ABCF->ab-angle b

Percentage Accurate: 18.8% → 51.4%

Time: 23.8s

Alternatives: 13

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: 18.8% 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: 51.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := \left(4 \cdot A\right) \cdot C - {B\_m}^{2}\\ t_2 := \frac{\sqrt{\left(\sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot t\_1\right)\right)}}{t\_1}\\ t_3 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := -t\_3\\ t_5 := F \cdot t\_3\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) - t\_0}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot \left(A + \left(C - t\_0\right)\right)\right)}}{t\_4}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C)))
        (t_1 (- (* (* 4.0 A) C) (pow B_m 2.0)))
        (t_2
         (/
          (sqrt
           (*
            (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))
            (* 2.0 (* F t_1))))
          t_1))
        (t_3 (fma B_m B_m (* A (* C -4.0))))
        (t_4 (- t_3))
        (t_5 (* F t_3)))
   (if (<= t_2 (- INFINITY))
     (*
      (sqrt (* F (/ (- (+ A C) t_0) (fma -4.0 (* A C) (pow B_m 2.0)))))
      (- (sqrt 2.0)))
     (if (<= t_2 -2e-134)
       (/ (sqrt (* t_5 (* 2.0 (+ A (- C t_0))))) t_4)
       (if (<= t_2 INFINITY)
         (/
          (sqrt (* t_5 (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
          t_4)
         (/ -1.0 (/ B_m (sqrt (* (* 2.0 F) (- A (hypot 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 = hypot(B_m, (A - C));
	double t_1 = ((4.0 * A) * C) - pow(B_m, 2.0);
	double t_2 = sqrt(((sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * t_1)))) / t_1;
	double t_3 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_4 = -t_3;
	double t_5 = F * t_3;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sqrt((F * (((A + C) - t_0) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_2 <= -2e-134) {
		tmp = sqrt((t_5 * (2.0 * (A + (C - t_0))))) / t_4;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_5 * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / t_4;
	} else {
		tmp = -1.0 / (B_m / sqrt(((2.0 * F) * (A - hypot(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 = hypot(B_m, Float64(A - C))
	t_1 = Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0))
	t_2 = Float64(sqrt(Float64(Float64(sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))) - Float64(A + C)) * Float64(2.0 * Float64(F * t_1)))) / t_1)
	t_3 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_4 = Float64(-t_3)
	t_5 = Float64(F * t_3)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) - t_0) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_2 <= -2e-134)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(C - t_0))))) / t_4);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / t_4);
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(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[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[(F * t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, -2e-134], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * N[(A + N[(C - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(t$95$5 * 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$4), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Simplified 16.5%

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

      1. clear-num 16.5%

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

      2. inv-pow 16.5%

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

   6. Applied egg-rr 15.9%

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

      1. unpow-1 15.9%

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

      2. associate-*l* 15.9%

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

      3. associate-+r- 16.5%

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

      4. hypot-undefine 3.3%

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

      5. unpow2 3.3%

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

      6. unpow2 3.3%

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

      7. +-commutative 3.3%

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

      8. unpow2 3.3%

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

      9. unpow2 3.3%

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

      10. hypot-undefine 16.5%

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

   8. Simplified 16.5%

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

   9. Taylor expanded in F around 0 21.7%

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

       1. mul-1-neg 21.7%

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

       2. associate-/l* 31.9%

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

       3. unpow2 31.9%

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

       4. unpow2 31.9%

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

       5. hypot-undefine 73.0%

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

       6. fma-define 73.0%

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

   11. Simplified 73.0%

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

   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))) < -2.00000000000000008e-134

   1. Initial program 98.7%

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

   3. Simplified 98.7%

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

   if -2.00000000000000008e-134 < (/.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 17.4%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in C around inf 31.5%

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

      1. mul-1-neg 31.5%

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

   7. Simplified 31.5%

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

   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 1.9%

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

      1. mul-1-neg 1.9%

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

      2. +-commutative 1.9%

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

      3. unpow2 1.9%

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

      4. unpow2 1.9%

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

      5. hypot-define 17.8%

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

   5. Simplified 17.8%

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

      1. add-cbrt-cube 12.5%

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

      2. pow1/3 11.9%

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

      3. add-sqr-sqrt 11.9%

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

      4. pow1 11.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 12.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 12.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\color{blue}{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 12.0%

