ABCF->ab-angle b

Percentage Accurate: 18.9% → 47.8%

Time: 26.3s

Alternatives: 11

Speedup: 5.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% 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: 47.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_1}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{B \cdot B + \left(A \cdot A - A \cdot A\right)}{C}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_1)))
   (if (<= t_2 -2e-184)
     (/
      (*
       (sqrt (fma B B (* C (* A -4.0))))
       (- (sqrt (* (+ A (- C (hypot B (- A C)))) (* 2.0 F)))))
      (fma B B (* A (* C -4.0))))
     (if (<= t_2 INFINITY)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            (* F t_0)
            (+ A (+ A (* -0.5 (/ (+ (* B B) (- (* A A) (* A A))) C))))))))
        t_0)
       (/ -1.0 (/ B (sqrt (* F (* 2.0 (- A (hypot B A)))))))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double tmp;
	if (t_2 <= -2e-184) {
		tmp = (sqrt(fma(B, B, (C * (A * -4.0)))) * -sqrt(((A + (C - hypot(B, (A - C)))) * (2.0 * F)))) / fma(B, B, (A * (C * -4.0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -sqrt((2.0 * ((F * t_0) * (A + (A + (-0.5 * (((B * B) + ((A * A) - (A * A))) / C))))))) / t_0;
	} else {
		tmp = -1.0 / (B / sqrt((F * (2.0 * (A - hypot(B, A))))));
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_1)
	tmp = 0.0
	if (t_2 <= -2e-184)
		tmp = Float64(Float64(sqrt(fma(B, B, Float64(C * Float64(A * -4.0)))) * Float64(-sqrt(Float64(Float64(A + Float64(C - hypot(B, Float64(A - C)))) * Float64(2.0 * F))))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64(Float64(B * B) + Float64(Float64(A * A) - Float64(A * A))) / C)))))))) / t_0);
	else
		tmp = Float64(-1.0 / Float64(B / sqrt(Float64(F * Float64(2.0 * Float64(A - hypot(B, A)))))));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-184], N[(N[(N[Sqrt[N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(A + N[(C - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(A + N[(A + N[(-0.5 * N[(N[(N[(B * B), $MachinePrecision] + N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-1.0 / N[(B / N[Sqrt[N[(F * N[(2.0 * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -2.0000000000000001e-184

   1. Initial program 44.5%

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

   3. Simplified 55.5%

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

      1. sqrt-prod 70.5%

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

      2. associate-*r* 70.5%

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

      3. *-commutative 70.5%

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

      4. associate--r- 70.3%

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

      5. +-commutative 70.3%

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

      6. *-commutative 70.3%

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

   5. Applied egg-rr 70.3%

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

   if -2.0000000000000001e-184 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in C around inf 20.7%

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

      1. associate--l+ 20.7%

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

      2. associate--l+ 20.8%

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

      3. unpow2 20.8%

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

      4. unpow2 20.8%

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

      5. unpow2 20.8%

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

      6. mul-1-neg 20.8%

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

      7. mul-1-neg 20.8%

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

      8. sqr-neg 20.8%

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

      9. mul-1-neg 20.8%

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

   6. Simplified 20.8%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 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. Step-by-step derivation

   3. Simplified 0.9%

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

   4. Taylor expanded in C around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. +-commutative 1.8%

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

      3. unpow2 1.8%

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

      4. unpow2 1.8%

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

      5. hypot-def 18.9%

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

   6. Simplified 18.9%

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

      1. associate-*l/ 18.9%

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

   8. Applied egg-rr 18.9%

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

      1. clear-num 18.9%

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

      2. inv-pow 18.9%

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

      3. sqrt-unprod 19.0%

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

   10. Applied egg-rr 19.0%

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

       1. unpow-1 19.0%

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

       2. *-commutative 19.0%

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

       3. *-commutative 19.0%

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

       4. associate-*l* 19.0%

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

       5. hypot-def 1.8%

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

       6. unpow2 1.8%

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

       7. unpow2 1.8%

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

       8. +-commutative 1.8%

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

       9. unpow2 1.8%

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

       10. unpow2 1.8%

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

       11. hypot-def 19.0%

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

   12. Simplified 19.0%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{B \cdot B + \left(A \cdot A - A \cdot A\right)}{C}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}}\\ \end{array} \]

