ABCF->ab-angle b

Percentage Accurate: 18.9% → 49.6%

Time: 25.9s

Alternatives: 18

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 49.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := F \cdot \left(4 \cdot A - \frac{{B\_m}^{2}}{C}\right)\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}}\\ t_4 := A \cdot \left(C \cdot -4\right)\\ t_5 := \mathsf{fma}\left(B\_m, B\_m, t\_4\right)\\ t_6 := -t\_5\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot \mathsf{hypot}\left(B\_m, \sqrt{t\_4}\right)}{t\_6}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-221}:\\ \;\;\;\;\frac{-1}{\frac{t\_2}{\sqrt{\left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right) \cdot \left(2 \cdot t\_2\right)}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot t\_5}}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (* F (- (* 4.0 A) (/ (pow B_m 2.0) C))))
        (t_2 (fma C (* A -4.0) (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_0) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B_m 2.0))))
        (t_4 (* A (* C -4.0)))
        (t_5 (fma B_m B_m t_4))
        (t_6 (- t_5)))
   (if (<= t_3 (- INFINITY))
     (/ (* (sqrt t_1) (hypot B_m (sqrt t_4))) t_6)
     (if (<= t_3 -2e-221)
       (/
        -1.0
        (/ t_2 (sqrt (* (* F (+ A (- C (hypot B_m (- A C))))) (* 2.0 t_2)))))
       (if (<= t_3 INFINITY)
         (/ (sqrt (* t_1 t_5)) t_6)
         (*
          (/ (exp (* (log 2.0) 0.5)) B_m)
          (- (sqrt (* F (- A (hypot B_m A)))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = F * ((4.0 * A) - (pow(B_m, 2.0) / C));
	double t_2 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0));
	double t_4 = A * (C * -4.0);
	double t_5 = fma(B_m, B_m, t_4);
	double t_6 = -t_5;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(t_1) * hypot(B_m, sqrt(t_4))) / t_6;
	} else if (t_3 <= -2e-221) {
		tmp = -1.0 / (t_2 / sqrt(((F * (A + (C - hypot(B_m, (A - C))))) * (2.0 * t_2))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * t_5)) / t_6;
	} else {
		tmp = (exp((log(2.0) * 0.5)) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(F * Float64(Float64(4.0 * A) - Float64((B_m ^ 2.0) / C)))
	t_2 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0)))
	t_4 = Float64(A * Float64(C * -4.0))
	t_5 = fma(B_m, B_m, t_4)
	t_6 = Float64(-t_5)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_1) * hypot(B_m, sqrt(t_4))) / t_6);
	elseif (t_3 <= -2e-221)
		tmp = Float64(-1.0 / Float64(t_2 / sqrt(Float64(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C))))) * Float64(2.0 * t_2)))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * t_5)) / t_6);
	else
		tmp = Float64(Float64(exp(Float64(log(2.0) * 0.5)) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(F * N[(N[(4.0 * A), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(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$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B$95$m * B$95$m + t$95$4), $MachinePrecision]}, Block[{t$95$6 = (-t$95$5)}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[B$95$m ^ 2 + N[Sqrt[t$95$4], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$3, -2e-221], N[(-1.0 / N[(t$95$2 / N[Sqrt[N[(N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$1 * t$95$5), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], N[(N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.2%

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

   3. Simplified 16.6%

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

   5. Taylor expanded in C around inf 20.8%

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

      1. mul-1-neg 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in C around 0 20.8%

