ABCF->ab-angle b

Percentage Accurate: 19.4% → 45.4%

Time: 31.1s

Alternatives: 11

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.4% 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: 45.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left({B\_m}^{2} + {A}^{2}\right) - {A}^{2}}{C}\right)\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+114}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(F \cdot \left(2 \cdot \left(C + \left(A - \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)\right)\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e-35)
     (/
      (sqrt
       (*
        (* 2.0 (* (- (pow B_m 2.0) t_0) F))
        (+
         A
         (+ A (* -0.5 (/ (- (+ (pow B_m 2.0) (pow A 2.0)) (pow A 2.0)) C))))))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e+114)
       (/ (sqrt (* t_1 (* F (* 2.0 (+ C (- A (hypot (- A C) B_m))))))) (- t_1))
       (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-35) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (A + (A + (-0.5 * (((pow(B_m, 2.0) + pow(A, 2.0)) - pow(A, 2.0)) / C)))))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e+114) {
		tmp = sqrt((t_1 * (F * (2.0 * (C + (A - hypot((A - C), B_m))))))) / -t_1;
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-35)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64(Float64((B_m ^ 2.0) + (A ^ 2.0)) - (A ^ 2.0)) / C)))))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e+114)
		tmp = Float64(sqrt(Float64(t_1 * Float64(F * Float64(2.0 * Float64(C + Float64(A - hypot(Float64(A - C), B_m))))))) / Float64(-t_1));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-35], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + N[(-0.5 * N[(N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+114], N[(N[Sqrt[N[(t$95$1 * N[(F * N[(2.0 * N[(C + N[(A - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 4.99999999999999964e-35

   1. Initial program 29.6%

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

   3. Taylor expanded in C around inf 23.6%

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

      1. associate--l+ 23.6%

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

      2. +-commutative 23.6%

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

      3. unpow2 23.6%

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

      4. mul-1-neg 23.6%

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

      5. mul-1-neg 23.6%

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

      6. sqr-neg 23.6%

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

      7. unpow2 23.6%

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

      8. mul-1-neg 23.6%

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

   5. Simplified 23.6%

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

   if 4.99999999999999964e-35 < (pow.f64 B 2) < 1e114

   1. Initial program 38.0%

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

   3. Simplified 53.3%

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

      1. distribute-frac-neg2 53.3%

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

   6. Applied egg-rr 54.9%

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

   if 1e114 < (pow.f64 B 2)

   1. Initial program 8.6%

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

   3. Taylor expanded in C around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. +-commutative 9.5%

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

      3. unpow2 9.5%

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

      4. unpow2 9.5%

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

      5. hypot-define 23.6%

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

   5. Simplified 23.6%

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

      1. associate-*l/ 23.6%

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

      2. pow1/2 23.6%

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

      3. pow1/2 23.6%

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

      4. pow-prod-down 23.7%

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

      5. hypot-undefine 9.5%

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

      6. unpow2 9.5%

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

      7. unpow2 9.5%

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

      8. +-commutative 9.5%

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

      9. unpow2 9.5%

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

      10. unpow2 9.5%

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

      11. hypot-define 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. unpow1/2 23.7%

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

      2. associate-*r* 23.7%

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

      3. *-commutative 23.7%

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

      4. hypot-undefine 9.5%

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

      5. unpow2 9.5%

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

      6. unpow2 9.5%

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

      7. +-commutative 9.5%

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

      8. unpow2 9.5%

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

      9. unpow2 9.5%

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

      10. hypot-define 23.7%

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

   9. Simplified 23.7%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {A}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+114}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(C + \left(A - \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+114}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(F \cdot \left(2 \cdot \left(C + \left(A - \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)\right)\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e-35)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e+114)
       (/ (sqrt (* t_1 (* F (* 2.0 (+ C (- A (hypot (- A C) B_m))))))) (- t_1))
       (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-35) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e+114) {
		tmp = sqrt((t_1 * (F * (2.0 * (C + (A - hypot((A - C), B_m))))))) / -t_1;
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-35)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e+114)
		tmp = Float64(sqrt(Float64(t_1 * Float64(F * Float64(2.0 * Float64(C + Float64(A - hypot(Float64(A - C), B_m))))))) / Float64(-t_1));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-35], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+114], N[(N[Sqrt[N[(t$95$1 * N[(F * N[(2.0 * N[(C + N[(A - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 4.99999999999999964e-35

