ABCF->ab-angle b

Percentage Accurate: 19.6% → 51.3%

Time: 20.1s

Alternatives: 16

Speedup: 5.6×

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: 19.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 51.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \frac{2 \cdot \left(C \cdot \left(A + A\right)\right) - {B\_m}^{2}}{C}}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\_m\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 -4e-154)
     (*
      (sqrt
       (*
        F
        (/ (+ A (- C (hypot B_m (- A C)))) (fma -4.0 (* A C) (pow B_m 2.0)))))
      (- (sqrt 2.0)))
     (if (<= t_2 INFINITY)
       (/
        (sqrt (* (* F t_0) (/ (- (* 2.0 (* C (+ A A))) (pow B_m 2.0)) C)))
        (- t_0))
       (/
        -1.0
        (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot A B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -4e-154) {
		tmp = sqrt((F * ((A + (C - hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (((2.0 * (C * (A + A))) - pow(B_m, 2.0)) / C))) / -t_0;
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(A, B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -4e-154)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C - hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(Float64(Float64(2.0 * Float64(C * Float64(A + A))) - (B_m ^ 2.0)) / C))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-154], N[(N[Sqrt[N[(F * N[(N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(N[(N[(2.0 * N[(C * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.9999999999999999e-154

   1. Initial program 32.0%

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

   3. Simplified 36.3%

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

      1. add-cbrt-cube 19.3%

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

      2. pow3 19.3%

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

      3. *-commutative 19.3%

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

   6. Applied egg-rr 19.3%

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

   7. Taylor expanded in F around 0 49.0%

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

      1. mul-1-neg 49.0%

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

   9. Simplified 69.2%

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

   if -3.9999999999999999e-154 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 25.3%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in C around inf 37.1%

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

   6. Taylor expanded in C around 0 37.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 1.9%

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

      1. mul-1-neg 1.9%

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

      2. +-commutative 1.9%

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

      3. unpow2 1.9%

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

      4. unpow2 1.9%

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

      5. hypot-define 14.6%

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

   5. Simplified 14.6%

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

      1. pow1/2 14.6%

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

      2. pow-to-exp 14.5%

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

   7. Applied egg-rr 14.5%

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

      1. associate-*l/ 14.5%

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

      2. exp-to-pow 14.6%

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

      3. pow1/2 14.6%

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

      4. sqrt-prod 14.6%

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

      5. distribute-frac-neg2 14.6%

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

      6. clear-num 14.6%

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

      7. associate-*r* 14.6%

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

   9. Applied egg-rr 14.6%

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

   10. Taylor expanded in F around 0 1.6%

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

       1. mul-1-neg 1.6%

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

       2. distribute-rgt-neg-in 1.6%

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

       3. associate-/r* 1.6%

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

       4. unpow2 1.6%

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

       5. unpow2 1.6%

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

       6. hypot-define 14.5%

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

   12. Simplified 14.5%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \frac{2 \cdot \left(C \cdot \left(A + A\right)\right) - {B}^{2}}{C}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(2 \cdot A\right) - \frac{{B\_m}^{2}}{C}\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\_m\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e+66)
     (/ (sqrt (* (* F t_0) (- (* 2.0 (* 2.0 A)) (/ (pow B_m 2.0) C)))) (- t_0))
     (/
      -1.0
      (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot A B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e+66) {
		tmp = sqrt(((F * t_0) * ((2.0 * (2.0 * A)) - (pow(B_m, 2.0) / C)))) / -t_0;
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(A, B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+66)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(Float64(2.0 * Float64(2.0 * A)) - Float64((B_m ^ 2.0) / C)))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+66], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(N[(2.0 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999989e66

   1. Initial program 24.0%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in C around inf 31.4%

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

      1. *-un-lft-identity 31.4%

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

      2. *-commutative 31.4%

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

      3. *-commutative 31.4%

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

      4. +-commutative 31.4%

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

      5. fma-define 31.4%

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

      6. cancel-sign-sub-inv 31.4%

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

      7. metadata-eval 31.4%

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

      8. *-un-lft-identity 31.4%

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

      9. mul-1-neg 31.4%

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

   7. Applied egg-rr 31.4%

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

      1. *-lft-identity 31.4%

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

      2. fmm-undef 31.4%

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

      3. count-2 31.4%

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

   9. Simplified 31.4%

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

   if 1.99999999999999989e66 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.6%