Alternative 2: 47.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 8e-9)
   (/
    (- (sqrt (* 2.0 (* (+ (* B B) (* -4.0 (* A C))) (* F (* 2.0 A))))))
    (- (* B B) (* 4.0 (* A C))))
   (/ (- (sqrt (* F (* 2.0 (- A (hypot B A)))))) B)))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 8e-9) {
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -sqrt((F * (2.0 * (A - hypot(B, A))))) / B;
	}
	return tmp;
}

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 8e-9) {
		tmp = -Math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -Math.sqrt((F * (2.0 * (A - Math.hypot(B, A))))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 8e-9:
		tmp = -math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -math.sqrt((F * (2.0 * (A - math.hypot(B, A))))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 8e-9)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))) * Float64(F * Float64(2.0 * A)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(A - hypot(B, A)))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 8e-9)
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -sqrt((F * (2.0 * (A - hypot(B, A))))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 8e-9], N[((-N[Sqrt[N[(2.0 * N[(N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(F * N[(2.0 * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.0000000000000005e-9

   1. Initial program 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around -inf 16.3%

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

      1. *-un-lft-identity 16.3%

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

      2. associate-*l* 15.7%

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

      3. *-commutative 15.7%

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

   6. Applied egg-rr 15.7%

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

      1. *-lft-identity 15.7%

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

      2. unpow2 15.7%

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

      3. cancel-sign-sub-inv 15.7%

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

      4. unpow2 15.7%

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

      5. metadata-eval 15.7%

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

   8. Simplified 15.7%

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

   if 8.0000000000000005e-9 < B

   1. Initial program 15.0%

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

   3. Simplified 19.8%

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

   4. Taylor expanded in C around 0 13.8%

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

      1. mul-1-neg 13.8%

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

      2. +-commutative 13.8%

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

      3. unpow2 13.8%

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

      4. unpow2 13.8%

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

      5. hypot-def 43.5%

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

   6. Simplified 43.5%

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

      1. associate-*l/ 43.4%

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

   8. Applied egg-rr 43.4%

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

      1. expm1-log1p-u 42.0%

         \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F}\right)\right)}}{B} \]

      2. expm1-udef 31.1%

         \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F}\right)} - 1}}{B} \]

      3. sqrt-unprod 31.1%

         \[\leadsto -\frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right)}}\right)} - 1}{B} \]

   10. Applied egg-rr 31.1%

       \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right)}\right)} - 1}}{B} \]
   11. Step-by-step derivation

       1. expm1-def 42.0%

          \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right)}\right)\right)}}{B} \]

       2. expm1-log1p 43.5%

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

       3. *-commutative 43.5%

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

       4. *-commutative 43.5%

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

       5. associate-*l* 43.5%

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

       6. hypot-def 13.8%

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

       7. unpow2 13.8%

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

       8. unpow2 13.8%

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

       9. +-commutative 13.8%

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

       10. unpow2 13.8%

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

       11. unpow2 13.8%

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

       12. hypot-def 43.5%

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

   12. Simplified 43.5%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 3: 43.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 0.00097:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 0.00097)
   (/
    (- (sqrt (* 2.0 (* (+ (* B B) (* -4.0 (* A C))) (* F (* 2.0 A))))))
    (- (* B B) (* 4.0 (* A C))))
   (- (* (/ (sqrt 2.0) B) (sqrt (* F (- A B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 0.00097) {
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -((sqrt(2.0) / B) * sqrt((F * (A - B))));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (b <= 0.00097d0) then
        tmp = -sqrt((2.0d0 * (((b * b) + ((-4.0d0) * (a * c))) * (f * (2.0d0 * a))))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = -((sqrt(2.0d0) / b) * sqrt((f * (a - b))))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 0.00097) {
		tmp = -Math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -((Math.sqrt(2.0) / B) * Math.sqrt((F * (A - B))));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 0.00097:
		tmp = -math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -((math.sqrt(2.0) / B) * math.sqrt((F * (A - B))))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 0.00097)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))) * Float64(F * Float64(2.0 * A)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(A - B)))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 0.00097)
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -((sqrt(2.0) / B) * sqrt((F * (A - B))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 0.00097], N[((-N[Sqrt[N[(2.0 * N[(N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 9.70000000000000051e-4