   1. Initial program 29.6%

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

   3. Taylor expanded in A around -inf 25.0%

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

   if 4.99999999999999964e-35 < (pow.f64 B 2) < 1e114

   1. Initial program 38.0%

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

   3. Simplified 53.3%

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

      1. distribute-frac-neg2 53.3%

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

   6. Applied egg-rr 54.9%

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

   if 1e114 < (pow.f64 B 2)

   1. Initial program 8.6%

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

   3. Taylor expanded in C around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. +-commutative 9.5%

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

      3. unpow2 9.5%

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

      4. unpow2 9.5%

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

      5. hypot-define 23.6%

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

   5. Simplified 23.6%

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

      1. associate-*l/ 23.6%

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

      2. pow1/2 23.6%

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

      3. pow1/2 23.6%

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

      4. pow-prod-down 23.7%

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

      5. hypot-undefine 9.5%

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

      6. unpow2 9.5%

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

      7. unpow2 9.5%

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

      8. +-commutative 9.5%

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

      9. unpow2 9.5%

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

      10. unpow2 9.5%

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

      11. hypot-define 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. unpow1/2 23.7%

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

      2. associate-*r* 23.7%

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

      3. *-commutative 23.7%

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

      4. hypot-undefine 9.5%

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

      5. unpow2 9.5%

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

      6. unpow2 9.5%

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

      7. +-commutative 9.5%

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

      8. unpow2 9.5%

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

      9. unpow2 9.5%

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

      10. hypot-define 23.7%

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

   9. Simplified 23.7%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+114}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(C + \left(A - \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 4.25 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 4.25e-16)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 4.25e-16) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 4.25e-16) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * F) * (A - Math.hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 4.25e-16:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt(((2.0 * F) * (A - math.hypot(B_m, A)))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 4.25e-16)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 4.25e-16)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 4.25e-16], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4.25e-16

   1. Initial program 24.3%

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

   3. Taylor expanded in A around -inf 17.4%

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

   if 4.25e-16 < B

   1. Initial program 11.2%

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

   3. Taylor expanded in C around 0 17.5%

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

      1. mul-1-neg 17.5%

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

      2. +-commutative 17.5%

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

      3. unpow2 17.5%

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

      4. unpow2 17.5%

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

      5. hypot-define 40.7%

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

   5. Simplified 40.7%

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

      1. associate-*l/ 40.6%

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

      2. pow1/2 40.6%

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

      3. pow1/2 40.6%

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

      4. pow-prod-down 40.8%

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

      5. hypot-undefine 17.6%

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

      6. unpow2 17.6%

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

      7. unpow2 17.6%

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

      8. +-commutative 17.6%

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

      9. unpow2 17.6%

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

      10. unpow2 17.6%

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

      11. hypot-define 40.8%

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

   7. Applied egg-rr 40.8%

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

      1. unpow1/2 40.8%

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

      2. associate-*r* 40.8%

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

      3. *-commutative 40.8%

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

      4. hypot-undefine 17.5%

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

      5. unpow2 17.5%

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

      6. unpow2 17.5%

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

      7. +-commutative 17.5%

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

      8. unpow2 17.5%

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

      9. unpow2 17.5%

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

      10. hypot-define 40.8%

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

   9. Simplified 40.8%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.25 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-57)
   (/
    (sqrt (* 2.0 (* -4.0 (* A (* C (* F (* 2.0 A)))))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-57) {
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (2.0 * A))))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-57) {
		tmp = Math.sqrt((2.0 * (-4.0 * (A * (C * (F * (2.0 * A))))))) / ((4.0 * (A * C)) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * F) * (A - Math.hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-57:
		tmp = math.sqrt((2.0 * (-4.0 * (A * (C * (F * (2.0 * A))))))) / ((4.0 * (A * C)) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt(((2.0 * F) * (A - math.hypot(B_m, A)))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-57)
		tmp = Float64(sqrt(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * Float64(F * Float64(2.0 * A))))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-57)
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (2.0 * A))))))) / ((4.0 * (A * C)) - (B_m ^ 2.0));
	else
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-57], N[(N[Sqrt[N[(2.0 * N[(-4.0 * N[(A * N[(C * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.0000000000000002e-57

   1. Initial program 26.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in C around inf 22.9%

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

      1. mul-1-neg 22.9%

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

   7. Simplified 22.9%

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

      1. pow1 22.9%

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

      2. neg-mul-1 22.9%

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

      3. cancel-sign-sub-inv 22.9%

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

      4. metadata-eval 22.9%

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

      5. *-un-lft-identity 22.9%

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

   9. Applied egg-rr 22.9%

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

       1. unpow1 22.9%

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

       2. count-2 22.9%

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

       3. *-commutative 22.9%

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

   11. Simplified 22.9%

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

   if 5.0000000000000002e-57 < (pow.f64 B 2)