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

      1. *-un-lft-identity 20.8%

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

      2. associate-*l* 19.0%

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

      3. +-commutative 19.0%

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

      4. fma-define 19.0%

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

      5. mul-1-neg 19.0%

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

   10. Applied egg-rr 19.0%

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

       1. *-lft-identity 19.0%

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

       2. fma-neg 19.0%

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

       3. associate-*r* 19.0%

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

   12. Simplified 19.0%

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

       1. *-un-lft-identity 19.0%

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

       2. sqrt-prod 20.8%

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

       3. fma-undefine 20.8%

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

       4. add-sqr-sqrt 16.8%

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

       5. hypot-define 16.8%

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

       6. associate-*l* 16.8%

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

       7. fma-neg 16.8%

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

   14. Applied egg-rr 16.8%

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

       1. *-lft-identity 16.8%

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

       2. *-commutative 16.8%

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

       3. fma-neg 16.8%

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

       4. div-sub 16.8%

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

       5. associate-*r/ 16.8%

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

       6. *-commutative 16.8%

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

       7. associate-/l* 16.8%

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

       8. *-inverses 16.8%

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

       9. *-rgt-identity 16.8%

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

       10. *-commutative 16.8%

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

   16. Simplified 16.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000003e-221

   1. Initial program 98.6%

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

   3. Simplified 83.0%

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

      1. clear-num 83.0%

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

      2. inv-pow 83.0%

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

   6. Applied egg-rr 98.8%

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

      1. unpow-1 98.8%

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

      2. associate-+r- 98.8%

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

      3. hypot-undefine 98.8%

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

      4. unpow2 98.8%

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

      5. unpow2 98.8%

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

      6. +-commutative 98.8%

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

      7. unpow2 98.8%

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

      8. unpow2 98.8%

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

      9. hypot-undefine 98.8%

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

   8. Simplified 98.8%

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

   if -2.00000000000000003e-221 < (/.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 20.3%

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

   3. Simplified 31.0%

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

   5. Taylor expanded in C around inf 22.5%

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

      1. mul-1-neg 22.5%

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

   7. Simplified 22.5%

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

   8. Taylor expanded in C around 0 22.5%

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

      1. *-un-lft-identity 22.5%

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

      2. associate-*l* 22.4%

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

      3. +-commutative 22.4%

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

      4. fma-define 22.4%

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

      5. mul-1-neg 22.4%

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

   10. Applied egg-rr 22.4%

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

       1. *-lft-identity 22.4%

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

       2. fma-neg 22.4%

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

       3. associate-*r* 22.4%

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

   12. Simplified 22.4%

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

   13. Taylor expanded in F around 0 22.4%

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

       1. fma-neg 22.4%

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

       2. associate-*r/ 22.4%

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

       3. fma-neg 22.4%

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

       4. div-sub 22.4%

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

       5. associate-*r/ 22.4%

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

       6. *-commutative 22.4%

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

       7. associate-/l* 22.4%

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

       8. *-inverses 22.4%

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

       9. *-rgt-identity 22.4%

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

       10. *-commutative 22.4%

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

   15. Simplified 22.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. +-commutative 1.8%

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

      3. unpow2 1.8%

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

      4. unpow2 1.8%

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

      5. hypot-define 14.8%

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

   5. Simplified 14.8%

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

      1. pow1/2 14.8%

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

      2. pow-to-exp 14.7%

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

   7. Applied egg-rr 14.7%

      \[\leadsto -\frac{\color{blue}{e^{\log 2 \cdot 0.5}}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(4 \cdot A - \frac{{B}^{2}}{C}\right)} \cdot \mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-221}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}{\sqrt{\left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(4 \cdot A - \frac{{B}^{2}}{C}\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 55:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \frac{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}{C}}}{A \cdot \left(C \cdot \left(--4\right) - \frac{{B\_m}^{2}}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 55.0)
   (/
    (sqrt
     (*
      (* F (fma B_m B_m (* A (* C -4.0))))
      (/ (- (* 4.0 (* A C)) (pow B_m 2.0)) C)))
    (* A (- (* C (- -4.0)) (/ (pow B_m 2.0) A))))
   (* (/ (exp (* (log 2.0) 0.5)) B_m) (- (sqrt (* F (- A (hypot B_m A))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 55.0) {
		tmp = sqrt(((F * fma(B_m, B_m, (A * (C * -4.0)))) * (((4.0 * (A * C)) - pow(B_m, 2.0)) / C))) / (A * ((C * -(-4.0)) - (pow(B_m, 2.0) / A)));
	} else {
		tmp = (exp((log(2.0) * 0.5)) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 55.0)
		tmp = Float64(sqrt(Float64(Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * Float64(Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)) / C))) / Float64(A * Float64(Float64(C * Float64(-(-4.0))) - Float64((B_m ^ 2.0) / A))));
	else
		tmp = Float64(Float64(exp(Float64(log(2.0) * 0.5)) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 55.0], N[(N[Sqrt[N[(N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(A * N[(N[(C * (--4.0)), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 55

