Quadratic roots, narrow range

Percentage Accurate: 55.5% → 91.1%

Time: 13.7s

Alternatives: 9

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: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.044)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.5 a))
     (-
      (-
       (fma
        -0.25
        (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 20.0)))
        (* -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))))
       (/ c b))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.044) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.5 / a);
	} else {
		tmp = (fma(-0.25, (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0))), (-2.0 * (pow(c, 3.0) / (pow(b, 5.0) / (a * a))))) - (c / b)) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.044)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(fma(-0.25, Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 20.0))), Float64(-2.0 * Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))))) - Float64(c / b)) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $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], -0.044], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $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)) < -0.043999999999999997

   1. Initial program 83.0%

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

      1. /-rgt-identity 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 83.0%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 83.0%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 83.1%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 82.9%

         \[\leadsto \left(\sqrt{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) \cdot \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right) \cdot \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}} - b\right) \cdot \frac{0.5}{a} \]

      2. pow3 82.8%

         \[\leadsto \left(\sqrt{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}^{3}}}} - b\right) \cdot \frac{0.5}{a} \]

      3. associate-*l* 82.8%

         \[\leadsto \left(\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -4\right)}\right)\right)}^{3}}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 82.8%

      \[\leadsto \left(\sqrt{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{3}}}} - b\right) \cdot \frac{0.5}{a} \]
   6. Step-by-step derivation

      1. flip-- 82.4%

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

      2. add-sqr-sqrt 83.2%

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

      3. rem-cbrt-cube 84.0%

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

      4. rem-cbrt-cube 84.0%

         \[\leadsto \frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} + b} \cdot \frac{0.5}{a} \]

   7. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}} \cdot \frac{0.5}{a} \]

   if -0.043999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.2%

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

      1. /-rgt-identity 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 46.2%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 46.2%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 46.2%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 46.2%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 46.3%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 46.3%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 46.3%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 46.3%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]

   4. Taylor expanded in a around 0 96.1%

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

   6. Simplified 96.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{b}{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]

   7. Taylor expanded in b around 0 96.1%

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

      1. associate-/l* 96.1%

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

      2. distribute-rgt-out 96.1%

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

      3. metadata-eval 96.1%

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

   9. Simplified 96.1%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 2: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \mathsf{fma}\left(0.03125, \frac{\left(a \cdot a\right) \cdot {\left(c \cdot -4\right)}^{3}}{{b}^{5}}, \frac{\left(a \cdot \left(\left(c \cdot c\right) \cdot 16\right)\right) \cdot -0.0625}{{b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.044)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.5 a))
     (fma
      0.25
      (/ (* c -4.0) b)
      (fma
       0.03125
       (/ (* (* a a) (pow (* c -4.0) 3.0)) (pow b 5.0))
       (/ (* (* a (* (* c c) 16.0)) -0.0625) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.044) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.5 / a);
	} else {
		tmp = fma(0.25, ((c * -4.0) / b), fma(0.03125, (((a * a) * pow((c * -4.0), 3.0)) / pow(b, 5.0)), (((a * ((c * c) * 16.0)) * -0.0625) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.044)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.5 / a));
	else
		tmp = fma(0.25, Float64(Float64(c * -4.0) / b), fma(0.03125, Float64(Float64(Float64(a * a) * (Float64(c * -4.0) ^ 3.0)) / (b ^ 5.0)), Float64(Float64(Float64(a * Float64(Float64(c * c) * 16.0)) * -0.0625) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $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], -0.044], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(c * -4.0), $MachinePrecision] / b), $MachinePrecision] + N[(0.03125 * N[(N[(N[(a * a), $MachinePrecision] * N[Power[N[(c * -4.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(N[(c * c), $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $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)) < -0.043999999999999997

   1. Initial program 83.0%

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

      1. /-rgt-identity 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 83.0%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 83.0%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 83.1%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 82.9%

         \[\leadsto \left(\sqrt{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) \cdot \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right) \cdot \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}} - b\right) \cdot \frac{0.5}{a} \]

      2. pow3 82.8%

         \[\leadsto \left(\sqrt{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}^{3}}}} - b\right) \cdot \frac{0.5}{a} \]

      3. associate-*l* 82.8%

         \[\leadsto \left(\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -4\right)}\right)\right)}^{3}}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 82.8%

      \[\leadsto \left(\sqrt{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{3}}}} - b\right) \cdot \frac{0.5}{a} \]
   6. Step-by-step derivation