   1. Initial program 16.5%

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

   3. Taylor expanded in C around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. +-commutative 9.0%

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

      3. unpow2 9.0%

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

      4. unpow2 9.0%

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

      5. hypot-define 20.1%

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

   5. Simplified 20.1%

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

      1. associate-*l/ 20.1%

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

      2. pow1/2 20.1%

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

      3. pow1/2 20.1%

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

      4. pow-prod-down 20.2%

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

      5. hypot-undefine 9.0%

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

      6. unpow2 9.0%

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

      7. unpow2 9.0%

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

      8. +-commutative 9.0%

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

      9. unpow2 9.0%

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

      10. unpow2 9.0%

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

      11. hypot-define 20.2%

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

   7. Applied egg-rr 20.2%

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

      1. unpow1/2 20.2%

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

      2. associate-*r* 20.2%

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

      3. *-commutative 20.2%

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

      4. hypot-undefine 9.0%

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

      5. unpow2 9.0%

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

      6. unpow2 9.0%

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

      7. +-commutative 9.0%

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

      8. unpow2 9.0%

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

      9. unpow2 9.0%

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

      10. hypot-define 20.2%

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

   9. Simplified 20.2%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(2 \cdot A\right) \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-57)
   (/
    (sqrt (* -8.0 (* (* 2.0 A) (* A (* C F)))))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-57) {
		tmp = sqrt((-8.0 * ((2.0 * A) * (A * (C * F))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-57) {
		tmp = Math.sqrt((-8.0 * ((2.0 * A) * (A * (C * F))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * F) * (A - Math.hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-57:
		tmp = math.sqrt((-8.0 * ((2.0 * A) * (A * (C * F))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt(((2.0 * F) * (A - math.hypot(B_m, A)))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-57)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(2.0 * A) * Float64(A * Float64(C * F))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-57)
		tmp = sqrt((-8.0 * ((2.0 * A) * (A * (C * F))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	else
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-57], N[(N[Sqrt[N[(-8.0 * N[(N[(2.0 * A), $MachinePrecision] * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.0000000000000002e-57

   1. Initial program 26.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in C around inf 22.9%

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

      1. mul-1-neg 22.9%

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

   7. Simplified 22.9%

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

      1. div-inv 22.8%

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

      2. associate-*r* 22.8%

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

      3. metadata-eval 22.8%

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

      4. associate-*r* 22.9%

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

      5. neg-mul-1 22.9%

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

      6. cancel-sign-sub-inv 22.9%

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

      7. metadata-eval 22.9%

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

      8. *-un-lft-identity 22.9%

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

   9. Applied egg-rr 22.9%

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

       1. associate-*r/ 22.9%

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

       2. *-commutative 22.9%

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

       3. *-lft-identity 22.9%

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

       4. associate-*r* 23.9%

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

       5. associate-*r* 20.2%

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

       6. *-commutative 20.2%

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

       7. count-2 20.2%

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

       8. *-commutative 20.2%

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

       9. associate-*r* 20.2%

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

   11. Simplified 20.2%

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

   if 5.0000000000000002e-57 < (pow.f64 B 2)

