Quadratic roots, narrow range

Percentage Accurate: 55.5% → 92.1%

Time: 15.2s

Alternatives: 12

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

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

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

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

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot {\left(a \cdot c\right)}^{2}\\ t_1 := \sqrt{a \cdot c}\\ t_2 := \mathsf{fma}\left(2, t_1, b\right) \cdot \mathsf{fma}\left(-2, t_1, b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -100:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t_2}^{1.5}}{{\left(-b\right)}^{2} + \left(t_2 + b \cdot \sqrt{t_2}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, -0.25 \cdot \left(\frac{t_0}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot t_0\right)}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow (* a c) 2.0)))
        (t_1 (sqrt (* a c)))
        (t_2 (* (fma 2.0 t_1 b) (fma -2.0 t_1 b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -100.0)
     (/
      (/
       (+ (pow (- b) 3.0) (pow t_2 1.5))
       (+ (pow (- b) 2.0) (+ t_2 (* b (sqrt t_2)))))
      (* a 2.0))
     (-
      (fma
       -0.25
       (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0)))
       (*
        -0.25
        (+
         (/ t_0 (* a (pow b 3.0)))
         (/ (* 2.0 (* (* a c) t_0)) (* a (pow b 5.0))))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * pow((a * c), 2.0);
	double t_1 = sqrt((a * c));
	double t_2 = fma(2.0, t_1, b) * fma(-2.0, t_1, b);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -100.0) {
		tmp = ((pow(-b, 3.0) + pow(t_2, 1.5)) / (pow(-b, 2.0) + (t_2 + (b * sqrt(t_2))))) / (a * 2.0);
	} else {
		tmp = fma(-0.25, ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0))), (-0.25 * ((t_0 / (a * pow(b, 3.0))) + ((2.0 * ((a * c) * t_0)) / (a * pow(b, 5.0)))))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * (Float64(a * c) ^ 2.0))
	t_1 = sqrt(Float64(a * c))
	t_2 = Float64(fma(2.0, t_1, b) * fma(-2.0, t_1, b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -100.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_2 ^ 1.5)) / Float64((Float64(-b) ^ 2.0) + Float64(t_2 + Float64(b * sqrt(t_2))))) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-0.25, Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0))), Float64(-0.25 * Float64(Float64(t_0 / Float64(a * (b ^ 3.0))) + Float64(Float64(2.0 * Float64(Float64(a * c) * t_0)) / Float64(a * (b ^ 5.0)))))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * t$95$1 + b), $MachinePrecision] * N[(-2.0 * t$95$1 + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -100.0], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$2, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$2 + N[(b * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(t$95$0 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(a * c), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -100

   1. Initial program 88.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\sqrt{\left(4 \cdot a\right) \cdot c} \cdot \sqrt{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. difference-of-squares 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + \sqrt{\left(4 \cdot a\right) \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}}{a \cdot 2} \]

      3. associate-*l* 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      4. sqrt-prod 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      5. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{2} \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      6. associate-*l* 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}}{a \cdot 2} \]

      7. sqrt-prod 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right)}}{a \cdot 2} \]

      8. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{2} \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

   5. Applied egg-rr 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - 2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{\sqrt{a \cdot c} \cdot 2}\right) \cdot \left(b - 2 \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

      2. cancel-sign-sub-inv 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \color{blue}{\left(b + \left(-2\right) \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

      3. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + \color{blue}{-2} \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

   7. Simplified 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. flip3-+ 88.6%

         \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}\right)}^{3}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)} \cdot \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)} - \left(-b\right) \cdot \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}\right)}}}{a \cdot 2} \]

   9. Applied egg-rr 89.9%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}\right)}}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. cancel-sign-sub 89.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \color{blue}{\left(\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right) + b \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}\right)}}}{a \cdot 2} \]

   11. Simplified 89.9%

       \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right) + b \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}\right)}}}{a \cdot 2} \]

   if -100 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 47.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 47.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 47.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. flip3-- 47.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      2. sqrt-div 46.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      3. pow2 46.5%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{\color{blue}{\left({b}^{2}\right)}}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      4. pow-pow 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{\color{blue}{{b}^{\left(2 \cdot 3\right)}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      5. metadata-eval 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{\color{blue}{6}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      6. associate-*l* 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - {\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      7. unpow-prod-down 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{{4}^{3} \cdot {\left(a \cdot c\right)}^{3}}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      8. metadata-eval 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   5. Applied egg-rr 46.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in b around inf 94.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(2 \cdot \left(a \cdot \left(c \cdot \left(-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.25 \cdot \frac{-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