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

   7. Applied egg-rr 12.0%

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

      1. unpow1/3 12.6%

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

   9. Simplified 12.6%

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

       1. associate-*l/ 12.5%

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

       2. pow1/2 12.5%

          \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}}{B} \]

       3. pow1/3 12.0%

          \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{B} \]

       4. pow-pow 17.8%

          \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

       5. metadata-eval 17.8%

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

       6. unpow-prod-down 17.9%

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

       7. clear-num 17.9%

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

       8. unpow1/2 17.9%

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

       9. associate-*r* 17.9%

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

   11. Applied egg-rr 17.9%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 4e-34)
     (/ (sqrt (* (* 4.0 A) (* F t_0))) (- t_0))
     (/ -1.0 (/ B_m (sqrt (* (* 2.0 F) (- A (hypot 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) <= 4e-34) {
		tmp = sqrt(((4.0 * A) * (F * t_0))) / -t_0;
	} else {
		tmp = -1.0 / (B_m / sqrt(((2.0 * F) * (A - hypot(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) <= 4e-34)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * t_0))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(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], 4e-34], 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[(B$95$m / N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999971e-34

   1. Initial program 21.3%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in A around -inf 24.2%

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

   if 3.99999999999999971e-34 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.4%

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

   3. Taylor expanded in C around 0 8.0%

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

      1. mul-1-neg 8.0%

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

      2. +-commutative 8.0%

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

      3. unpow2 8.0%

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

      4. unpow2 8.0%

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

      5. hypot-define 22.4%

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

   5. Simplified 22.4%

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

      1. add-cbrt-cube 15.9%

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

      2. pow1/3 15.0%

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

      3. add-sqr-sqrt 15.0%

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

      4. pow1 15.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 15.1%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 15.1%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\color{blue}{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 15.1%

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

   7. Applied egg-rr 15.1%

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

      1. unpow1/3 15.9%

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

   9. Simplified 15.9%

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

       1. associate-*l/ 15.9%

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

       2. pow1/2 15.9%

          \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}}{B} \]

       3. pow1/3 15.1%

          \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{B} \]

       4. pow-pow 22.4%

          \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

       5. metadata-eval 22.4%

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

       6. unpow-prod-down 22.5%

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

       7. clear-num 22.5%

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

       8. unpow1/2 22.5%

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

       9. associate-*r* 22.5%

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

   11. Applied egg-rr 22.5%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e-34)
   (/
    -1.0
    (/
     (fma B_m B_m (* A (* C -4.0)))
     (sqrt (* -8.0 (* A (* C (* F (+ A A))))))))
   (/ -1.0 (/ B_m (sqrt (* (* 2.0 F) (- A (hypot 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) <= 4e-34) {
		tmp = -1.0 / (fma(B_m, B_m, (A * (C * -4.0))) / sqrt((-8.0 * (A * (C * (F * (A + A)))))));
	} else {
		tmp = -1.0 / (B_m / sqrt(((2.0 * F) * (A - hypot(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) <= 4e-34)
		tmp = Float64(-1.0 / Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) / sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(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], 4e-34], N[(-1.0 / N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999971e-34

   1. Initial program 21.3%

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

   3. Simplified 29.5%

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

      1. clear-num 29.5%

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

      2. inv-pow 29.5%

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

   6. Applied egg-rr 28.1%

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

      1. unpow-1 28.1%

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

      2. associate-*l* 28.1%

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

      3. associate-+r- 29.5%

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

      4. hypot-undefine 22.3%

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

      5. unpow2 22.3%

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

      6. unpow2 22.3%

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

      7. +-commutative 22.3%

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

      8. unpow2 22.3%

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

      9. unpow2 22.3%

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

      10. hypot-undefine 29.5%

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

   8. Simplified 29.5%

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

   9. Taylor expanded in C around inf 22.1%

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

   if 3.99999999999999971e-34 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.4%

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

   3. Taylor expanded in C around 0 8.0%

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

      1. mul-1-neg 8.0%

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

      2. +-commutative 8.0%

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

      3. unpow2 8.0%

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

      4. unpow2 8.0%

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

      5. hypot-define 22.4%

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

   5. Simplified 22.4%

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

      1. add-cbrt-cube 15.9%

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

      2. pow1/3 15.0%

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

      3. add-sqr-sqrt 15.0%

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

      4. pow1 15.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 15.1%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 15.1%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\color{blue}{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 15.1%