   1. Initial program 16.5%

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

   3. Taylor expanded in C around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. +-commutative 9.0%

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

      3. unpow2 9.0%

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

      4. unpow2 9.0%

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

      5. hypot-define 20.1%

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

   5. Simplified 20.1%

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

      1. associate-*l/ 20.1%

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

      2. pow1/2 20.1%

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

      3. pow1/2 20.1%

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

      4. pow-prod-down 20.2%

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

      5. hypot-undefine 9.0%

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

      6. unpow2 9.0%

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

      7. unpow2 9.0%

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

      8. +-commutative 9.0%

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

      9. unpow2 9.0%

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

      10. unpow2 9.0%

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

      11. hypot-define 20.2%

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

   7. Applied egg-rr 20.2%

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

      1. unpow1/2 20.2%

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

      2. associate-*r* 20.2%

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

      3. *-commutative 20.2%

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

      4. hypot-undefine 9.0%

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

      5. unpow2 9.0%

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

      6. unpow2 9.0%

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

      7. +-commutative 9.0%

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

      8. unpow2 9.0%

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

      9. unpow2 9.0%

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

      10. hypot-define 20.2%

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

   9. Simplified 20.2%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(2 \cdot A\right) \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(2 \cdot A\right)\right) \cdot \left(\left(A \cdot C\right) \cdot -8\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-57)
   (/
    (sqrt (* (* F (* 2.0 A)) (* (* A C) -8.0)))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-57) {
		tmp = sqrt(((F * (2.0 * A)) * ((A * C) * -8.0))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-57) {
		tmp = Math.sqrt(((F * (2.0 * A)) * ((A * C) * -8.0))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * F) * (A - Math.hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-57:
		tmp = math.sqrt(((F * (2.0 * A)) * ((A * C) * -8.0))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt(((2.0 * F) * (A - math.hypot(B_m, A)))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-57)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(2.0 * A)) * Float64(Float64(A * C) * -8.0))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-57)
		tmp = sqrt(((F * (2.0 * A)) * ((A * C) * -8.0))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	else
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-57], N[(N[Sqrt[N[(N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision] * N[(N[(A * C), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.0000000000000002e-57

   1. Initial program 26.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in C around inf 22.9%

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

      1. mul-1-neg 22.9%

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

   7. Simplified 22.9%

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

      1. div-inv 22.8%

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

      2. associate-*r* 22.8%

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

      3. metadata-eval 22.8%

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

      4. associate-*r* 22.9%

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

      5. neg-mul-1 22.9%

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

      6. cancel-sign-sub-inv 22.9%

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

      7. metadata-eval 22.9%

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

      8. *-un-lft-identity 22.9%

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

   9. Applied egg-rr 22.9%

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

       1. associate-*r/ 22.9%

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

       2. *-commutative 22.9%

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

       3. *-lft-identity 22.9%

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

       4. associate-*r* 22.9%

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

       5. count-2 22.9%

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

       6. *-commutative 22.9%

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

       7. associate-*r* 22.9%

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

   11. Simplified 22.9%

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

   if 5.0000000000000002e-57 < (pow.f64 B 2)

   1. Initial program 16.5%

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

   3. Taylor expanded in C around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. +-commutative 9.0%

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

      3. unpow2 9.0%

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

      4. unpow2 9.0%

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

      5. hypot-define 20.1%

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

   5. Simplified 20.1%

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

      1. associate-*l/ 20.1%

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

      2. pow1/2 20.1%

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

      3. pow1/2 20.1%

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

      4. pow-prod-down 20.2%

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

      5. hypot-undefine 9.0%

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

      6. unpow2 9.0%

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

      7. unpow2 9.0%

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

      8. +-commutative 9.0%

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

      9. unpow2 9.0%

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

      10. unpow2 9.0%

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

      11. hypot-define 20.2%

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

   7. Applied egg-rr 20.2%

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

      1. unpow1/2 20.2%

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

      2. associate-*r* 20.2%

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

      3. *-commutative 20.2%

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

      4. hypot-undefine 9.0%

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

      5. unpow2 9.0%

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

      6. unpow2 9.0%

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

      7. +-commutative 9.0%

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

      8. unpow2 9.0%

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

      9. unpow2 9.0%

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

      10. hypot-define 20.2%

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

   9. Simplified 20.2%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(2 \cdot A\right)\right) \cdot \left(\left(A \cdot C\right) \cdot -8\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{+244}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(2 \cdot A\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{B\_m \cdot \left(-F\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -4e+244)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (* 2.0 A)))))
   (* (sqrt (* B_m (- F))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -4e+244) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (2.0 * A)));
	} else {
		tmp = sqrt((B_m * -F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-4d+244)) then
        tmp = (sqrt(2.0d0) / b_m) * -sqrt((f * (2.0d0 * a)))
    else
        tmp = sqrt((b_m * -f)) * (sqrt(2.0d0) / -b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -4e+244) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (2.0 * A)));
	} else {
		tmp = Math.sqrt((B_m * -F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -4e+244:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (2.0 * A)))
	else:
		tmp = math.sqrt((B_m * -F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -4e+244)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(2.0 * A)))));
	else
		tmp = Float64(sqrt(Float64(B_m * Float64(-F))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -4e+244)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (2.0 * A)));
	else
		tmp = sqrt((B_m * -F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -4e+244], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.0000000000000003e244