   1. Initial program 20.6%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in C around inf 15.1%

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

      1. mul-1-neg 15.1%

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

   7. Simplified 15.1%

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

   8. Taylor expanded in C around 0 15.1%

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

   9. Taylor expanded in A around -inf 15.1%

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

       1. associate-*r* 15.1%

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

       2. mul-1-neg 15.1%

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

       3. cancel-sign-sub-inv 15.1%

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

       4. metadata-eval 15.1%

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

   11. Simplified 15.1%

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

   if 55 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

      1. pow1/2 43.8%

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

      2. pow-to-exp 43.6%

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

   7. Applied egg-rr 43.6%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 55:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \frac{4 \cdot \left(A \cdot C\right) - {B}^{2}}{C}}}{A \cdot \left(C \cdot \left(--4\right) - \frac{{B}^{2}}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 85:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \frac{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}{C}}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 85.0)
     (/ (sqrt (* (* F t_0) (/ (- (* 4.0 (* A C)) (pow B_m 2.0)) C))) (- t_0))
     (*
      (/ (exp (* (log 2.0) 0.5)) B_m)
      (- (sqrt (* F (- A (hypot B_m A)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 85.0) {
		tmp = sqrt(((F * t_0) * (((4.0 * (A * C)) - pow(B_m, 2.0)) / C))) / -t_0;
	} else {
		tmp = (exp((log(2.0) * 0.5)) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 85.0)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)) / C))) / Float64(-t_0));
	else
		tmp = Float64(Float64(exp(Float64(log(2.0) * 0.5)) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 85.0], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 85

   1. Initial program 20.6%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in C around inf 15.1%

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

      1. mul-1-neg 15.1%

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

   7. Simplified 15.1%

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

   8. Taylor expanded in C around 0 15.1%

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

   if 85 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

      1. pow1/2 43.8%

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

      2. pow-to-exp 43.6%

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

   7. Applied egg-rr 43.6%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 85:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \frac{4 \cdot \left(A \cdot C\right) - {B}^{2}}{C}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 62:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot A - \frac{{B\_m}^{2}}{C}\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 62.0)
     (/ (sqrt (* (* F t_0) (- (* 4.0 A) (/ (pow B_m 2.0) C)))) (- t_0))
     (*
      (/ (exp (* (log 2.0) 0.5)) B_m)
      (- (sqrt (* F (- A (hypot B_m A)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 62.0) {
		tmp = sqrt(((F * t_0) * ((4.0 * A) - (pow(B_m, 2.0) / C)))) / -t_0;
	} else {
		tmp = (exp((log(2.0) * 0.5)) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 62.0)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(Float64(4.0 * A) - Float64((B_m ^ 2.0) / C)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(exp(Float64(log(2.0) * 0.5)) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 62.0], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(N[(4.0 * A), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 62

   1. Initial program 20.6%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in C around inf 15.1%

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

      1. mul-1-neg 15.1%

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

   7. Simplified 15.1%

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

   8. Taylor expanded in A around 0 15.1%

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

   if 62 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

      1. pow1/2 43.8%

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

      2. pow-to-exp 43.6%

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

   7. Applied egg-rr 43.6%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 62:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot A - \frac{{B}^{2}}{C}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 125:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(4 \cdot A - \frac{{B\_m}^{2}}{C}\right)\right) \cdot t\_0}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 125.0)
     (/ (sqrt (* (* F (- (* 4.0 A) (/ (pow B_m 2.0) C))) t_0)) (- t_0))
     (*
      (/ (exp (* (log 2.0) 0.5)) B_m)
      (- (sqrt (* F (- A (hypot B_m A)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 125.0) {
		tmp = sqrt(((F * ((4.0 * A) - (pow(B_m, 2.0) / C))) * t_0)) / -t_0;
	} else {
		tmp = (exp((log(2.0) * 0.5)) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 125.0)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(4.0 * A) - Float64((B_m ^ 2.0) / C))) * t_0)) / Float64(-t_0));
	else
		tmp = Float64(Float64(exp(Float64(log(2.0) * 0.5)) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 125.0], N[(N[Sqrt[N[(N[(F * N[(N[(4.0 * A), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 125