      1. flip-- 82.4%

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

      2. add-sqr-sqrt 83.2%

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

      3. rem-cbrt-cube 84.0%

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

      4. rem-cbrt-cube 84.0%

         \[\leadsto \frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} + b} \cdot \frac{0.5}{a} \]

   7. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}} \cdot \frac{0.5}{a} \]

   if -0.043999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.2%

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

      1. /-rgt-identity 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 46.2%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 46.2%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 46.2%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 46.2%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 46.3%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 46.3%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 46.3%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 46.3%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 46.2%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b\right) \cdot \frac{0.5}{a} \]

      2. *-commutative 46.2%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      3. metadata-eval 46.2%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(-4\right)} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a} \]

      4. cancel-sign-sub-inv 46.2%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      5. associate-*l* 46.2%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}} - b\right) \cdot \frac{0.5}{a} \]

      6. *-un-lft-identity 46.2%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      7. prod-diff 46.3%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 46.2%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
   6. Step-by-step derivation

      1. +-commutative 46.2%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

      2. fma-udef 46.2%

         \[\leadsto \left(\sqrt{\color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      3. *-rgt-identity 46.2%

         \[\leadsto \left(\sqrt{\left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      4. *-rgt-identity 46.2%

         \[\leadsto \left(\sqrt{\left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      5. count-2 46.2%

         \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      6. *-commutative 46.2%

         \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(\left(c \cdot -4\right) \cdot a\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      7. *-commutative 46.2%

         \[\leadsto \left(\sqrt{2 \cdot \left(\color{blue}{\left(-4 \cdot c\right)} \cdot a\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      8. associate-*r* 46.2%

         \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(-4 \cdot \left(c \cdot a\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      9. *-rgt-identity 46.2%

         \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]

      10. fma-neg 46.1%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

      11. *-commutative 46.1%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{\left(c \cdot -4\right) \cdot a}\right)} - b\right) \cdot \frac{0.5}{a} \]

      12. *-commutative 46.1%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{\left(-4 \cdot c\right)} \cdot a\right)} - b\right) \cdot \frac{0.5}{a} \]

      13. associate-*r* 46.1%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]

   7. Simplified 46.1%

      \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

   8. Taylor expanded in a around 0 94.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{-8 \cdot c - -4 \cdot c}{b} + \left(-0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}} + 0.03125 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]
   9. Step-by-step derivation

      1. fma-def 94.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{-8 \cdot c - -4 \cdot c}{b}, -0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}} + 0.03125 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]

      2. distribute-rgt-out-- 94.4%

         \[\leadsto \mathsf{fma}\left(0.25, \frac{\color{blue}{c \cdot \left(-8 - -4\right)}}{b}, -0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}} + 0.03125 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{3} \cdot {a}^{2}}{{b}^{5}}\right) \]

      3. metadata-eval 94.4%

         \[\leadsto \mathsf{fma}\left(0.25, \frac{c \cdot \color{blue}{-4}}{b}, -0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}} + 0.03125 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{3} \cdot {a}^{2}}{{b}^{5}}\right) \]

      4. +-commutative 94.4%

         \[\leadsto \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \color{blue}{0.03125 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}}}\right) \]

      5. fma-def 94.4%

         \[\leadsto \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \color{blue}{\mathsf{fma}\left(0.03125, \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{3} \cdot {a}^{2}}{{b}^{5}}, -0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}}\right)}\right) \]

      6. *-commutative 94.4%

         \[\leadsto \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \mathsf{fma}\left(0.03125, \frac{\color{blue}{{a}^{2} \cdot {\left(-8 \cdot c - -4 \cdot c\right)}^{3}}}{{b}^{5}}, -0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}}\right)\right) \]

      7. unpow2 94.4%

         \[\leadsto \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \mathsf{fma}\left(0.03125, \frac{\color{blue}{\left(a \cdot a\right)} \cdot {\left(-8 \cdot c - -4 \cdot c\right)}^{3}}{{b}^{5}}, -0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}}\right)\right) \]