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

   3. Taylor expanded in C around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. +-commutative 11.3%

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

      3. unpow2 11.3%

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

      4. unpow2 11.3%

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

      5. hypot-define 21.5%

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

   5. Simplified 21.5%

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

      1. pow1/2 21.5%

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

      2. pow-to-exp 21.5%

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

   7. Applied egg-rr 21.5%

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

      1. associate-*l/ 21.5%

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

      2. exp-to-pow 21.6%

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

      3. pow1/2 21.6%

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

      4. sqrt-prod 21.6%

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

      5. distribute-frac-neg2 21.6%

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

      6. clear-num 21.6%

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

      7. associate-*r* 21.6%

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

   9. Applied egg-rr 21.6%

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

   10. Taylor expanded in F around 0 11.1%

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

       1. mul-1-neg 11.1%

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

       2. distribute-rgt-neg-in 11.1%

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

       3. associate-/r* 11.6%

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

       4. unpow2 11.6%

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

       5. unpow2 11.6%

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

       6. hypot-define 22.2%

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

   12. Simplified 22.2%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(2 \cdot A\right) - \frac{{B}^{2}}{C}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\_m\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e+66)
     (/ (sqrt (* (* 4.0 A) (* F t_0))) (- t_0))
     (/
      -1.0
      (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot A B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e+66) {
		tmp = sqrt(((4.0 * A) * (F * t_0))) / -t_0;
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(A, B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+66)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * t_0))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+66], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999989e66

   1. Initial program 24.0%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in A around -inf 31.2%

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

   if 1.99999999999999989e66 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.6%

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

   3. Taylor expanded in C around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. +-commutative 11.3%

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

      3. unpow2 11.3%

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

      4. unpow2 11.3%

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

      5. hypot-define 21.5%

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

   5. Simplified 21.5%

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

      1. pow1/2 21.5%

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

      2. pow-to-exp 21.5%

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

   7. Applied egg-rr 21.5%

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

      1. associate-*l/ 21.5%

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

      2. exp-to-pow 21.6%

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

      3. pow1/2 21.6%

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

      4. sqrt-prod 21.6%

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

      5. distribute-frac-neg2 21.6%

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

      6. clear-num 21.6%

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

      7. associate-*r* 21.6%

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

   9. Applied egg-rr 21.6%

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

   10. Taylor expanded in F around 0 11.1%

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

       1. mul-1-neg 11.1%

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

       2. distribute-rgt-neg-in 11.1%

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

       3. associate-/r* 11.6%

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

       4. unpow2 11.6%

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

       5. unpow2 11.6%

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

       6. hypot-define 22.2%

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

   12. Simplified 22.2%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\_m\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-9)
   (/
    (sqrt (* (* A -8.0) (* C (* F (* 2.0 A)))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (/ -1.0 (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot A B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-9) {
		tmp = sqrt(((A * -8.0) * (C * (F * (2.0 * A))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(A, B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-9)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(2.0 * A))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, 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_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-9], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-9

   1. Initial program 22.8%

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

   3. Simplified 33.6%

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

      1. add-cbrt-cube 24.4%

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

      2. pow3 24.4%

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

      3. *-commutative 24.4%

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

   6. Applied egg-rr 24.4%

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

   7. Taylor expanded in C around inf 30.0%

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

      1. associate-*r* 30.0%

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

      2. neg-mul-1 30.0%

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

      3. unsub-neg 30.0%

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

      4. remove-double-neg 30.0%

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

      5. count-2 30.0%

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

   9. Simplified 30.0%

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

   if 1.00000000000000006e-9 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.7%