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

   7. Applied egg-rr 15.1%

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

      1. unpow1/3 15.9%

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

   9. Simplified 15.9%

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

       1. associate-*l/ 15.9%

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

       2. pow1/2 15.9%

          \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}}{B} \]

       3. pow1/3 15.1%

          \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{B} \]

       4. pow-pow 22.4%

          \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

       5. metadata-eval 22.4%

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

       6. unpow-prod-down 22.5%

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

       7. clear-num 22.5%

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

       8. unpow1/2 22.5%

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

       9. associate-*r* 22.5%

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

   11. Applied egg-rr 22.5%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{-8 \cdot \left(A \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.5e-28)
   (/
    -1.0
    (/
     (fma B_m B_m (* A (* C -4.0)))
     (sqrt (* -8.0 (* A (* (* C F) (+ A A)))))))
   (/ -1.0 (/ B_m (sqrt (* (* 2.0 F) (- A (hypot 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 (B_m <= 2.5e-28) {
		tmp = -1.0 / (fma(B_m, B_m, (A * (C * -4.0))) / sqrt((-8.0 * (A * ((C * F) * (A + A))))));
	} else {
		tmp = -1.0 / (B_m / sqrt(((2.0 * F) * (A - hypot(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 <= 2.5e-28)
		tmp = Float64(-1.0 / Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) / sqrt(Float64(-8.0 * Float64(A * Float64(Float64(C * F) * Float64(A + A)))))));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(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[B$95$m, 2.5e-28], N[(-1.0 / N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-8.0 * N[(A * N[(N[(C * F), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.5000000000000001e-28

   1. Initial program 19.0%

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

   3. Simplified 26.4%

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

      1. clear-num 26.4%

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

      2. inv-pow 26.4%

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

   6. Applied egg-rr 25.3%

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

      1. unpow-1 25.3%

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

      2. associate-*l* 25.3%

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

      3. associate-+r- 26.4%

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

      4. hypot-undefine 19.8%

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

      5. unpow2 19.8%

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

      6. unpow2 19.8%

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

      7. +-commutative 19.8%

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

      8. unpow2 19.8%

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

      9. unpow2 19.8%

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

      10. hypot-undefine 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in C around inf 16.9%

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

       1. associate-*r* 16.0%

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

       2. mul-1-neg 16.0%

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

   11. Simplified 16.0%

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

   if 2.5000000000000001e-28 < B

   1. Initial program 11.4%

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

   3. Taylor expanded in C around 0 13.8%

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

      1. mul-1-neg 13.8%

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

      2. +-commutative 13.8%

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

      3. unpow2 13.8%

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

      4. unpow2 13.8%

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

      5. hypot-define 39.5%

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

   5. Simplified 39.5%

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

      1. add-cbrt-cube 27.6%

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

      2. pow1/3 26.0%

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

      3. add-sqr-sqrt 26.0%

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

      4. pow1 26.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 26.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 26.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\color{blue}{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 26.0%

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

   7. Applied egg-rr 26.0%

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

      1. unpow1/3 27.6%

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

   9. Simplified 27.6%

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

       1. associate-*l/ 27.6%

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

       2. pow1/2 27.6%

          \[\leadsto -\frac{\color{blue}{{2}^{0.5}} \cdot \sqrt[3]{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}}}{B} \]

       3. pow1/3 26.0%

          \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{B} \]

