot-undefine 21.1%

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

      6. unpow2 21.1%

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

      7. unpow2 21.1%

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

      8. pow-prod-down 21.2%

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

      9. unpow2 21.2%

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

      10. unpow2 21.2%

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

      11. hypot-undefine 46.8%

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

   7. Applied egg-rr 46.8%

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

      1. neg-sub0 46.8%

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

      2. distribute-neg-frac2 46.8%

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

      3. unpow1/2 46.8%

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

   9. Simplified 46.8%

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

   10. Taylor expanded in A around 0 44.4%

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

       1. +-commutative 44.4%

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

       2. mul-1-neg 44.4%

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

       3. unsub-neg 44.4%

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

       4. *-commutative 44.4%

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

   12. Simplified 44.4%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.55 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A \cdot F - B \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.9%

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

3. Taylor expanded in C around 0 8.5%

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

   1. mul-1-neg 8.5%

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

   2. +-commutative 8.5%

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

   3. unpow2 8.5%

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

   4. unpow2 8.5%

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

   5. hypot-define 16.6%

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

5. Simplified 16.6%

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

   1. neg-sub0 16.6%

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

   2. associate-*l/ 16.6%

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

   3. pow1/2 16.6%

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

   4. pow1/2 16.6%

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

   5. hypot-undefine 8.6%

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

   6. unpow2 8.6%

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

   7. unpow2 8.6%

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

   8. pow-prod-down 8.6%

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

   9. unpow2 8.6%

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

   10. unpow2 8.6%

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

   11. hypot-undefine 16.7%

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

7. Applied egg-rr 16.7%

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

   1. neg-sub0 16.7%

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

   2. distribute-neg-frac2 16.7%

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

   3. unpow1/2 16.6%

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

9. Simplified 16.6%

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

10. Taylor expanded in A around 0 14.4%

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

    1. +-commutative 14.4%

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

    2. mul-1-neg 14.4%

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

    3. unsub-neg 14.4%

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

    4. *-commutative 14.4%

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

12. Simplified 14.4%

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

13. Final simplification 14.4%

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

Alternative 11: 25.9% accurate, 5.9× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.9%

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

3. Taylor expanded in C around 0 8.5%

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

   1. mul-1-neg 8.5%

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

   2. +-commutative 8.5%

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

   3. unpow2 8.5%

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

   4. unpow2 8.5%

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

   5. hypot-define 16.6%

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

5. Simplified 16.6%

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

   1. neg-sub0 16.6%

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

   2. associate-*l/ 16.6%

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

   3. pow1/2 16.6%

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

   4. pow1/2 16.6%

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

   5. hypot-undefine 8.6%

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

   6. unpow2 8.6%

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

   7. unpow2 8.6%

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

   8. pow-prod-down 8.6%

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

   9. unpow2 8.6%

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

   10. unpow2 8.6%

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

   11. hypot-undefine 16.7%

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

7. Applied egg-rr 16.7%

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

   1. neg-sub0 16.7%

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

   2. distribute-neg-frac2 16.7%

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

   3. unpow1/2 16.6%

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

9. Simplified 16.6%

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

10. Taylor expanded in A around 0 15.2%

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

11. Final simplification 15.2%

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

Alternative 12: 8.7% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.9%

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

3. Taylor expanded in C around 0 8.5%

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

   1. mul-1-neg 8.5%

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

   2. +-commutative 8.5%

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

   3. unpow2 8.5%

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

   4. unpow2 8.5%

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

   5. hypot-define 16.6%

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

5. Simplified 16.6%

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

   1. add-exp-log 14.4%

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

7. Applied egg-rr 14.4%

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

8. Taylor expanded in A around -inf 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 3.0%

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

   5. rem-square-sqrt 3.1%

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

   6. metadata-eval 3.1%

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

10. Simplified 3.1%

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

11. Final simplification 3.1%

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

Alternative 13: 1.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.9%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.1%

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

5. Simplified 2.1%

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

6. Taylor expanded in F around 0 2.1%

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

   1. sqrt-unprod 2.1%

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

   2. pow1/2 2.2%

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

8. Applied egg-rr 2.2%

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

9. Final simplification 2.2%

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

Alternative 14: 1.6% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.9%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.1%

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

5. Simplified 2.1%

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

6. Taylor expanded in F around 0 2.1%

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

   1. pow1 2.1%

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

   2. sqrt-unprod 2.1%

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

8. Applied egg-rr 2.1%

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

   1. unpow1 2.1%

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

   2. *-commutative 2.1%

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

   3. associate-*r/ 2.1%

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

10. Simplified 2.1%

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

Reproduce

herbie shell --seed 2024129 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))