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

   3. Taylor expanded in C around 0 12.4%

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

      1. mul-1-neg 12.4%

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

      2. +-commutative 12.4%

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

      3. unpow2 12.4%

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

      4. unpow2 12.4%

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

      5. hypot-define 21.0%

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

   5. Simplified 21.0%

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

      1. pow1/2 21.0%

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

      2. pow-to-exp 21.0%

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

   7. Applied egg-rr 21.0%

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

      1. associate-*l/ 21.0%

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

      2. exp-to-pow 21.0%

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

      3. pow1/2 21.0%

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

      4. sqrt-prod 21.1%

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

      5. distribute-frac-neg2 21.1%

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

      6. clear-num 21.1%

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

      7. associate-*r* 21.1%

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

   9. Applied egg-rr 21.1%

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

   10. Taylor expanded in F around 0 12.2%

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

       1. mul-1-neg 12.2%

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

       2. distribute-rgt-neg-in 12.2%

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

       3. associate-/r* 12.6%

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

       4. unpow2 12.6%

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

       5. unpow2 12.6%

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

       6. hypot-define 21.4%

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

   12. Simplified 21.4%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.65:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.65)
   (/
    (sqrt (* (* A -8.0) (* C (* F (* 2.0 A)))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (/ -1.0 (* B_m (pow (* (* 2.0 F) (- A (hypot B_m A))) -0.5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.65) {
		tmp = sqrt(((A * -8.0) * (C * (F * (2.0 * A))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) * (A - hypot(B_m, A))), -0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.65)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(2.0 * A))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A))) ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.65], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.64999999999999991

   1. Initial program 18.6%

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

   3. Simplified 25.3%

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

      1. add-cbrt-cube 17.5%

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

      2. pow3 17.5%

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

      3. *-commutative 17.5%

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

   6. Applied egg-rr 17.5%

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

   7. Taylor expanded in C around inf 19.3%

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

      1. associate-*r* 19.3%

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

      2. neg-mul-1 19.3%

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

      3. unsub-neg 19.3%

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

      4. remove-double-neg 19.3%

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

      5. count-2 19.3%

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

   9. Simplified 19.3%

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

   if 3.64999999999999991 < B

   1. Initial program 14.6%

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

   3. Taylor expanded in C around 0 26.2%

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

      1. mul-1-neg 26.2%

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

      2. +-commutative 26.2%

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

      3. unpow2 26.2%

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

      4. unpow2 26.2%

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

      5. hypot-define 44.6%

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

   5. Simplified 44.6%

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

      1. pow1/2 44.6%

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

      2. pow-to-exp 44.5%

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

   7. Applied egg-rr 44.5%

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

      1. associate-*l/ 44.5%

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

      2. exp-to-pow 44.6%

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

      3. pow1/2 44.6%

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

      4. sqrt-prod 44.7%

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

      5. distribute-frac-neg2 44.7%

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

      6. clear-num 44.7%

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

      7. associate-*r* 44.7%

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

   9. Applied egg-rr 44.7%

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

       1. div-inv 44.6%

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

       2. pow1/2 44.6%

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

       3. pow-flip 44.7%

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

       4. metadata-eval 44.7%

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

   11. Applied egg-rr 44.7%

       \[\leadsto \frac{1}{\color{blue}{\left(-B\right) \cdot {\left(\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{-0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.65:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := 2 \cdot \left(A \cdot F\right)\\ \mathbf{if}\;A \leq -1.4 \cdot 10^{+245}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{t\_0 + -2 \cdot \left(B\_m \cdot F\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 2.0 (* A F))))
   (if (<= A -1.4e+245)
     (* (/ (sqrt 2.0) B_m) (- (sqrt t_0)))
     (/ -1.0 (/ B_m (sqrt (+ t_0 (* -2.0 (* B_m F)))))))))