   1. Initial program 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 16.2%

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

      1. *-un-lft-identity 16.2%

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

      2. associate-*l* 15.7%

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

      3. *-commutative 15.7%

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

   6. Applied egg-rr 15.7%

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

      1. *-lft-identity 15.7%

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

      2. unpow2 15.7%

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

      3. cancel-sign-sub-inv 15.7%

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

      4. unpow2 15.7%

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

      5. metadata-eval 15.7%

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

   8. Simplified 15.7%

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

   if 9.70000000000000051e-4 < B

   1. Initial program 15.2%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in C around 0 13.9%

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

      1. mul-1-neg 13.9%

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

      2. +-commutative 13.9%

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

      3. unpow2 13.9%

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

      4. unpow2 13.9%

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

      5. hypot-def 44.2%

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

   6. Simplified 44.2%

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

   7. Taylor expanded in A around 0 38.3%

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

      1. mul-1-neg 38.3%

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

      2. unsub-neg 38.3%

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

   9. Simplified 38.3%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.00097:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 4: 43.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 0.000145:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 0.000145)
   (/
    (- (sqrt (* 2.0 (* (+ (* B B) (* -4.0 (* A C))) (* F (* 2.0 A))))))
    (- (* B B) (* 4.0 (* A C))))
   (* (/ (sqrt 2.0) B) (- (sqrt (* B (- F)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 0.000145) {
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((B * -F));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (b <= 0.000145d0) then
        tmp = -sqrt((2.0d0 * (((b * b) + ((-4.0d0) * (a * c))) * (f * (2.0d0 * a))))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = (sqrt(2.0d0) / b) * -sqrt((b * -f))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 0.000145) {
		tmp = -Math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((B * -F));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 0.000145:
		tmp = -math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((B * -F))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 0.000145)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))) * Float64(F * Float64(2.0 * A)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(B * Float64(-F)))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 0.000145)
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = (sqrt(2.0) / B) * -sqrt((B * -F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 0.000145], N[((-N[Sqrt[N[(2.0 * N[(N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.45e-4