   1. Initial program 20.6%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in C around inf 15.1%

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

      1. mul-1-neg 15.1%

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

   7. Simplified 15.1%

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

   8. Taylor expanded in C around 0 15.1%

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

      1. *-un-lft-identity 15.1%

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

      2. associate-*l* 14.6%

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

      3. +-commutative 14.6%

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

      4. fma-define 14.6%

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

      5. mul-1-neg 14.6%

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

   10. Applied egg-rr 14.6%

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

       1. *-lft-identity 14.6%

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

       2. fma-neg 14.6%

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

       3. associate-*r* 14.6%

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

   12. Simplified 14.6%

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

   13. Taylor expanded in F around 0 14.6%

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

       1. fma-neg 14.6%

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

       2. associate-*r/ 14.6%

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

       3. fma-neg 14.6%

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

       4. div-sub 14.6%

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

       5. associate-*r/ 14.6%

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

       6. *-commutative 14.6%

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

       7. associate-/l* 14.6%

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

       8. *-inverses 14.6%

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

       9. *-rgt-identity 14.6%

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

       10. *-commutative 14.6%

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

   15. Simplified 14.6%

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

   if 125 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

      1. pow1/2 43.8%

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

      2. pow-to-exp 43.6%

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

   7. Applied egg-rr 43.6%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 125:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(4 \cdot A - \frac{{B}^{2}}{C}\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 38:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\left(4 \cdot A\right) \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 38.0)
     (/ (sqrt (* t_0 (* (* 4.0 A) F))) (- t_0))
     (*
      (/ (exp (* (log 2.0) 0.5)) B_m)
      (- (sqrt (* F (- A (hypot B_m A)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 38.0) {
		tmp = sqrt((t_0 * ((4.0 * A) * F))) / -t_0;
	} else {
		tmp = (exp((log(2.0) * 0.5)) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 38.0)
		tmp = Float64(sqrt(Float64(t_0 * Float64(Float64(4.0 * A) * F))) / Float64(-t_0));
	else
		tmp = Float64(Float64(exp(Float64(log(2.0) * 0.5)) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 38.0], N[(N[Sqrt[N[(t$95$0 * N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 38

   1. Initial program 20.6%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in C around inf 15.1%

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

      1. mul-1-neg 15.1%

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

   7. Simplified 15.1%

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

   8. Taylor expanded in C around 0 15.1%

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

      1. *-un-lft-identity 15.1%

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

      2. associate-*l* 14.6%

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

      3. +-commutative 14.6%

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

      4. fma-define 14.6%

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

      5. mul-1-neg 14.6%

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

   10. Applied egg-rr 14.6%

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

       1. *-lft-identity 14.6%

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

       2. fma-neg 14.6%

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

       3. associate-*r* 14.6%

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

   12. Simplified 14.6%

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

   13. Taylor expanded in A around inf 14.5%

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

   if 38 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

      1. pow1/2 43.8%

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

      2. pow-to-exp 43.6%

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

   7. Applied egg-rr 43.6%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 38:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(4 \cdot A\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 1000:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1000.0)
   (/
    (sqrt (* -8.0 (* A (* C (* F (+ A A))))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1000.0) {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1000.0)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1000.0], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e3

   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 32.8%

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

   5. Taylor expanded in B around 0 22.4%

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

   6. Taylor expanded in C around inf 20.2%

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

      1. sub-neg 20.2%

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

      2. neg-mul-1 20.2%

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

      3. remove-double-neg 20.2%

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

   8. Simplified 20.2%

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

   if 1e3 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.0%

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

   3. Taylor expanded in C around 0 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 22.8%

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

   5. Simplified 22.8%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1000:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 116:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\left(4 \cdot A\right) \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 116.0)
     (/ (sqrt (* t_0 (* (* 4.0 A) F))) (- t_0))
     (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 116.0) {
		tmp = sqrt((t_0 * ((4.0 * A) * F))) / -t_0;
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 116.0)
		tmp = Float64(sqrt(Float64(t_0 * Float64(Float64(4.0 * A) * F))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 116.0], N[(N[Sqrt[N[(t$95$0 * N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 116

   1. Initial program 20.6%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in C around inf 15.1%

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

      1. mul-1-neg 15.1%

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

   7. Simplified 15.1%

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

   8. Taylor expanded in C around 0 15.1%

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

      1. *-un-lft-identity 15.1%

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

      2. associate-*l* 14.6%

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

      3. +-commutative 14.6%

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

      4. fma-define 14.6%

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

      5. mul-1-neg 14.6%

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

   10. Applied egg-rr 14.6%

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

       1. *-lft-identity 14.6%

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

       2. fma-neg 14.6%

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

       3. associate-*r* 14.6%

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

   12. Simplified 14.6%

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

   13. Taylor expanded in A around inf 14.5%

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

   if 116 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 116:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(4 \cdot A\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 58:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 58.0)
     (/ (sqrt (* (* 4.0 A) (* F t_0))) (- t_0))
     (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 58.0) {
		tmp = sqrt(((4.0 * A) * (F * t_0))) / -t_0;
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 58.0)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * t_0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 58.0], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 58