   8. Simplified 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(2, 0 + \left(\left(2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)\right) \cdot c\right) \cdot a, {\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b}} \]

   9. Taylor expanded in c around 0 94.8%

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]
   10. Step-by-step derivation

       1. distribute-rgt-out 94.8%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

       2. associate-*r* 94.8%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

       3. *-commutative 94.8%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

       4. times-frac 94.8%

          \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

   11. Simplified 94.8%

       \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -100:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right) + b \cdot \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, -0.25 \cdot \left(\frac{4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b}\\ \end{array} \]

Alternative 2: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ t_1 := \mathsf{fma}\left(2, t_0, b\right) \cdot \mathsf{fma}\left(-2, t_0, b\right)\\ t_2 := 4 \cdot {\left(a \cdot c\right)}^{2}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -100:\\ \;\;\;\;\frac{\frac{{b}^{2} - t_1}{\left(-b\right) - \sqrt{t_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, -0.25 \cdot \left(\frac{t_2}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot t_2\right)}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c)))
        (t_1 (* (fma 2.0 t_0 b) (fma -2.0 t_0 b)))
        (t_2 (* 4.0 (pow (* a c) 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -100.0)
     (/ (/ (- (pow b 2.0) t_1) (- (- b) (sqrt t_1))) (* a 2.0))
     (-
      (fma
       -0.25
       (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0)))
       (*
        -0.25
        (+
         (/ t_2 (* a (pow b 3.0)))
         (/ (* 2.0 (* (* a c) t_2)) (* a (pow b 5.0))))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double t_1 = fma(2.0, t_0, b) * fma(-2.0, t_0, b);
	double t_2 = 4.0 * pow((a * c), 2.0);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -100.0) {
		tmp = ((pow(b, 2.0) - t_1) / (-b - sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = fma(-0.25, ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0))), (-0.25 * ((t_2 / (a * pow(b, 3.0))) + ((2.0 * ((a * c) * t_2)) / (a * pow(b, 5.0)))))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	t_1 = Float64(fma(2.0, t_0, b) * fma(-2.0, t_0, b))
	t_2 = Float64(4.0 * (Float64(a * c) ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -100.0)
		tmp = Float64(Float64(Float64((b ^ 2.0) - t_1) / Float64(Float64(-b) - sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-0.25, Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0))), Float64(-0.25 * Float64(Float64(t_2 / Float64(a * (b ^ 3.0))) + Float64(Float64(2.0 * Float64(Float64(a * c) * t_2)) / Float64(a * (b ^ 5.0)))))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * t$95$0 + b), $MachinePrecision] * N[(-2.0 * t$95$0 + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -100.0], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(t$95$2 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(a * c), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -100

   1. Initial program 88.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\sqrt{\left(4 \cdot a\right) \cdot c} \cdot \sqrt{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. difference-of-squares 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + \sqrt{\left(4 \cdot a\right) \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}}{a \cdot 2} \]

      3. associate-*l* 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      4. sqrt-prod 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      5. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{2} \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      6. associate-*l* 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}}{a \cdot 2} \]

      7. sqrt-prod 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right)}}{a \cdot 2} \]

      8. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{2} \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

   5. Applied egg-rr 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - 2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{\sqrt{a \cdot c} \cdot 2}\right) \cdot \left(b - 2 \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

      2. cancel-sign-sub-inv 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \color{blue}{\left(b + \left(-2\right) \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

      3. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + \color{blue}{-2} \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

   7. Simplified 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. flip-+ 88.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)} \cdot \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}}{a \cdot 2} \]

      2. pow2 88.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)} \cdot \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

      3. add-sqr-sqrt 89.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

      4. +-commutative 89.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt{a \cdot c} \cdot 2 + b\right)} \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

      5. *-commutative 89.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{2 \cdot \sqrt{a \cdot c}} + b\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

      6. fma-def 89.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right)} \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

      7. +-commutative 89.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \color{blue}{\left(-2 \cdot \sqrt{a \cdot c} + b\right)}}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

      8. fma-def 89.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

   9. Applied egg-rr 89.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}}}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. unpow2 89.8%

          \[\leadsto \frac{\frac{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}}}{a \cdot 2} \]

       2. sqr-neg 89.8%

          \[\leadsto \frac{\frac{\color{blue}{b \cdot b} - \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}}}{a \cdot 2} \]