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 = 2.0 * (A * F);
	double tmp;
	if (A <= -1.4e+245) {
		tmp = (sqrt(2.0) / B_m) * -sqrt(t_0);
	} else {
		tmp = -1.0 / (B_m / sqrt((t_0 + (-2.0 * (B_m * F)))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (a * f)
    if (a <= (-1.4d+245)) then
        tmp = (sqrt(2.0d0) / b_m) * -sqrt(t_0)
    else
        tmp = (-1.0d0) / (b_m / sqrt((t_0 + ((-2.0d0) * (b_m * f)))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 2.0 * (A * F);
	double tmp;
	if (A <= -1.4e+245) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt(t_0);
	} else {
		tmp = -1.0 / (B_m / Math.sqrt((t_0 + (-2.0 * (B_m * F)))));
	}
	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 = 2.0 * (A * F)
	tmp = 0
	if A <= -1.4e+245:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt(t_0)
	else:
		tmp = -1.0 / (B_m / math.sqrt((t_0 + (-2.0 * (B_m * F)))))
	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(2.0 * Float64(A * F))
	tmp = 0.0
	if (A <= -1.4e+245)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(t_0)));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(t_0 + Float64(-2.0 * Float64(B_m * F))))));
	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 = 2.0 * (A * F);
	tmp = 0.0;
	if (A <= -1.4e+245)
		tmp = (sqrt(2.0) / B_m) * -sqrt(t_0);
	else
		tmp = -1.0 / (B_m / sqrt((t_0 + (-2.0 * (B_m * F)))));
	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[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -1.4e+245], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(t$95$0 + N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < -1.39999999999999989e245

   1. Initial program 1.3%

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

   3. Taylor expanded in C around 0 0.8%

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

      1. mul-1-neg 0.8%

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

      2. +-commutative 0.8%

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

      3. unpow2 0.8%

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

      4. unpow2 0.8%

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

      5. hypot-define 6.9%

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

   5. Simplified 6.9%

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

   6. Taylor expanded in A around -inf 6.9%

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

   if -1.39999999999999989e245 < A

   1. Initial program 18.7%

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

   3. Taylor expanded in C around 0 9.9%

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

      1. mul-1-neg 9.9%

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

      2. +-commutative 9.9%

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

      3. unpow2 9.9%

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

      4. unpow2 9.9%

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

      5. hypot-define 15.4%

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

   5. Simplified 15.4%

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

      1. pow1/2 15.4%

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

      2. pow-to-exp 15.4%

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

   7. Applied egg-rr 15.4%

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

      1. associate-*l/ 15.4%

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

      2. exp-to-pow 15.4%

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

      3. pow1/2 15.4%

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

      4. sqrt-prod 15.5%

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

      5. distribute-frac-neg2 15.5%

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

      6. clear-num 15.5%

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

      7. associate-*r* 15.5%

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

   9. Applied egg-rr 15.5%

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

   10. Taylor expanded in A around 0 13.1%

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

4. Final simplification 12.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.4 \cdot 10^{+245}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{2 \cdot \left(A \cdot F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(A \cdot F\right) + -2 \cdot \left(B \cdot F\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-1}{B\_m \cdot {\left(\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}^{-0.5}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/ -1.0 (* B_m (pow (* (* 2.0 F) (- A (hypot B_m A))) -0.5))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -1.0 / (B_m * pow(((2.0 * F) * (A - hypot(B_m, A))), -0.5));
}

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 -1.0 / (B_m * Math.pow(((2.0 * F) * (A - Math.hypot(B_m, A))), -0.5));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -1.0 / (B_m * math.pow(((2.0 * F) * (A - math.hypot(B_m, A))), -0.5))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A))) ^ -0.5)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -1.0 / (B_m * (((2.0 * F) * (A - hypot(B_m, A))) ^ -0.5));
end

Wolfram

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

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in C around 0 9.3%

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

   1. mul-1-neg 9.3%

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

   2. +-commutative 9.3%

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

   3. unpow2 9.3%

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

   4. unpow2 9.3%

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

   5. hypot-define 14.9%

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

5. Simplified 14.9%

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

   1. pow1/2 14.9%

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

   2. pow-to-exp 14.8%

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

7. Applied egg-rr 14.8%

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

   1. associate-*l/ 14.8%

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

   2. exp-to-pow 14.9%

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

   3. pow1/2 14.9%

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

   4. sqrt-prod 14.9%

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

   5. distribute-frac-neg2 14.9%

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

   6. clear-num 14.9%

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

   7. associate-*r* 14.9%

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

9. Applied egg-rr 14.9%

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

    1. div-inv 14.9%

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

    2. pow1/2 14.9%

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

    3. pow-flip 15.0%

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

    4. metadata-eval 15.0%

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

11. Applied egg-rr 15.0%

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

12. Final simplification 15.0%

    \[\leadsto \frac{-1}{B \cdot {\left(\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{-0.5}} \]
13. Add Preprocessing