       4. pow-pow 39.4%

          \[\leadsto -\frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

       5. metadata-eval 39.4%

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

       6. unpow-prod-down 39.6%

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

       7. clear-num 39.6%

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

       8. unpow1/2 39.6%

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

       9. associate-*r* 39.6%

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

   11. Applied egg-rr 39.6%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{-8 \cdot \left(A \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 27.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.8 \cdot 10^{+218}:\\ \;\;\;\;\frac{{\left(4 \cdot \left(A \cdot F\right)\right)}^{0.5}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -1.8e+218)
   (/ (pow (* 4.0 (* A F)) 0.5) (- B_m))
   (/ (sqrt (* 2.0 (* F (- C (hypot C B_m))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -1.8e+218) {
		tmp = pow((4.0 * (A * F)), 0.5) / -B_m;
	} else {
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -1.8e+218) {
		tmp = Math.pow((4.0 * (A * F)), 0.5) / -B_m;
	} else {
		tmp = Math.sqrt((2.0 * (F * (C - Math.hypot(C, B_m))))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -1.8e+218:
		tmp = math.pow((4.0 * (A * F)), 0.5) / -B_m
	else:
		tmp = math.sqrt((2.0 * (F * (C - math.hypot(C, B_m))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -1.8e+218)
		tmp = Float64((Float64(4.0 * Float64(A * F)) ^ 0.5) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - hypot(C, B_m))))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -1.8e+218)
		tmp = ((4.0 * (A * F)) ^ 0.5) / -B_m;
	else
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -1.8e+218], N[(N[Power[N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.79999999999999995e218

   1. Initial program 2.7%

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

   3. Taylor expanded in C around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. +-commutative 1.4%

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

      3. unpow2 1.4%

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

      4. unpow2 1.4%

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

      5. hypot-define 9.2%

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

   5. Simplified 9.2%

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

      1. associate-*l/ 9.2%

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

      2. pow1/2 9.2%

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

      3. pow1/2 10.2%

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

      4. pow-prod-down 10.2%

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

   7. Applied egg-rr 10.2%

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

   8. Taylor expanded in A around -inf 10.3%

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

   if -1.79999999999999995e218 < A

   1. Initial program 18.7%

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

   3. Taylor expanded in A around 0 6.8%

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

      1. mul-1-neg 6.8%

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

      2. +-commutative 6.8%

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

      3. unpow2 6.8%

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

      4. unpow2 6.8%

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

      5. hypot-define 14.3%

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

   5. Simplified 14.3%

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

      1. neg-sub0 14.3%

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

      2. associate-*l/ 14.3%

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

      3. pow1/2 14.3%

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

      4. pow1/2 14.3%

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

      5. pow-prod-down 14.4%

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

   7. Applied egg-rr 14.4%

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

      1. neg-sub0 14.4%

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

      2. distribute-neg-frac2 14.4%

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

      3. unpow1/2 14.3%

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

   9. Simplified 14.3%

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

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.8 \cdot 10^{+218}:\\ \;\;\;\;\frac{{\left(4 \cdot \left(A \cdot F\right)\right)}^{0.5}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.9% accurate, 3.0× speedup

Math

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

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
 (/ (pow (* 2.0 (* F (- A (hypot B_m A)))) 0.5) (- B_m)))

C

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

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 * (A - Math.hypot(B_m, A)))), 0.5) / -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.pow((2.0 * (F * (A - math.hypot(B_m, A)))), 0.5) / -B_m

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64((Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A)))) ^ 0.5) / 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 = ((2.0 * (F * (A - hypot(B_m, A)))) ^ 0.5) / -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[Power[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in C around 0 6.1%

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

   1. mul-1-neg 6.1%

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

   2. +-commutative 6.1%

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

   3. unpow2 6.1%

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

   4. unpow2 6.1%

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

   5. hypot-define 13.6%

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

5. Simplified 13.6%

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

   1. associate-*l/ 13.6%

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

   2. pow1/2 13.6%

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

   3. pow1/2 13.7%

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

   4. pow-prod-down 13.7%

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

7. Applied egg-rr 13.7%

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

8. Final simplification 13.7%

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

Alternative 7: 31.9% accurate, 3.0× speedup

Math

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

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 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) {
	return sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
}

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) * (A - Math.hypot(B_m, A)))) / -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) * (A - math.hypot(B_m, A)))) / -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(2.0 * F) * Float64(A - hypot(B_m, A)))) / 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 * F) * (A - hypot(B_m, A)))) / -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[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in C around 0 6.1%