       3. unpow2 89.8%

          \[\leadsto \frac{\frac{\color{blue}{{b}^{2}} - \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}}}{a \cdot 2} \]

   11. Simplified 89.8%

       \[\leadsto \frac{\color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}}}}{a \cdot 2} \]

   if -100 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 47.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 47.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 47.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. flip3-- 47.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      2. sqrt-div 46.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      3. pow2 46.5%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{\color{blue}{\left({b}^{2}\right)}}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      4. pow-pow 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{\color{blue}{{b}^{\left(2 \cdot 3\right)}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      5. metadata-eval 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{\color{blue}{6}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      6. associate-*l* 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - {\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      7. unpow-prod-down 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{{4}^{3} \cdot {\left(a \cdot c\right)}^{3}}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      8. metadata-eval 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   5. Applied egg-rr 46.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in b around inf 94.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(2 \cdot \left(a \cdot \left(c \cdot \left(-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.25 \cdot \frac{-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

   8. Simplified 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(2, 0 + \left(\left(2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)\right) \cdot c\right) \cdot a, {\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b}} \]

   9. Taylor expanded in c around 0 94.8%

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]
   10. Step-by-step derivation

       1. distribute-rgt-out 94.8%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

       2. associate-*r* 94.8%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

       3. *-commutative 94.8%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

       4. times-frac 94.8%

          \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

   11. Simplified 94.8%

       \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -100:\\ \;\;\;\;\frac{\frac{{b}^{2} - \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, -0.25 \cdot \left(\frac{4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b}\\ \end{array} \]

Alternative 3: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot {\left(a \cdot c\right)}^{2}\\ \mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, -0.25 \cdot \left(\frac{t_0}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot t_0\right)}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow (* a c) 2.0))))
   (-
    (fma
     -0.25
     (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0)))
     (*
      -0.25
      (+
       (/ t_0 (* a (pow b 3.0)))
       (/ (* 2.0 (* (* a c) t_0)) (* a (pow b 5.0))))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * pow((a * c), 2.0);
	return fma(-0.25, ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0))), (-0.25 * ((t_0 / (a * pow(b, 3.0))) + ((2.0 * ((a * c) * t_0)) / (a * pow(b, 5.0)))))) - (c / b);
}

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * (Float64(a * c) ^ 2.0))
	return Float64(fma(-0.25, Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0))), Float64(-0.25 * Float64(Float64(t_0 / Float64(a * (b ^ 3.0))) + Float64(Float64(2.0 * Float64(Float64(a * c) * t_0)) / Float64(a * (b ^ 5.0)))))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(t$95$0 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(a * c), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 49.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 49.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 49.5%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Step-by-step derivation

   1. flip3-- 49.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

   2. sqrt-div 48.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

   3. pow2 48.9%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{\color{blue}{\left({b}^{2}\right)}}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   4. pow-pow 49.0%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{\color{blue}{{b}^{\left(2 \cdot 3\right)}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   5. metadata-eval 49.0%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{\color{blue}{6}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   6. associate-*l* 49.0%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - {\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   7. unpow-prod-down 49.0%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{{4}^{3} \cdot {\left(a \cdot c\right)}^{3}}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   8. metadata-eval 49.0%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

5. Applied egg-rr 49.2%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

6. Taylor expanded in b around inf 92.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(2 \cdot \left(a \cdot \left(c \cdot \left(-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.25 \cdot \frac{-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

8. Simplified 92.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(2, 0 + \left(\left(2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)\right) \cdot c\right) \cdot a, {\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b}} \]

9. Taylor expanded in c around 0 92.9%

   \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]
10. Step-by-step derivation

    1. distribute-rgt-out 92.9%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

    2. associate-*r* 92.9%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

    3. *-commutative 92.9%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

    4. times-frac 92.9%

       \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

11. Simplified 92.9%

    \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}}, -0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

12. Final simplification 92.9%

    \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, -0.25 \cdot \left(\frac{4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}}\right)\right) - \frac{c}{b} \]

Alternative 4: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (* -0.25 (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0))))
    (/ (* a (pow c 2.0)) (pow b 3.0)))
   (/ c b))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0)))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * ((((a * c) ** 4.0d0) / a) * (20.0d0 / (b ** 7.0d0)))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * ((Math.pow((a * c), 4.0) / a) * (20.0 / Math.pow(b, 7.0)))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) - (c / b));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * ((math.pow((a * c), 4.0) / a) * (20.0 / math.pow(b, 7.0)))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) - (c / b))