Alternative 8: 32.3% accurate, 3.0× speedup

Math

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

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

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in C around 0 9.3%

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

   1. mul-1-neg 9.3%

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

   2. +-commutative 9.3%

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

   3. unpow2 9.3%

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

   4. unpow2 9.3%

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

   5. hypot-define 14.9%

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

5. Simplified 14.9%

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

   1. neg-sub0 14.9%

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

   2. associate-*l/ 14.9%

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

   3. pow1/2 14.9%

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

   4. pow1/2 15.0%

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

   5. pow-prod-down 15.0%

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

7. Applied egg-rr 15.0%

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

   1. neg-sub0 15.0%

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

   2. distribute-neg-frac2 15.0%

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

   3. unpow1/2 14.9%

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

9. Simplified 14.9%

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

Alternative 9: 27.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.6 \cdot 10^{+249}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(A \cdot F\right) + -2 \cdot \left(B\_m \cdot F\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -1.6e+249)
   (/ (sqrt (* (* 4.0 A) F)) (- B_m))
   (/ -1.0 (/ B_m (sqrt (+ (* 2.0 (* A F)) (* -2.0 (* B_m F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -1.6e+249) {
		tmp = sqrt(((4.0 * A) * F)) / -B_m;
	} else {
		tmp = -1.0 / (B_m / sqrt(((2.0 * (A * F)) + (-2.0 * (B_m * F)))));
	}
	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 <= (-1.6d+249)) then
        tmp = sqrt(((4.0d0 * a) * f)) / -b_m
    else
        tmp = (-1.0d0) / (b_m / sqrt(((2.0d0 * (a * f)) + ((-2.0d0) * (b_m * f)))))
    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 <= -1.6e+249) {
		tmp = Math.sqrt(((4.0 * A) * F)) / -B_m;
	} else {
		tmp = -1.0 / (B_m / Math.sqrt(((2.0 * (A * F)) + (-2.0 * (B_m * F)))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -1.6e+249:
		tmp = math.sqrt(((4.0 * A) * F)) / -B_m
	else:
		tmp = -1.0 / (B_m / math.sqrt(((2.0 * (A * F)) + (-2.0 * (B_m * F)))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -1.6e+249)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * F)) / Float64(-B_m));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(Float64(2.0 * Float64(A * F)) + Float64(-2.0 * Float64(B_m * F))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -1.6e+249)
		tmp = sqrt(((4.0 * A) * F)) / -B_m;
	else
		tmp = -1.0 / (B_m / sqrt(((2.0 * (A * F)) + (-2.0 * (B_m * F)))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -1.6e+249], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.60000000000000007e249