   1. Initial program 20.6%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in A around -inf 14.5%

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

   if 58 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 58:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 63:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 63.0)
   (/
    (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
    (- (fma C (* A -4.0) (pow B_m 2.0))))
   (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 63.0) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 63.0)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(-fma(C, Float64(A * -4.0), (B_m ^ 2.0))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 63.0], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 63

   1. Initial program 20.6%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in C around inf 14.4%

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

      1. associate-*r* 14.4%

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

      2. mul-1-neg 14.4%

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

   7. Simplified 14.4%

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

   if 63 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 63:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 27:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 27.0)
   (/
    (sqrt (* -8.0 (* A (* C (* F (+ A A))))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (* (sqrt 2.0) (/ (sqrt (* F (- A (hypot B_m A)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 27.0) {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(B_m, A)))) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 27.0)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 27.0], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 27

   1. Initial program 20.6%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in B around 0 16.1%

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

   6. Taylor expanded in C around inf 14.4%

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

      1. sub-neg 14.4%

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

      2. neg-mul-1 14.4%

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

      3. remove-double-neg 14.4%

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

   8. Simplified 14.4%

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

   if 27 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

      1. add-cbrt-cube 35.2%

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

      2. pow1/3 33.2%

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

      3. add-sqr-sqrt 33.2%

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

      4. pow1 33.2%

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

      5. pow1/2 33.2%

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

      6. pow-prod-up 33.2%

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

      7. metadata-eval 33.2%

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

   7. Applied egg-rr 33.2%

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

      1. unpow1/3 35.2%

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

   9. Simplified 35.2%

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

   10. Taylor expanded in F around 0 21.8%

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

       1. +-commutative 21.8%

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

       2. unpow2 21.8%

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

       3. unpow2 21.8%

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

       4. hypot-undefine 43.8%

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

       5. associate-*l/ 43.7%

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

       6. associate-*r/ 43.7%

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

   12. Simplified 43.7%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 27:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 38:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 38.0)
   (/
    (sqrt (* -8.0 (* A (* C (* F (+ A A))))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 38.0) {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 38.0)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 38.0], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 38

   1. Initial program 20.6%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in B around 0 16.1%

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

   6. Taylor expanded in C around inf 14.4%

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

      1. sub-neg 14.4%

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

      2. neg-mul-1 14.4%

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

      3. remove-double-neg 14.4%

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

   8. Simplified 14.4%

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

   if 38 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in C around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. +-commutative 21.8%

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

      3. unpow2 21.8%

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

      4. unpow2 21.8%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

      1. neg-sub0 43.8%

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

      2. associate-*l/ 43.7%

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

      3. pow1/2 43.7%

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

      4. pow1/2 43.7%

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

      5. pow-prod-down 42.1%

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

   7. Applied egg-rr 42.1%

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

      1. neg-sub0 42.1%

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

      2. distribute-neg-frac2 42.1%

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

      3. unpow1/2 42.1%

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

   9. Simplified 42.1%

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

Alternative 13: 31.5% 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 19.4%

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

3. Taylor expanded in C around 0 9.0%

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

   1. mul-1-neg 9.0%

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

   2. +-commutative 9.0%

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

   3. unpow2 9.0%

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

   4. unpow2 9.0%

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

   5. hypot-define 14.8%

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

5. Simplified 14.8%

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

   1. neg-sub0 14.8%

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

   2. associate-*l/ 14.8%

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

   3. pow1/2 14.8%

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

   4. pow1/2 14.9%

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

   5. pow-prod-down 14.5%

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

7. Applied egg-rr 14.5%

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

   1. neg-sub0 14.5%

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

   2. distribute-neg-frac2 14.5%

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

   3. unpow1/2 14.4%

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

9. Simplified 14.4%

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

Alternative 14: 26.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

3. Taylor expanded in C around 0 9.0%

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

   1. mul-1-neg 9.0%

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

   2. +-commutative 9.0%

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

   3. unpow2 9.0%

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

   4. unpow2 9.0%

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

   5. hypot-define 14.8%

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

5. Simplified 14.8%

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

6. Taylor expanded in A around 0 13.8%

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

   1. mul-1-neg 13.8%

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

8. Simplified 13.8%

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

9. Final simplification 13.8%

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

Alternative 15: 4.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{A \cdot F} \cdot \frac{\sqrt{8}}{-B\_m} \end{array} \]