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * ((((a * c) ^ 4.0) / a) * (20.0 / (b ^ 7.0)))) - ((a * (c ^ 2.0)) / (b ^ 3.0))) - (c / b));
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 49.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 49.5%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

4. Taylor expanded in b around inf 92.9%

   \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

5. Taylor expanded in c around 0 92.9%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
6. Step-by-step derivation

   1. distribute-rgt-out 92.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

   2. associate-*r* 92.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}\right)\right) \]

   3. *-commutative 92.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}\right)\right) \]

   4. times-frac 92.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}\right)}\right)\right) \]

7. Simplified 92.9%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right)}\right)\right) \]

8. Final simplification 92.9%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \]

Alternative 5: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot {\left(a \cdot c\right)}^{2}\\ t_1 := 4 \cdot \left(a \cdot c\right)\\ t_2 := \mathsf{fma}\left(b, b, t_1\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -100:\\ \;\;\;\;\frac{\sqrt{\frac{{b}^{4}}{t_2} - \frac{{t_1}^{2}}{t_2}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{t_0}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot t_0\right)}{a \cdot {b}^{5}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow (* a c) 2.0)))
        (t_1 (* 4.0 (* a c)))
        (t_2 (fma b b t_1)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -100.0)
     (/ (- (sqrt (- (/ (pow b 4.0) t_2) (/ (pow t_1 2.0) t_2))) b) (* a 2.0))
     (-
      (*
       -0.25
       (+
        (/ t_0 (* a (pow b 3.0)))
        (/ (* 2.0 (* (* a c) t_0)) (* a (pow b 5.0)))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * pow((a * c), 2.0);
	double t_1 = 4.0 * (a * c);
	double t_2 = fma(b, b, t_1);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -100.0) {
		tmp = (sqrt(((pow(b, 4.0) / t_2) - (pow(t_1, 2.0) / t_2))) - b) / (a * 2.0);
	} else {
		tmp = (-0.25 * ((t_0 / (a * pow(b, 3.0))) + ((2.0 * ((a * c) * t_0)) / (a * pow(b, 5.0))))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * (Float64(a * c) ^ 2.0))
	t_1 = Float64(4.0 * Float64(a * c))
	t_2 = fma(b, b, t_1)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -100.0)
		tmp = Float64(Float64(sqrt(Float64(Float64((b ^ 4.0) / t_2) - Float64((t_1 ^ 2.0) / t_2))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-0.25 * Float64(Float64(t_0 / Float64(a * (b ^ 3.0))) + Float64(Float64(2.0 * Float64(Float64(a * c) * t_0)) / Float64(a * (b ^ 5.0))))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * b + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -100.0], N[(N[(N[Sqrt[N[(N[(N[Power[b, 4.0], $MachinePrecision] / t$95$2), $MachinePrecision] - N[(N[Power[t$95$1, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(t$95$0 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(a * c), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -100

   1. Initial program 88.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 88.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. *-commutative 88.5%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      2. metadata-eval 88.5%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{\left(-4\right)} \cdot a\right)\right)} - b}{a \cdot 2} \]

      3. distribute-lft-neg-in 88.5%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      4. distribute-rgt-neg-in 88.5%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{-c \cdot \left(4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      5. *-commutative 88.5%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      6. fma-neg 88.4%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      7. flip-- 88.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}} - b}{a \cdot 2} \]

      8. div-sub 88.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c} - \frac{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}} - b}{a \cdot 2} \]

      9. pow2 88.4%

         \[\leadsto \frac{\sqrt{\frac{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c} - \frac{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      10. pow2 88.4%

          \[\leadsto \frac{\sqrt{\frac{{b}^{2} \cdot \color{blue}{{b}^{2}}}{b \cdot b + \left(4 \cdot a\right) \cdot c} - \frac{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      11. pow-prod-up 88.5%