   1. Initial program 1.3%

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

   3. Taylor expanded in C around 0 0.8%

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

      1. mul-1-neg 0.8%

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

      2. +-commutative 0.8%

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

      3. unpow2 0.8%

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

      4. unpow2 0.8%

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

      5. hypot-define 7.3%

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

   5. Simplified 7.3%

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

      1. neg-sub0 7.3%

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

      2. associate-*l/ 7.3%

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

      3. pow1/2 7.3%

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

      4. pow1/2 7.8%

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

      5. pow-prod-down 7.8%

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

   7. Applied egg-rr 7.8%

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

      1. neg-sub0 7.8%

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

      2. distribute-neg-frac2 7.8%

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

      3. unpow1/2 7.3%

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

   9. Simplified 7.3%

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

   10. Taylor expanded in A around -inf 7.3%

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

       1. associate-*r* 7.3%

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

       2. *-commutative 7.3%

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

   12. Simplified 7.3%

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

   if -1.60000000000000007e249 < A

   1. Initial program 18.6%

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

   3. Taylor expanded in C around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. unpow2 9.8%

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

      4. unpow2 9.8%

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

      5. hypot-define 15.4%

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

   5. Simplified 15.4%

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

      1. pow1/2 15.4%

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

      2. pow-to-exp 15.3%

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

   7. Applied egg-rr 15.3%

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

      1. associate-*l/ 15.3%

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

      2. exp-to-pow 15.4%

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

      3. pow1/2 15.4%

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

      4. sqrt-prod 15.4%

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

      5. distribute-frac-neg2 15.4%

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

      6. clear-num 15.4%

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

      7. associate-*r* 15.4%

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

   9. Applied egg-rr 15.4%

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

   10. Taylor expanded in A around 0 13.1%

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

4. Final simplification 12.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.6 \cdot 10^{+249}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(A \cdot F\right) + -2 \cdot \left(B \cdot F\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A \cdot F\right) + -2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -1.5e+254)
   (/ (sqrt (* (* 4.0 A) F)) (- B_m))
   (/ (sqrt (+ (* 2.0 (* A F)) (* -2.0 (* B_m F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -1.5e+254) {
		tmp = sqrt(((4.0 * A) * F)) / -B_m;
	} else {
		tmp = sqrt(((2.0 * (A * F)) + (-2.0 * (B_m * F)))) / -B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-1.5d+254)) then
        tmp = sqrt(((4.0d0 * a) * f)) / -b_m
    else
        tmp = sqrt(((2.0d0 * (a * f)) + ((-2.0d0) * (b_m * f)))) / -b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -1.5e+254) {
		tmp = Math.sqrt(((4.0 * A) * F)) / -B_m;
	} else {
		tmp = Math.sqrt(((2.0 * (A * F)) + (-2.0 * (B_m * F)))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -1.5e+254:
		tmp = math.sqrt(((4.0 * A) * F)) / -B_m
	else:
		tmp = math.sqrt(((2.0 * (A * F)) + (-2.0 * (B_m * F)))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -1.5e+254)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * F)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(A * F)) + Float64(-2.0 * Float64(B_m * F)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -1.5e+254)
		tmp = sqrt(((4.0 * A) * F)) / -B_m;
	else
		tmp = sqrt(((2.0 * (A * F)) + (-2.0 * (B_m * F)))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -1.5e+254], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.50000000000000003e254