   1. Initial program 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 16.2%

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

      1. *-un-lft-identity 16.2%

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

      2. associate-*l* 15.7%

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

      3. *-commutative 15.7%

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

   6. Applied egg-rr 15.7%

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

      1. *-lft-identity 15.7%

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

      2. unpow2 15.7%

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

      3. cancel-sign-sub-inv 15.7%

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

      4. unpow2 15.7%

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

      5. metadata-eval 15.7%

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

   8. Simplified 15.7%

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

   if 1.45e-4 < B

   1. Initial program 15.2%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in C around 0 13.9%

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

      1. mul-1-neg 13.9%

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

      2. +-commutative 13.9%

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

      3. unpow2 13.9%

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

      4. unpow2 13.9%

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

      5. hypot-def 44.2%

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

   6. Simplified 44.2%

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

   7. Taylor expanded in A around 0 38.1%

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

      1. associate-*r* 38.1%

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

      2. mul-1-neg 38.1%

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

   9. Simplified 38.1%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.000145:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 5: 29.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 4.7e-5)
     (/
      (- (sqrt (* 2.0 (* (+ (* B B) (* -4.0 (* A C))) (* F (* 2.0 A))))))
      t_0)
     (if (<= B 7.5e+76)
       (/ (- (sqrt (* 2.0 (* (* F t_0) (+ A (- C B)))))) t_0)
       (* -2.0 (/ (pow (* A F) 0.5) B))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 4.7e-5) {
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / t_0;
	} else if (B <= 7.5e+76) {
		tmp = -sqrt((2.0 * ((F * t_0) * (A + (C - B))))) / t_0;
	} else {
		tmp = -2.0 * (pow((A * F), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= 4.7d-5) then
        tmp = -sqrt((2.0d0 * (((b * b) + ((-4.0d0) * (a * c))) * (f * (2.0d0 * a))))) / t_0
    else if (b <= 7.5d+76) then
        tmp = -sqrt((2.0d0 * ((f * t_0) * (a + (c - b))))) / t_0
    else
        tmp = (-2.0d0) * (((a * f) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 4.7e-5) {
		tmp = -Math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / t_0;
	} else if (B <= 7.5e+76) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (A + (C - B))))) / t_0;
	} else {
		tmp = -2.0 * (Math.pow((A * F), 0.5) / B);
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 4.7e-5:
		tmp = -math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / t_0
	elif B <= 7.5e+76:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (A + (C - B))))) / t_0
	else:
		tmp = -2.0 * (math.pow((A * F), 0.5) / B)
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 4.7e-5)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))) * Float64(F * Float64(2.0 * A)))))) / t_0);
	elseif (B <= 7.5e+76)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(A + Float64(C - B)))))) / t_0);
	else
		tmp = Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 4.7e-5)
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / t_0;
	elseif (B <= 7.5e+76)
		tmp = -sqrt((2.0 * ((F * t_0) * (A + (C - B))))) / t_0;
	else
		tmp = -2.0 * (((A * F) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 4.7e-5], N[((-N[Sqrt[N[(2.0 * N[(N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 7.5e+76], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(A + N[(C - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.69999999999999972e-5

   1. Initial program 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 16.2%

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

      1. *-un-lft-identity 16.2%

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

      2. associate-*l* 15.7%

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

      3. *-commutative 15.7%

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

   6. Applied egg-rr 15.7%

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

      1. *-lft-identity 15.7%

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

      2. unpow2 15.7%

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

      3. cancel-sign-sub-inv 15.7%

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

      4. unpow2 15.7%

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

      5. metadata-eval 15.7%

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

   8. Simplified 15.7%

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

   if 4.69999999999999972e-5 < B < 7.4999999999999995e76

   1. Initial program 37.6%

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

   3. Simplified 37.6%

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

      1. associate--l+ 37.4%

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

      2. unpow2 37.4%

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

      3. hypot-udef 42.6%

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

      4. add-cbrt-cube 31.9%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 31.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*l* 31.9%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. cube-unmult 31.9%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{3}}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 31.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{3}}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in B around inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. unsub-neg 26.5%

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

   10. Simplified 26.5%

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

   if 7.4999999999999995e76 < B

   1. Initial program 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in A around -inf 3.8%

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

   5. Taylor expanded in B around inf 12.5%

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

      1. associate-*r/ 12.5%

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

      2. *-rgt-identity 12.5%

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

      3. *-commutative 12.5%

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

   7. Simplified 12.5%

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

      1. pow1/2 12.6%

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

      2. *-commutative 12.6%

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

   9. Applied egg-rr 12.6%

      \[\leadsto -2 \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 6: 27.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 3.2e-8)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
     (* -2.0 (/ (pow (* A F) 0.5) B)))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 3.2e-8) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = -2.0 * (pow((A * F), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= 3.2d-8) then
        tmp = -sqrt((2.0d0 * ((f * t_0) * (2.0d0 * a)))) / t_0
    else
        tmp = (-2.0d0) * (((a * f) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 3.2e-8) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = -2.0 * (Math.pow((A * F), 0.5) / B);
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 3.2e-8:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0
	else:
		tmp = -2.0 * (math.pow((A * F), 0.5) / B)
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 3.2e-8)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 3.2e-8)
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	else
		tmp = -2.0 * (((A * F) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 3.2e-8], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.2000000000000002e-8