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{{b}^{\left(2 + 2\right)}}}{b \cdot b + \left(4 \cdot a\right) \cdot c} - \frac{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      12. metadata-eval 88.5%

          \[\leadsto \frac{\sqrt{\frac{{b}^{\color{blue}{4}}}{b \cdot b + \left(4 \cdot a\right) \cdot c} - \frac{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      13. fma-def 88.6%

          \[\leadsto \frac{\sqrt{\frac{{b}^{4}}{\color{blue}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}} - \frac{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      14. associate-*l* 88.6%

          \[\leadsto \frac{\sqrt{\frac{{b}^{4}}{\mathsf{fma}\left(b, b, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - \frac{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      15. pow2 88.6%

          \[\leadsto \frac{\sqrt{\frac{{b}^{4}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)} - \frac{\color{blue}{{\left(\left(4 \cdot a\right) \cdot c\right)}^{2}}}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      16. associate-*l* 88.6%

          \[\leadsto \frac{\sqrt{\frac{{b}^{4}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)} - \frac{{\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{2}}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      17. fma-def 88.6%

          \[\leadsto \frac{\sqrt{\frac{{b}^{4}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)} - \frac{{\left(4 \cdot \left(a \cdot c\right)\right)}^{2}}{\color{blue}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}} - b}{a \cdot 2} \]

      18. associate-*l* 88.6%

          \[\leadsto \frac{\sqrt{\frac{{b}^{4}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)} - \frac{{\left(4 \cdot \left(a \cdot c\right)\right)}^{2}}{\mathsf{fma}\left(b, b, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}} - b}{a \cdot 2} \]

   5. Applied egg-rr 88.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{{b}^{4}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)} - \frac{{\left(4 \cdot \left(a \cdot c\right)\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}} - b}{a \cdot 2} \]

   if -100 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 47.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 47.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 47.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. flip3-- 47.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      2. sqrt-div 46.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      3. pow2 46.5%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{\color{blue}{\left({b}^{2}\right)}}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      4. pow-pow 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{\color{blue}{{b}^{\left(2 \cdot 3\right)}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      5. metadata-eval 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{\color{blue}{6}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      6. associate-*l* 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - {\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      7. unpow-prod-down 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{{4}^{3} \cdot {\left(a \cdot c\right)}^{3}}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      8. metadata-eval 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   5. Applied egg-rr 46.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in b around inf 92.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.25 \cdot \frac{-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)} \]

   8. Simplified 92.6%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right) - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -100:\\ \;\;\;\;\frac{\sqrt{\frac{{b}^{4}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)} - \frac{{\left(4 \cdot \left(a \cdot c\right)\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}}\right) - \frac{c}{b}\\ \end{array} \]

Alternative 6: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot {\left(a \cdot c\right)}^{2}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -100:\\ \;\;\;\;\frac{\sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + -2 \cdot \left(\sqrt{c} \cdot \sqrt{a}\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{t_0}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot t_0\right)}{a \cdot {b}^{5}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow (* a c) 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -100.0)
     (/
      (-
       (sqrt
        (* (+ b (* 2.0 (sqrt (* a c)))) (+ b (* -2.0 (* (sqrt c) (sqrt a))))))
       b)
      (* a 2.0))
     (-
      (*
       -0.25
       (+
        (/ t_0 (* a (pow b 3.0)))
        (/ (* 2.0 (* (* a c) t_0)) (* a (pow b 5.0)))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * pow((a * c), 2.0);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -100.0) {
		tmp = (sqrt(((b + (2.0 * sqrt((a * c)))) * (b + (-2.0 * (sqrt(c) * sqrt(a)))))) - b) / (a * 2.0);
	} else {
		tmp = (-0.25 * ((t_0 / (a * pow(b, 3.0))) + ((2.0 * ((a * c) * t_0)) / (a * pow(b, 5.0))))) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * ((a * c) ** 2.0d0)
    if (((sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)) <= (-100.0d0)) then
        tmp = (sqrt(((b + (2.0d0 * sqrt((a * c)))) * (b + ((-2.0d0) * (sqrt(c) * sqrt(a)))))) - b) / (a * 2.0d0)
    else
        tmp = ((-0.25d0) * ((t_0 / (a * (b ** 3.0d0))) + ((2.0d0 * ((a * c) * t_0)) / (a * (b ** 5.0d0))))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 4.0 * Math.pow((a * c), 2.0);
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -100.0) {
		tmp = (Math.sqrt(((b + (2.0 * Math.sqrt((a * c)))) * (b + (-2.0 * (Math.sqrt(c) * Math.sqrt(a)))))) - b) / (a * 2.0);
	} else {
		tmp = (-0.25 * ((t_0 / (a * Math.pow(b, 3.0))) + ((2.0 * ((a * c) * t_0)) / (a * Math.pow(b, 5.0))))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = 4.0 * math.pow((a * c), 2.0)
	tmp = 0
	if ((math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -100.0:
		tmp = (math.sqrt(((b + (2.0 * math.sqrt((a * c)))) * (b + (-2.0 * (math.sqrt(c) * math.sqrt(a)))))) - b) / (a * 2.0)
	else:
		tmp = (-0.25 * ((t_0 / (a * math.pow(b, 3.0))) + ((2.0 * ((a * c) * t_0)) / (a * math.pow(b, 5.0))))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * (Float64(a * c) ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -100.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + Float64(2.0 * sqrt(Float64(a * c)))) * Float64(b + Float64(-2.0 * Float64(sqrt(c) * sqrt(a)))))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-0.25 * Float64(Float64(t_0 / Float64(a * (b ^ 3.0))) + Float64(Float64(2.0 * Float64(Float64(a * c) * t_0)) / Float64(a * (b ^ 5.0))))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = 4.0 * ((a * c) ^ 2.0);
	tmp = 0.0;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -100.0)
		tmp = (sqrt(((b + (2.0 * sqrt((a * c)))) * (b + (-2.0 * (sqrt(c) * sqrt(a)))))) - b) / (a * 2.0);
	else
		tmp = (-0.25 * ((t_0 / (a * (b ^ 3.0))) + ((2.0 * ((a * c) * t_0)) / (a * (b ^ 5.0))))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -100.0], N[(N[(N[Sqrt[N[(N[(b + N[(2.0 * N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b + N[(-2.0 * N[(N[Sqrt[c], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(t$95$0 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(a * c), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -100