   1. Initial program 0.4%

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

   3. Taylor expanded in C around 0 0.9%

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

      1. mul-1-neg 0.9%

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

      2. +-commutative 0.9%

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

      3. unpow2 0.9%

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

      4. unpow2 0.9%

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

      5. hypot-define 20.5%

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

   5. Simplified 20.5%

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

   6. Taylor expanded in A around -inf 20.5%

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

      1. associate-*r* 20.5%

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

      2. count-2 20.5%

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

      3. *-commutative 20.5%

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

      4. count-2 20.5%

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

      5. *-commutative 20.5%

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

   8. Simplified 20.5%

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

   if -4.0000000000000003e244 < A

   1. Initial program 21.6%

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

   3. Taylor expanded in C around 0 7.8%

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

      1. mul-1-neg 7.8%

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

      2. +-commutative 7.8%

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

      3. unpow2 7.8%

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

      4. unpow2 7.8%

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

      5. hypot-define 14.5%

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

   5. Simplified 14.5%

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

   6. Taylor expanded in A around 0 13.3%

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

      1. mul-1-neg 13.3%

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

   8. Simplified 13.3%

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

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{+244}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(2 \cdot A\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.8%

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

3. Taylor expanded in C around 0 7.6%

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

   1. mul-1-neg 7.6%

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

   2. +-commutative 7.6%

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

   3. unpow2 7.6%

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

   4. unpow2 7.6%

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

   5. hypot-define 14.7%

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

5. Simplified 14.7%

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

   1. associate-*l/ 14.7%

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

   2. pow1/2 14.7%

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

   3. pow1/2 14.8%

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

   4. pow-prod-down 14.9%

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

   5. hypot-undefine 7.7%

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

   6. unpow2 7.7%

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

   7. unpow2 7.7%

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

   8. +-commutative 7.7%

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

   9. unpow2 7.7%

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

   10. unpow2 7.7%

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

   11. hypot-define 14.9%

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

7. Applied egg-rr 14.9%

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

   1. unpow1/2 14.8%

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

   2. associate-*r* 14.8%

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

   3. *-commutative 14.8%

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

   4. hypot-undefine 7.6%

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

   5. unpow2 7.6%

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

   6. unpow2 7.6%

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

   7. +-commutative 7.6%

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

   8. unpow2 7.6%

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

   9. unpow2 7.6%

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

   10. hypot-define 14.8%

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

9. Simplified 14.8%

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

10. Final simplification 14.8%

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

Alternative 9: 26.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.8%

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

3. Taylor expanded in C around 0 7.6%

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

   1. mul-1-neg 7.6%

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

   2. +-commutative 7.6%

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

   3. unpow2 7.6%

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

   4. unpow2 7.6%

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

   5. hypot-define 14.7%

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

5. Simplified 14.7%

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

6. Taylor expanded in A around 0 12.9%

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

   1. mul-1-neg 12.9%

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

8. Simplified 12.9%

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

9. Final simplification 12.9%

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

Alternative 10: 0.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.8%

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

3. Taylor expanded in C around 0 7.6%

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

   1. mul-1-neg 7.6%

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

   2. +-commutative 7.6%

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

   3. unpow2 7.6%

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

   4. unpow2 7.6%

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

   5. hypot-define 14.7%

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

5. Simplified 14.7%

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

6. Taylor expanded in B around -inf 1.5%

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

7. Final simplification 1.5%

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

Alternative 11: 0.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \left(0.25 \cdot \sqrt{\frac{F}{C}}\right) \cdot \sqrt{-16} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (* 0.25 (sqrt (/ F C))) (sqrt -16.0)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return (0.25 * sqrt((F / C))) * sqrt(-16.0);
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (0.25d0 * sqrt((f / c))) * sqrt((-16.0d0))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return (0.25 * Math.sqrt((F / C))) * Math.sqrt(-16.0);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return (0.25 * math.sqrt((F / C))) * math.sqrt(-16.0)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(0.25 * sqrt(Float64(F / C))) * sqrt(-16.0))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (0.25 * sqrt((F / C))) * sqrt(-16.0);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[(0.25 * N[Sqrt[N[(F / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[-16.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.8%

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

3. Simplified 20.4%

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

5. Taylor expanded in C around inf 12.4%

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

   1. mul-1-neg 12.4%

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

7. Simplified 12.4%

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

8. Taylor expanded in A around inf 0.0%

   \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-16}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 0.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot \sqrt{\frac{F}{C}}\right) \cdot \sqrt{-16}} \]

10. Simplified 0.0%

    \[\leadsto \color{blue}{\left(0.25 \cdot \sqrt{\frac{F}{C}}\right) \cdot \sqrt{-16}} \]

11. Final simplification 0.0%

    \[\leadsto \left(0.25 \cdot \sqrt{\frac{F}{C}}\right) \cdot \sqrt{-16} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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))))