   1. Initial program 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around -inf 16.3%

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

   if 3.2000000000000002e-8 < B

   1. Initial program 15.0%

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

   3. Simplified 15.0%

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

   4. Taylor expanded in A around -inf 3.4%

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

   5. Taylor expanded in B around inf 9.5%

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

      1. associate-*r/ 9.5%

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

      2. *-rgt-identity 9.5%

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

      3. *-commutative 9.5%

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

   7. Simplified 9.5%

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

      1. pow1/2 9.6%

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

      2. *-commutative 9.6%

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

   9. Applied egg-rr 9.6%

      \[\leadsto -2 \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 7: 27.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 3.2e-8)
   (/
    (- (sqrt (* 2.0 (* (+ (* B B) (* -4.0 (* A C))) (* F (* 2.0 A))))))
    (- (* B B) (* 4.0 (* A C))))
   (* -2.0 (/ (pow (* A F) 0.5) B))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 3.2e-8) {
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (pow((A * F), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (b <= 3.2d-8) then
        tmp = -sqrt((2.0d0 * (((b * b) + ((-4.0d0) * (a * c))) * (f * (2.0d0 * a))))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = (-2.0d0) * (((a * f) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 3.2e-8) {
		tmp = -Math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (Math.pow((A * F), 0.5) / B);
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 3.2e-8:
		tmp = -math.sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -2.0 * (math.pow((A * F), 0.5) / B)
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 3.2e-8)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))) * Float64(F * Float64(2.0 * A)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 3.2e-8)
		tmp = -sqrt((2.0 * (((B * B) + (-4.0 * (A * C))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -2.0 * (((A * F) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 3.2e-8], N[((-N[Sqrt[N[(2.0 * N[(N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.2000000000000002e-8

   1. Initial program 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around -inf 16.3%

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

      1. *-un-lft-identity 16.3%

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

      2. associate-*l* 15.7%

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

      3. *-commutative 15.7%

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

   6. Applied egg-rr 15.7%

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

      1. *-lft-identity 15.7%

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

      2. unpow2 15.7%

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

      3. cancel-sign-sub-inv 15.7%

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

      4. unpow2 15.7%

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

      5. metadata-eval 15.7%

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

   8. Simplified 15.7%

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

   if 3.2000000000000002e-8 < B

   1. Initial program 15.0%

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

   3. Simplified 15.0%

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

   4. Taylor expanded in A around -inf 3.4%

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

   5. Taylor expanded in B around inf 9.5%

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

      1. associate-*r/ 9.5%

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

      2. *-rgt-identity 9.5%

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

      3. *-commutative 9.5%

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

   7. Simplified 9.5%

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

      1. pow1/2 9.6%

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

      2. *-commutative 9.6%

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

   9. Applied egg-rr 9.6%

      \[\leadsto -2 \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 8: 22.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 2.3e-8)
   (/
    (- (sqrt (* 2.0 (* (* 2.0 A) (* -4.0 (* A (* C F)))))))
    (- (* B B) (* 4.0 (* A C))))
   (* -2.0 (/ (pow (* A F) 0.5) B))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 2.3e-8) {
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (pow((A * F), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (b <= 2.3d-8) then
        tmp = -sqrt((2.0d0 * ((2.0d0 * a) * ((-4.0d0) * (a * (c * f)))))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = (-2.0d0) * (((a * f) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 2.3e-8) {
		tmp = -Math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (Math.pow((A * F), 0.5) / B);
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 2.3e-8:
		tmp = -math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -2.0 * (math.pow((A * F), 0.5) / B)
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 2.3e-8)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(2.0 * A) * Float64(-4.0 * Float64(A * Float64(C * F))))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 2.3e-8)
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -2.0 * (((A * F) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 2.3e-8], N[((-N[Sqrt[N[(2.0 * N[(N[(2.0 * A), $MachinePrecision] * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.3000000000000001e-8