   1. Initial program 88.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\sqrt{\left(4 \cdot a\right) \cdot c} \cdot \sqrt{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. difference-of-squares 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + \sqrt{\left(4 \cdot a\right) \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}}{a \cdot 2} \]

      3. associate-*l* 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      4. sqrt-prod 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      5. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{2} \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      6. associate-*l* 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}}{a \cdot 2} \]

      7. sqrt-prod 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right)}}{a \cdot 2} \]

      8. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{2} \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

   5. Applied egg-rr 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - 2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{\sqrt{a \cdot c} \cdot 2}\right) \cdot \left(b - 2 \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

      2. cancel-sign-sub-inv 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \color{blue}{\left(b + \left(-2\right) \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

      3. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + \color{blue}{-2} \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

   7. Simplified 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. pow1/2 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \color{blue}{{\left(a \cdot c\right)}^{0.5}}\right)}}{a \cdot 2} \]

      2. *-commutative 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot {\color{blue}{\left(c \cdot a\right)}}^{0.5}\right)}}{a \cdot 2} \]

      3. unpow-prod-down 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \color{blue}{\left({c}^{0.5} \cdot {a}^{0.5}\right)}\right)}}{a \cdot 2} \]

      4. pow1/2 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \left(\color{blue}{\sqrt{c}} \cdot {a}^{0.5}\right)\right)}}{a \cdot 2} \]

      5. pow1/2 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \left(\sqrt{c} \cdot \color{blue}{\sqrt{a}}\right)\right)}}{a \cdot 2} \]

   9. Applied egg-rr 88.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \color{blue}{\left(\sqrt{c} \cdot \sqrt{a}\right)}\right)}}{a \cdot 2} \]

   if -100 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 47.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 47.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 47.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. flip3-- 47.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      2. sqrt-div 46.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      3. pow2 46.5%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{\color{blue}{\left({b}^{2}\right)}}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      4. pow-pow 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{\color{blue}{{b}^{\left(2 \cdot 3\right)}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      5. metadata-eval 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{\color{blue}{6}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      6. associate-*l* 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - {\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      7. unpow-prod-down 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{{4}^{3} \cdot {\left(a \cdot c\right)}^{3}}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      8. metadata-eval 46.6%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   5. Applied egg-rr 46.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in b around inf 92.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.25 \cdot \frac{-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)} \]

   8. Simplified 92.6%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{0 + 4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(0 + \left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right) + 0}{a \cdot {b}^{5}}\right) - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -100:\\ \;\;\;\;\frac{\sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + -2 \cdot \left(\sqrt{c} \cdot \sqrt{a}\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{2 \cdot \left(\left(a \cdot c\right) \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}}\right) - \frac{c}{b}\\ \end{array} \]