      8. distribute-rgt-out-- 94.4%

         \[\leadsto \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \mathsf{fma}\left(0.03125, \frac{\left(a \cdot a\right) \cdot {\color{blue}{\left(c \cdot \left(-8 - -4\right)\right)}}^{3}}{{b}^{5}}, -0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}}\right)\right) \]

      9. metadata-eval 94.4%

         \[\leadsto \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \mathsf{fma}\left(0.03125, \frac{\left(a \cdot a\right) \cdot {\left(c \cdot \color{blue}{-4}\right)}^{3}}{{b}^{5}}, -0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}}\right)\right) \]

   10. Simplified 94.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \mathsf{fma}\left(0.03125, \frac{\left(a \cdot a\right) \cdot {\left(c \cdot -4\right)}^{3}}{{b}^{5}}, \frac{\left(a \cdot \left(16 \cdot \left(c \cdot c\right)\right)\right) \cdot -0.0625}{{b}^{3}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \mathsf{fma}\left(0.03125, \frac{\left(a \cdot a\right) \cdot {\left(c \cdot -4\right)}^{3}}{{b}^{5}}, \frac{\left(a \cdot \left(\left(c \cdot c\right) \cdot 16\right)\right) \cdot -0.0625}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 3: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.044)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.5 a))
     (-
      (fma -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a))) (/ (- c) b))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.044) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.5 / a);
	} else {
		tmp = fma(-2.0, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), (-c / b)) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.044)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.5 / a));
	else
		tmp = Float64(fma(-2.0, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), Float64(Float64(-c) / b)) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $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], -0.044], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $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)) < -0.043999999999999997

   1. Initial program 83.0%

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

      1. /-rgt-identity 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 83.0%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 83.0%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 83.1%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 82.9%

         \[\leadsto \left(\sqrt{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) \cdot \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right) \cdot \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}} - b\right) \cdot \frac{0.5}{a} \]

      2. pow3 82.8%

         \[\leadsto \left(\sqrt{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}^{3}}}} - b\right) \cdot \frac{0.5}{a} \]

      3. associate-*l* 82.8%

         \[\leadsto \left(\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -4\right)}\right)\right)}^{3}}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 82.8%

      \[\leadsto \left(\sqrt{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{3}}}} - b\right) \cdot \frac{0.5}{a} \]
   6. Step-by-step derivation

      1. flip-- 82.4%

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

      2. add-sqr-sqrt 83.2%

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

      3. rem-cbrt-cube 84.0%

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

      4. rem-cbrt-cube 84.0%

         \[\leadsto \frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} + b} \cdot \frac{0.5}{a} \]

   7. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}} \cdot \frac{0.5}{a} \]

   if -0.043999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.2%

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

      1. /-rgt-identity 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 46.2%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 46.2%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 46.2%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 46.2%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 46.3%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 46.3%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 46.3%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 46.3%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 46.3%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]

   4. Taylor expanded in b around inf 94.4%

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

      1. +-commutative 94.4%

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

      2. mul-1-neg 94.4%

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

      3. unsub-neg 94.4%

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

      4. +-commutative 94.4%

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

      5. fma-def 94.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, -1 \cdot \frac{c}{b}\right)} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. associate-/l* 94.4%

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, -1 \cdot \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      7. unpow2 94.4%

         \[\leadsto \mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, -1 \cdot \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      8. mul-1-neg 94.4%

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

      9. distribute-neg-frac 94.4%

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

      10. associate-/l* 94.4%

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

      11. unpow2 94.4%

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

   6. Simplified 94.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 4: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.044)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.5 a))
     (/ (/ (* c (* a -4.0)) (+ b (+ b (* -2.0 (/ c (/ b a)))))) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.044) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.5 / a);
	} else {
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.044)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(c * Float64(a * -4.0)) / Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a)))))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $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], -0.044], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $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)) < -0.043999999999999997

   1. Initial program 83.0%

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

      1. /-rgt-identity 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 83.0%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 83.0%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 83.1%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 82.9%

         \[\leadsto \left(\sqrt{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) \cdot \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right) \cdot \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}} - b\right) \cdot \frac{0.5}{a} \]

      2. pow3 82.8%

         \[\leadsto \left(\sqrt{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}^{3}}}} - b\right) \cdot \frac{0.5}{a} \]

      3. associate-*l* 82.8%

         \[\leadsto \left(\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -4\right)}\right)\right)}^{3}}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 82.8%

      \[\leadsto \left(\sqrt{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{3}}}} - b\right) \cdot \frac{0.5}{a} \]
   6. Step-by-step derivation