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

   1. mul-1-neg 6.1%

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

   2. +-commutative 6.1%

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

   3. unpow2 6.1%

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

   4. unpow2 6.1%

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

   5. hypot-define 13.6%

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

5. Simplified 13.6%

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

   1. neg-sub0 13.6%

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

   2. associate-*l/ 13.6%

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

   3. pow1/2 13.6%

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

   4. pow1/2 13.7%

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

   5. pow-prod-down 13.7%

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

7. Applied egg-rr 13.7%

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

   1. neg-sub0 13.7%

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

   2. distribute-neg-frac2 13.7%

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

   3. unpow1/2 13.6%

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

   4. associate-*r* 13.6%

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

9. Simplified 13.6%

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

Alternative 8: 26.8% accurate, 5.4× speedup

Math

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

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
 (/ (pow (* B_m (+ (* F -2.0) (* 2.0 (/ (* A F) B_m)))) 0.5) (- B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return pow((B_m * ((F * -2.0) + (2.0 * ((A * F) / B_m)))), 0.5) / -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 = ((b_m * ((f * (-2.0d0)) + (2.0d0 * ((a * f) / b_m)))) ** 0.5d0) / -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.pow((B_m * ((F * -2.0) + (2.0 * ((A * F) / B_m)))), 0.5) / -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.pow((B_m * ((F * -2.0) + (2.0 * ((A * F) / B_m)))), 0.5) / -B_m

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64((Float64(B_m * Float64(Float64(F * -2.0) + Float64(2.0 * Float64(Float64(A * F) / B_m)))) ^ 0.5) / 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 = ((B_m * ((F * -2.0) + (2.0 * ((A * F) / B_m)))) ^ 0.5) / -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[Power[N[(B$95$m * N[(N[(F * -2.0), $MachinePrecision] + N[(2.0 * N[(N[(A * F), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in C around 0 6.1%

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

   1. mul-1-neg 6.1%

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

   2. +-commutative 6.1%

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

   3. unpow2 6.1%

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

   4. unpow2 6.1%

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

   5. hypot-define 13.6%

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

5. Simplified 13.6%

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

   1. associate-*l/ 13.6%

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

   2. pow1/2 13.6%

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

   3. pow1/2 13.7%

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

   4. pow-prod-down 13.7%

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

7. Applied egg-rr 13.7%

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

8. Taylor expanded in B around inf 11.3%

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

9. Final simplification 11.3%

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

Alternative 9: 27.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -3.3 \cdot 10^{+218}:\\ \;\;\;\;\frac{{\left(4 \cdot \left(A \cdot F\right)\right)}^{0.5}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

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 (<= A -3.3e+218)
   (/ (pow (* 4.0 (* A F)) 0.5) (- B_m))
   (/ (sqrt (* -2.0 (* B_m F))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -3.3e+218) {
		tmp = pow((4.0 * (A * F)), 0.5) / -B_m;
	} else {
		tmp = sqrt((-2.0 * (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 (a <= (-3.3d+218)) then
        tmp = ((4.0d0 * (a * f)) ** 0.5d0) / -b_m
    else
        tmp = sqrt(((-2.0d0) * (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 (A <= -3.3e+218) {
		tmp = Math.pow((4.0 * (A * F)), 0.5) / -B_m;
	} else {
		tmp = Math.sqrt((-2.0 * (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 A <= -3.3e+218:
		tmp = math.pow((4.0 * (A * F)), 0.5) / -B_m
	else:
		tmp = math.sqrt((-2.0 * (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 (A <= -3.3e+218)
		tmp = Float64((Float64(4.0 * Float64(A * F)) ^ 0.5) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(-2.0 * 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 (A <= -3.3e+218)
		tmp = ((4.0 * (A * F)) ^ 0.5) / -B_m;
	else
		tmp = sqrt((-2.0 * (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[A, -3.3e+218], N[(N[Power[N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -3.29999999999999998e218