   1. Initial program 1.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 7.8%

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

   5. Simplified 7.8%

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

      1. neg-sub0 7.8%

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

      2. associate-*l/ 7.8%

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

      3. pow1/2 7.8%

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

      4. pow1/2 8.3%

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

      5. pow-prod-down 8.3%

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

   7. Applied egg-rr 8.3%

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

      1. neg-sub0 8.3%

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

      2. distribute-neg-frac2 8.3%

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

      3. unpow1/2 7.8%

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

   9. Simplified 7.8%

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

   10. Taylor expanded in A around -inf 7.8%

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

       1. associate-*r* 7.8%

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

       2. *-commutative 7.8%

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

   12. Simplified 7.8%

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

   if -1.50000000000000003e254 < A

   1. Initial program 18.5%

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

   3. Taylor expanded in C around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. unpow2 9.8%

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

      4. unpow2 9.8%

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

      5. hypot-define 15.3%

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

   5. Simplified 15.3%

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

      1. neg-sub0 15.3%

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

      2. associate-*l/ 15.3%

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

      3. pow1/2 15.3%

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

      4. pow1/2 15.4%

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

      5. pow-prod-down 15.4%

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

   7. Applied egg-rr 15.4%

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

      1. neg-sub0 15.4%

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

      2. distribute-neg-frac2 15.4%

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

      3. unpow1/2 15.3%

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

   9. Simplified 15.3%

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

   10. Taylor expanded in A around 0 13.0%

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

4. Final simplification 12.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A \cdot F\right) + -2 \cdot \left(B \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2.9 \cdot 10^{+249}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B\_m \cdot -2\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -2.9e+249)
   (/ (sqrt (* (* 4.0 A) F)) (- B_m))
   (/ (sqrt (* F (* B_m -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 <= -2.9e+249) {
		tmp = sqrt(((4.0 * A) * F)) / -B_m;
	} else {
		tmp = sqrt((F * (B_m * -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 <= (-2.9d+249)) then
        tmp = sqrt(((4.0d0 * a) * f)) / -b_m
    else
        tmp = sqrt((f * (b_m * (-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 <= -2.9e+249) {
		tmp = Math.sqrt(((4.0 * A) * F)) / -B_m;
	} else {
		tmp = Math.sqrt((F * (B_m * -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 <= -2.9e+249:
		tmp = math.sqrt(((4.0 * A) * F)) / -B_m
	else:
		tmp = math.sqrt((F * (B_m * -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 <= -2.9e+249)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * F)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(F * Float64(B_m * -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 <= -2.9e+249)
		tmp = sqrt(((4.0 * A) * F)) / -B_m;
	else
		tmp = sqrt((F * (B_m * -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, -2.9e+249], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(B$95$m * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -2.90000000000000017e249

   1. Initial program 1.3%

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

   3. Taylor expanded in C around 0 0.8%

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

      1. mul-1-neg 0.8%

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

      2. +-commutative 0.8%

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

      3. unpow2 0.8%

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

      4. unpow2 0.8%

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

      5. hypot-define 7.3%

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

   5. Simplified 7.3%

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

      1. neg-sub0 7.3%

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

      2. associate-*l/ 7.3%

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

      3. pow1/2 7.3%

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

      4. pow1/2 7.8%

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

      5. pow-prod-down 7.8%

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

   7. Applied egg-rr 7.8%

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

      1. neg-sub0 7.8%

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

      2. distribute-neg-frac2 7.8%

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

      3. unpow1/2 7.3%

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

   9. Simplified 7.3%

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

   10. Taylor expanded in A around -inf 7.3%

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

       1. associate-*r* 7.3%

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

       2. *-commutative 7.3%

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

   12. Simplified 7.3%

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

   if -2.90000000000000017e249 < A

   1. Initial program 18.6%

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

   3. Taylor expanded in C around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. unpow2 9.8%

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

      4. unpow2 9.8%

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

      5. hypot-define 15.4%

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

   5. Simplified 15.4%

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

      1. neg-sub0 15.4%

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

      2. associate-*l/ 15.4%

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

      3. pow1/2 15.4%

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

      4. pow1/2 15.4%

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

      5. pow-prod-down 15.5%

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

   7. Applied egg-rr 15.5%

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

      1. neg-sub0 15.5%

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

      2. distribute-neg-frac2 15.5%

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

      3. unpow1/2 15.4%

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

   9. Simplified 15.4%

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

   10. Taylor expanded in A around 0 13.9%

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

       1. associate-*r* 13.9%

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

   12. Simplified 13.9%

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

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.9 \cdot 10^{+249}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B \cdot -2\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2.4 \cdot 10^{+249}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B\_m \cdot -2\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -2.4e+249)
   (* (sqrt (* A F)) (/ -2.0 B_m))
   (/ (sqrt (* F (* B_m -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 <= -2.4e+249) {
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = sqrt((F * (B_m * -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 <= (-2.4d+249)) then
        tmp = sqrt((a * f)) * ((-2.0d0) / b_m)
    else
        tmp = sqrt((f * (b_m * (-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 <= -2.4e+249) {
		tmp = Math.sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = Math.sqrt((F * (B_m * -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 <= -2.4e+249:
		tmp = math.sqrt((A * F)) * (-2.0 / B_m)
	else:
		tmp = math.sqrt((F * (B_m * -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 <= -2.4e+249)
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m));
	else
		tmp = Float64(sqrt(Float64(F * Float64(B_m * -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 <= -2.4e+249)
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	else
		tmp = sqrt((F * (B_m * -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, -2.4e+249], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(B$95$m * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -2.4e249