   1. Initial program 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around -inf 16.3%

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

   5. Taylor expanded in B around 0 12.9%

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

      1. *-commutative 12.9%

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

   7. Simplified 12.9%

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

   if 2.3000000000000001e-8 < B

   1. Initial program 15.0%

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

   3. Simplified 15.0%

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

   4. Taylor expanded in A around -inf 3.4%

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

   5. Taylor expanded in B around inf 9.5%

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

      1. associate-*r/ 9.5%

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

      2. *-rgt-identity 9.5%

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

      3. *-commutative 9.5%

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

   7. Simplified 9.5%

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

      1. pow1/2 9.6%

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

      2. *-commutative 9.6%

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

   9. Applied egg-rr 9.6%

      \[\leadsto -2 \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 9: 18.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C \cdot F\right) \cdot \left(\left(A \cdot A\right) \cdot -8\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 3.1e-9)
   (/
    (- (sqrt (* 2.0 (* (* C F) (* (* A A) -8.0)))))
    (- (* B B) (* 4.0 (* A C))))
   (* -2.0 (/ (pow (* A F) 0.5) B))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 3.1e-9) {
		tmp = -sqrt((2.0 * ((C * F) * ((A * A) * -8.0)))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (pow((A * F), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (b <= 3.1d-9) then
        tmp = -sqrt((2.0d0 * ((c * f) * ((a * a) * (-8.0d0))))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = (-2.0d0) * (((a * f) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 3.1e-9) {
		tmp = -Math.sqrt((2.0 * ((C * F) * ((A * A) * -8.0)))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (Math.pow((A * F), 0.5) / B);
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 3.1e-9:
		tmp = -math.sqrt((2.0 * ((C * F) * ((A * A) * -8.0)))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -2.0 * (math.pow((A * F), 0.5) / B)
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 3.1e-9)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(C * F) * Float64(Float64(A * A) * -8.0))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 3.1e-9)
		tmp = -sqrt((2.0 * ((C * F) * ((A * A) * -8.0)))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -2.0 * (((A * F) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 3.1e-9], N[((-N[Sqrt[N[(2.0 * N[(N[(C * F), $MachinePrecision] * N[(N[(A * A), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.10000000000000005e-9

   1. Initial program 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around -inf 16.3%

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

   5. Taylor expanded in B around 0 10.4%

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

      1. associate-*r* 10.4%

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

      2. unpow2 10.4%

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

      3. *-commutative 10.4%

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

   7. Simplified 10.4%

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

   if 3.10000000000000005e-9 < B

   1. Initial program 15.0%

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

   3. Simplified 15.0%

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

   4. Taylor expanded in A around -inf 3.4%

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

   5. Taylor expanded in B around inf 9.5%

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

      1. associate-*r/ 9.5%

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

      2. *-rgt-identity 9.5%

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

      3. *-commutative 9.5%

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

   7. Simplified 9.5%

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

      1. pow1/2 9.6%

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

      2. *-commutative 9.6%

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

   9. Applied egg-rr 9.6%

      \[\leadsto -2 \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C \cdot F\right) \cdot \left(\left(A \cdot A\right) \cdot -8\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 10: 9.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    code = (-2.0d0) * (((a * f) ** 0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Simplified 19.8%

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

4. Taylor expanded in A around -inf 13.3%

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

5. Taylor expanded in B around inf 3.3%

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

   1. associate-*r/ 3.3%

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

   2. *-rgt-identity 3.3%

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

   3. *-commutative 3.3%

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

7. Simplified 3.3%

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

   1. pow1/2 3.5%

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

   2. *-commutative 3.5%

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

9. Applied egg-rr 3.5%

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

10. Final simplification 3.5%

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

Alternative 11: 9.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    code = (-2.0d0) * (sqrt((a * f)) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Simplified 19.8%

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

4. Taylor expanded in A around -inf 13.3%

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

5. Taylor expanded in B around inf 3.3%

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

   1. associate-*r/ 3.3%

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

   2. *-rgt-identity 3.3%

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

   3. *-commutative 3.3%

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

7. Simplified 3.3%

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

8. Final simplification 3.3%

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

Reproduce

herbie shell --seed 2023257 
(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))))