   1. Initial program 2.7%

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

   3. Taylor expanded in C around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. +-commutative 1.4%

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

      3. unpow2 1.4%

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

      4. unpow2 1.4%

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

      5. hypot-define 9.2%

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

   5. Simplified 9.2%

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

      1. associate-*l/ 9.2%

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

      2. pow1/2 9.2%

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

      3. pow1/2 10.2%

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

      4. pow-prod-down 10.2%

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

   7. Applied egg-rr 10.2%

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

   8. Taylor expanded in A around -inf 10.3%

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

   if -3.29999999999999998e218 < A

   1. Initial program 18.7%

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

   3. Taylor expanded in C around 0 6.7%

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

      1. mul-1-neg 6.7%

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

      2. +-commutative 6.7%

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

      3. unpow2 6.7%

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

      4. unpow2 6.7%

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

      5. hypot-define 14.1%

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

   5. Simplified 14.1%

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

      1. neg-sub0 14.1%

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

      2. associate-*l/ 14.1%

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

      3. pow1/2 14.1%

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

      4. pow1/2 14.1%

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

      5. pow-prod-down 14.1%

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

   7. Applied egg-rr 14.1%

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

      1. neg-sub0 14.1%

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

      2. distribute-neg-frac2 14.1%

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

      3. unpow1/2 14.1%

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

      4. associate-*r* 14.1%

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

   9. Simplified 14.1%

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

   10. Taylor expanded in A around 0 12.6%

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

4. Final simplification 12.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.3 \cdot 10^{+218}:\\ \;\;\;\;\frac{{\left(4 \cdot \left(A \cdot F\right)\right)}^{0.5}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.9% accurate, 5.7× speedup

Math

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

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
 (/ (pow (* 2.0 (* F (- A B_m))) 0.5) (- B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return pow((2.0 * (F * (A - B_m))), 0.5) / -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 = ((2.0d0 * (f * (a - b_m))) ** 0.5d0) / -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.pow((2.0 * (F * (A - B_m))), 0.5) / -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.pow((2.0 * (F * (A - B_m))), 0.5) / -B_m

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64((Float64(2.0 * Float64(F * Float64(A - B_m))) ^ 0.5) / 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 = ((2.0 * (F * (A - B_m))) ^ 0.5) / -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[Power[N[(2.0 * N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in C around 0 6.1%

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

   1. mul-1-neg 6.1%

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

   2. +-commutative 6.1%

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

   3. unpow2 6.1%

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

   4. unpow2 6.1%

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

   5. hypot-define 13.6%

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

5. Simplified 13.6%

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

   1. associate-*l/ 13.6%

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

   2. pow1/2 13.6%

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

   3. pow1/2 13.7%

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

   4. pow-prod-down 13.7%

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

7. Applied egg-rr 13.7%

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

8. Taylor expanded in A around 0 11.3%

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

9. Final simplification 11.3%

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

Alternative 11: 26.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in C around 0 6.1%

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

   1. mul-1-neg 6.1%

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

   2. +-commutative 6.1%

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

   3. unpow2 6.1%

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

   4. unpow2 6.1%

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

   5. hypot-define 13.6%

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

5. Simplified 13.6%

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

   1. neg-sub0 13.6%

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

   2. associate-*l/ 13.6%

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

   3. pow1/2 13.6%

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

   4. pow1/2 13.7%

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

   5. pow-prod-down 13.7%

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

7. Applied egg-rr 13.7%

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

   1. neg-sub0 13.7%

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

   2. distribute-neg-frac2 13.7%

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

   3. unpow1/2 13.6%

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

   4. associate-*r* 13.6%

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

9. Simplified 13.6%

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

10. Taylor expanded in A around 0 11.7%

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

Alternative 12: 2.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in B around -inf 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 1.9%

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

5. Simplified 1.9%

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

6. Taylor expanded in F around 0 1.9%

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

   1. pow1 1.9%

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

   2. sqrt-unprod 1.9%

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

8. Applied egg-rr 1.9%

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

   1. unpow1 1.9%

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

10. Simplified 1.9%

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

    1. associate-*l/ 1.9%

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

    2. *-commutative 1.9%

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

    3. frac-2neg 1.9%

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

    4. add-sqr-sqrt 1.1%

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

    5. sqrt-unprod 2.4%

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

    6. sqr-neg 2.4%

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

    7. sqrt-unprod 1.1%

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

    8. add-sqr-sqrt 1.8%

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

12. Applied egg-rr 1.8%

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

    1. distribute-lft-neg-in 1.8%

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

    2. metadata-eval 1.8%

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

14. Simplified 1.8%

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

15. Final simplification 1.8%

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

Alternative 13: 1.6% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in B around -inf 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 1.9%

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

5. Simplified 1.9%

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

6. Taylor expanded in F around 0 1.9%

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

   1. pow1 1.9%

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

   2. sqrt-unprod 1.9%

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

8. Applied egg-rr 1.9%

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

   1. unpow1 1.9%

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

10. Simplified 1.9%

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

11. Final simplification 1.9%

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

Reproduce

herbie shell --seed 2024139 
(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))))