Alternative 7: 85.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.26:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-c}{b} - {c}^{2} \cdot \frac{a}{{b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.26)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (log1p (expm1 (- (/ (- c) b) (* (pow c 2.0) (/ a (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.26) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = log1p(expm1(((-c / b) - (pow(c, 2.0) * (a / pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.26)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = log1p(expm1(Float64(Float64(Float64(-c) / b) - Float64((c ^ 2.0) * Float64(a / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.26], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(N[((-c) / b), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.26000000000000001

   1. Initial program 81.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]

   if -0.26000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 44.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 89.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 89.5%

         \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

      2. unsub-neg 89.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 89.5%

         \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. distribute-neg-frac 89.5%

         \[\leadsto \color{blue}{\frac{-c}{b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 89.5%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]

   6. Simplified 89.5%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 89.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\right)\right)} \]

      2. associate-/r/ 89.5%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-c}{b} - \color{blue}{\frac{a}{{b}^{3}} \cdot {c}^{2}}\right)\right) \]

   8. Applied egg-rr 89.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot {c}^{2}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.26:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-c}{b} - {c}^{2} \cdot \frac{a}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 8: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.26:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{a}{\frac{b}{{\left(-2 \cdot \frac{c}{b}\right)}^{2}}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.26)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (* -0.25 (/ a (/ b (pow (* -2.0 (/ c b)) 2.0)))) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.26) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-0.25 * (a / (b / pow((-2.0 * (c / b)), 2.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.26)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-0.25 * Float64(a / Float64(b / (Float64(-2.0 * Float64(c / b)) ^ 2.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.26], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(a / N[(b / N[Power[N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.26000000000000001

   1. Initial program 81.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]

   if -0.26000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 44.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. flip3-- 44.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      2. sqrt-div 44.0%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      3. pow2 44.0%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{\color{blue}{\left({b}^{2}\right)}}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      4. pow-pow 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{\color{blue}{{b}^{\left(2 \cdot 3\right)}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      5. metadata-eval 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{\color{blue}{6}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      6. associate-*l* 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - {\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      7. unpow-prod-down 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{{4}^{3} \cdot {\left(a \cdot c\right)}^{3}}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      8. metadata-eval 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   5. Applied egg-rr 44.4%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in a around 0 89.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -0.25 \cdot \frac{a \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{c}^{2}}{{b}^{2}} + {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)\right)}{b}} \]
   7. Step-by-step derivation

      1. +-commutative 89.5%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{c}^{2}}{{b}^{2}} + {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)\right)}{b} + -1 \cdot \frac{c}{b}} \]

      2. mul-1-neg 89.5%

         \[\leadsto -0.25 \cdot \frac{a \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{c}^{2}}{{b}^{2}} + {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)\right)}{b} + \color{blue}{\left(-\frac{c}{b}\right)} \]

      3. unsub-neg 89.5%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{c}^{2}}{{b}^{2}} + {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)\right)}{b} - \frac{c}{b}} \]

   8. Simplified 89.5%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a}{\frac{b}{0 + {\left(\frac{c}{b} \cdot -2\right)}^{2}}} - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.26:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{a}{\frac{b}{{\left(-2 \cdot \frac{c}{b}\right)}^{2}}} - \frac{c}{b}\\ \end{array} \]

Alternative 9: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.26:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{a}{\frac{b}{{\left(-2 \cdot \frac{c}{b}\right)}^{2}}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.26)
     t_0
     (- (* -0.25 (/ a (/ b (pow (* -2.0 (/ c b)) 2.0)))) (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.26) {
		tmp = t_0;
	} else {
		tmp = (-0.25 * (a / (b / pow((-2.0 * (c / b)), 2.0)))) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.26d0)) then
        tmp = t_0
    else
        tmp = ((-0.25d0) * (a / (b / (((-2.0d0) * (c / b)) ** 2.0d0)))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.26) {
		tmp = t_0;
	} else {
		tmp = (-0.25 * (a / (b / Math.pow((-2.0 * (c / b)), 2.0)))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.26:
		tmp = t_0
	else:
		tmp = (-0.25 * (a / (b / math.pow((-2.0 * (c / b)), 2.0)))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.26)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.25 * Float64(a / Float64(b / (Float64(-2.0 * Float64(c / b)) ^ 2.0)))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.26)
		tmp = t_0;
	else
		tmp = (-0.25 * (a / (b / ((-2.0 * (c / b)) ^ 2.0)))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.26], t$95$0, N[(N[(-0.25 * N[(a / N[(b / N[Power[N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.26000000000000001