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)) (/ (sqrt 8.0) (- 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)) * (sqrt(8.0) / -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)) * (sqrt(8.0d0) / -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)) * (Math.sqrt(8.0) / -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)) * (math.sqrt(8.0) / -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(sqrt(8.0) / Float64(-B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((A * F)) * (sqrt(8.0) / -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[(N[Sqrt[8.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.4%

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

3. Simplified 20.9%

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

5. Taylor expanded in C around inf 9.5%

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

6. Taylor expanded in A around 0 1.8%

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

   1. mul-1-neg 1.8%

      \[\leadsto \color{blue}{-\sqrt{A \cdot F} \cdot \frac{\sqrt{8}}{B}} \]

8. Simplified 1.8%

   \[\leadsto \color{blue}{-\sqrt{A \cdot F} \cdot \frac{\sqrt{8}}{B}} \]

9. Final simplification 1.8%

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

Alternative 16: 2.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ {\left({\left(2 \cdot \frac{F}{B\_m}\right)}^{3}\right)}^{0.16666666666666666} \end{array} \]

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 (pow (* 2.0 (/ F B_m)) 3.0) 0.16666666666666666))

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(pow((2.0 * (F / B_m)), 3.0), 0.16666666666666666);
}

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)) ** 3.0d0) ** 0.16666666666666666d0
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(Math.pow((2.0 * (F / B_m)), 3.0), 0.16666666666666666);
}

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(math.pow((2.0 * (F / B_m)), 3.0), 0.16666666666666666)

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)) ^ 3.0) ^ 0.16666666666666666
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)) ^ 3.0) ^ 0.16666666666666666;
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[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]

Derivation

1. Initial program 19.4%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.9%

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

5. Simplified 1.9%

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

6. Taylor expanded in F around 0 1.9%

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

   1. pow1 1.9%

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

   2. sqrt-unprod 1.9%

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

8. Applied egg-rr 1.9%

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

   1. unpow1 1.9%

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

10. Simplified 1.9%

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

    1. rem-cbrt-cube 1.9%

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

    2. unpow1/3 1.9%

       \[\leadsto \color{blue}{{\left({\left(\sqrt{\frac{F}{B} \cdot 2}\right)}^{3}\right)}^{0.3333333333333333}} \]

    3. sqr-pow 1.9%

       \[\leadsto \color{blue}{{\left({\left(\sqrt{\frac{F}{B} \cdot 2}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left(\sqrt{\frac{F}{B} \cdot 2}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

    4. pow-prod-down 1.9%

       \[\leadsto \color{blue}{{\left({\left(\sqrt{\frac{F}{B} \cdot 2}\right)}^{3} \cdot {\left(\sqrt{\frac{F}{B} \cdot 2}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

    5. pow-prod-down 1.9%

       \[\leadsto {\color{blue}{\left({\left(\sqrt{\frac{F}{B} \cdot 2} \cdot \sqrt{\frac{F}{B} \cdot 2}\right)}^{3}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)} \]

    6. add-sqr-sqrt 2.9%

       \[\leadsto {\left({\color{blue}{\left(\frac{F}{B} \cdot 2\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

    7. *-commutative 2.9%

       \[\leadsto {\left({\color{blue}{\left(2 \cdot \frac{F}{B}\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

    8. metadata-eval 2.9%

       \[\leadsto {\left({\left(2 \cdot \frac{F}{B}\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

12. Applied egg-rr 2.9%

    \[\leadsto \color{blue}{{\left({\left(2 \cdot \frac{F}{B}\right)}^{3}\right)}^{0.16666666666666666}} \]
13. Add Preprocessing

Alternative 17: 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 19.4%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.9%

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

5. Simplified 1.9%

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

6. Taylor expanded in F around 0 1.9%

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

   1. sqrt-unprod 1.9%

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

   2. pow1/2 2.1%

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

8. Applied egg-rr 2.1%

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

9. Final simplification 2.1%

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

Alternative 18: 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 19.4%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.9%

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

5. Simplified 1.9%

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

6. Taylor expanded in F around 0 1.9%

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

   1. pow1 1.9%

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

   2. sqrt-unprod 1.9%

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

8. Applied egg-rr 1.9%

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

   1. unpow1 1.9%

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

10. Simplified 1.9%

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

11. Final simplification 1.9%

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

Reproduce

herbie shell --seed 2024132 
(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))))