   1. Initial program 1.3%

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

   3. Taylor expanded in C around 0 0.8%

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

      1. mul-1-neg 0.8%

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

      2. +-commutative 0.8%

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

      3. unpow2 0.8%

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

      4. unpow2 0.8%

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

      5. hypot-define 7.3%

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

   5. Simplified 7.3%

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

      1. pow1/2 7.3%

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

      2. pow-to-exp 7.2%

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

   7. Applied egg-rr 7.2%

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

   8. Taylor expanded in A around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 7.2%

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

      4. unpow2 7.2%

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

      5. rem-square-sqrt 7.2%

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

      6. metadata-eval 7.2%

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

   10. Simplified 7.2%

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

   if -2.4e249 < A

   1. Initial program 18.6%

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

   3. Taylor expanded in C around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. unpow2 9.8%

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

      4. unpow2 9.8%

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

      5. hypot-define 15.4%

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

   5. Simplified 15.4%

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

      1. neg-sub0 15.4%

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

      2. associate-*l/ 15.4%

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

      3. pow1/2 15.4%

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

      4. pow1/2 15.4%

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

      5. pow-prod-down 15.5%

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

   7. Applied egg-rr 15.5%

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

      1. neg-sub0 15.5%

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

      2. distribute-neg-frac2 15.5%

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

      3. unpow1/2 15.4%

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

   9. Simplified 15.4%

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

   10. Taylor expanded in A around 0 13.9%

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

       1. associate-*r* 13.9%

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

   12. Simplified 13.9%

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

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.4 \cdot 10^{+249}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B \cdot -2\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -3.1 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -3.1e+253)
   (* (sqrt (* A F)) (/ -2.0 B_m))
   (/ (sqrt (* -2.0 (* B_m F))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -3.1e+253) {
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-3.1d+253)) then
        tmp = sqrt((a * f)) * ((-2.0d0) / b_m)
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / -b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -3.1e+253) {
		tmp = Math.sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -3.1e+253:
		tmp = math.sqrt((A * F)) * (-2.0 / B_m)
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -3.1e+253)
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -3.1e+253)
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	else
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -3.1e+253], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -3.10000000000000006e253

   1. Initial program 1.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 7.8%

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

   5. Simplified 7.8%

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

      1. pow1/2 7.8%

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

      2. pow-to-exp 7.6%

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

   7. Applied egg-rr 7.6%

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

   8. Taylor expanded in A around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 7.6%

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

      4. unpow2 7.6%

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

      5. rem-square-sqrt 7.7%

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

      6. metadata-eval 7.7%

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

   10. Simplified 7.7%

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

   if -3.10000000000000006e253 < A

   1. Initial program 18.5%

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

   3. Taylor expanded in C around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. unpow2 9.8%

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

      4. unpow2 9.8%

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

      5. hypot-define 15.3%

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

   5. Simplified 15.3%

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

      1. neg-sub0 15.3%

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

      2. associate-*l/ 15.3%

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

      3. pow1/2 15.3%

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

      4. pow1/2 15.4%

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

      5. pow-prod-down 15.4%

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

   7. Applied egg-rr 15.4%

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

      1. neg-sub0 15.4%

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

      2. distribute-neg-frac2 15.4%

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

      3. unpow1/2 15.3%

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

   9. Simplified 15.3%

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

   10. Taylor expanded in A around 0 13.9%

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

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.1 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 9.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in C around 0 9.3%

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

   1. mul-1-neg 9.3%

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

   2. +-commutative 9.3%

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

   3. unpow2 9.3%

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

   4. unpow2 9.3%

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

   5. hypot-define 14.9%

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

5. Simplified 14.9%

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

   1. pow1/2 14.9%

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

   2. pow-to-exp 14.8%

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

7. Applied egg-rr 14.8%

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

8. Taylor expanded in A around -inf 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.9%

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

   4. unpow2 2.9%

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

   5. rem-square-sqrt 2.9%

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

   6. metadata-eval 2.9%

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

10. Simplified 2.9%

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

11. Final simplification 2.9%

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

Alternative 15: 1.7% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.1%

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

5. Simplified 2.1%

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

6. Taylor expanded in F around 0 2.1%

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

   1. sqrt-unprod 2.1%

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

   2. pow1/2 2.2%

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

8. Applied egg-rr 2.2%

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

9. Final simplification 2.2%

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

Alternative 16: 1.6% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.1%

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

5. Simplified 2.1%

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

6. Taylor expanded in F around 0 2.1%

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

   1. sqrt-unprod 2.1%

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

8. Applied egg-rr 2.1%

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

9. Final simplification 2.1%

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

Reproduce

herbie shell --seed 2024186 
(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))))