   1. Initial program 81.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   if -0.26000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 44.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Step-by-step derivation

      1. flip3-- 44.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      2. sqrt-div 44.0%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

      3. pow2 44.0%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{\color{blue}{\left({b}^{2}\right)}}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      4. pow-pow 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{\color{blue}{{b}^{\left(2 \cdot 3\right)}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      5. metadata-eval 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{\color{blue}{6}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      6. associate-*l* 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - {\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      7. unpow-prod-down 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{{4}^{3} \cdot {\left(a \cdot c\right)}^{3}}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      8. metadata-eval 44.2%

         \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   5. Applied egg-rr 44.4%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in a around 0 89.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -0.25 \cdot \frac{a \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{c}^{2}}{{b}^{2}} + {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)\right)}{b}} \]
   7. Step-by-step derivation

      1. +-commutative 89.5%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{c}^{2}}{{b}^{2}} + {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)\right)}{b} + -1 \cdot \frac{c}{b}} \]

      2. mul-1-neg 89.5%

         \[\leadsto -0.25 \cdot \frac{a \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{c}^{2}}{{b}^{2}} + {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)\right)}{b} + \color{blue}{\left(-\frac{c}{b}\right)} \]

      3. unsub-neg 89.5%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{c}^{2}}{{b}^{2}} + {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)\right)}{b} - \frac{c}{b}} \]

   8. Simplified 89.5%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a}{\frac{b}{0 + {\left(\frac{c}{b} \cdot -2\right)}^{2}}} - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.26:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{a}{\frac{b}{{\left(-2 \cdot \frac{c}{b}\right)}^{2}}} - \frac{c}{b}\\ \end{array} \]

Alternative 10: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -3.02 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -3.02e-5) t_0 (/ (- c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -3.02e-5) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-3.02d-5)) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -3.02e-5) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -3.02e-5:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -3.02e-5)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -3.02e-5)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -3.02e-5], t$95$0, N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -3.01999999999999988e-5

   1. Initial program 71.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   if -3.01999999999999988e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 33.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 33.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 82.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 82.0%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 82.0%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   6. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -3.02 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 11: 64.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (- c) b))

C

double code(double a, double b, double c) {
	return -c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -c / b
end function

Java

public static double code(double a, double b, double c) {
	return -c / b;
}

Python

def code(a, b, c):
	return -c / b

Julia

function code(a, b, c)
	return Float64(Float64(-c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = -c / b;
end

Wolfram

code[a_, b_, c_] := N[((-c) / b), $MachinePrecision]

Derivation

1. Initial program 49.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 49.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 49.5%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

4. Taylor expanded in b around inf 69.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
5. Step-by-step derivation

   1. mul-1-neg 69.1%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   2. distribute-neg-frac 69.1%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

6. Simplified 69.1%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]

7. Final simplification 69.1%

   \[\leadsto \frac{-c}{b} \]

Alternative 12: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 49.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 49.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 49.5%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 49.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\sqrt{\left(4 \cdot a\right) \cdot c} \cdot \sqrt{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. difference-of-squares 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + \sqrt{\left(4 \cdot a\right) \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}}{a \cdot 2} \]

   3. associate-*l* 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

   4. sqrt-prod 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

   5. metadata-eval 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{2} \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

   6. associate-*l* 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}}{a \cdot 2} \]

   7. sqrt-prod 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{\sqrt{4} \cdot \sqrt{a \cdot c}}\right)}}{a \cdot 2} \]

   8. metadata-eval 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - \color{blue}{2} \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

5. Applied egg-rr 49.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b - 2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. *-commutative 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \color{blue}{\sqrt{a \cdot c} \cdot 2}\right) \cdot \left(b - 2 \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

   2. cancel-sign-sub-inv 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \color{blue}{\left(b + \left(-2\right) \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

   3. metadata-eval 49.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + \color{blue}{-2} \cdot \sqrt{a \cdot c}\right)}}{a \cdot 2} \]

7. Simplified 49.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b + \sqrt{a \cdot c} \cdot 2\right) \cdot \left(b + -2 \cdot \sqrt{a \cdot c}\right)}}}{a \cdot 2} \]

8. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}\right)}{a}} \]

   2. distribute-rgt-out 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\sqrt{a \cdot c} \cdot \left(-2 + 2\right)\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{a \cdot c} \cdot \color{blue}{0}\right)}{a} \]

   4. mul0-rgt 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

10. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]

11. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023320 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))