      1. flip-- 82.4%

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

      2. add-sqr-sqrt 83.2%

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

      3. rem-cbrt-cube 84.0%

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

      4. rem-cbrt-cube 84.0%

         \[\leadsto \frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} + b} \cdot \frac{0.5}{a} \]

   7. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}} \cdot \frac{0.5}{a} \]

   if -0.043999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.2%

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

      1. *-commutative 46.2%

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

      2. +-commutative 46.2%

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

      3. unsub-neg 46.2%

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

      4. fma-neg 46.3%

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

      5. associate-*l* 46.3%

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

      6. *-commutative 46.3%

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

      7. distribute-rgt-neg-in 46.3%

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

      8. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in b around inf 36.3%

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

      1. flip-- 36.1%

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

      2. associate-/l* 36.1%

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

      3. associate-/l* 36.1%

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

      4. associate-/l* 36.1%

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

   6. Applied egg-rr 36.1%

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

   7. Taylor expanded in b around inf 90.9%

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

      1. *-commutative 90.9%

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

      2. associate-*r* 90.9%

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

   9. Simplified 90.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 5: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -8 \cdot \left(a \cdot c\right) + \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{t_0 - b \cdot b}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* -8.0 (* a c)) (+ (* b b) (* 4.0 (* a c))))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.044)
     (* (/ 0.5 a) (/ (- t_0 (* b b)) (+ b (sqrt t_0))))
     (/ (/ (* c (* a -4.0)) (+ b (+ b (* -2.0 (/ c (/ b a)))))) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (-8.0 * (a * c)) + ((b * b) + (4.0 * (a * c)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.044) {
		tmp = (0.5 / a) * ((t_0 - (b * b)) / (b + sqrt(t_0)));
	} else {
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	}
	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 = ((-8.0d0) * (a * c)) + ((b * b) + (4.0d0 * (a * c)))
    if (((sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)) <= (-0.044d0)) then
        tmp = (0.5d0 / a) * ((t_0 - (b * b)) / (b + sqrt(t_0)))
    else
        tmp = ((c * (a * (-4.0d0))) / (b + (b + ((-2.0d0) * (c / (b / a)))))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-8.0 * (a * c)) + ((b * b) + (4.0 * (a * c)));
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.044) {
		tmp = (0.5 / a) * ((t_0 - (b * b)) / (b + Math.sqrt(t_0)));
	} else {
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (-8.0 * (a * c)) + ((b * b) + (4.0 * (a * c)))
	tmp = 0
	if ((math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.044:
		tmp = (0.5 / a) * ((t_0 - (b * b)) / (b + math.sqrt(t_0)))
	else:
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-8.0 * Float64(a * c)) + Float64(Float64(b * b) + Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.044)
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(c * Float64(a * -4.0)) / Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a)))))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (-8.0 * (a * c)) + ((b * b) + (4.0 * (a * c)));
	tmp = 0.0;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.044)
		tmp = (0.5 / a) * ((t_0 - (b * b)) / (b + sqrt(t_0)));
	else
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-8.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] + N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $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], -0.044], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $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)) < -0.043999999999999997

   1. Initial program 83.0%

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

      1. /-rgt-identity 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 83.0%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 83.0%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 83.0%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 83.1%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 83.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 83.1%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 83.0%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b\right) \cdot \frac{0.5}{a} \]

      2. *-commutative 83.0%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      3. metadata-eval 83.0%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(-4\right)} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a} \]

      4. cancel-sign-sub-inv 83.0%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      5. associate-*l* 83.0%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}} - b\right) \cdot \frac{0.5}{a} \]

      6. *-un-lft-identity 83.0%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      7. prod-diff 83.1%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 82.9%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
   6. Step-by-step derivation

      1. +-commutative 82.9%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

      2. fma-udef 82.9%

         \[\leadsto \left(\sqrt{\color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      3. *-rgt-identity 82.9%

         \[\leadsto \left(\sqrt{\left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      4. *-rgt-identity 82.9%

         \[\leadsto \left(\sqrt{\left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      5. count-2 82.9%

         \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      6. *-commutative 82.9%

         \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(\left(c \cdot -4\right) \cdot a\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      7. *-commutative 82.9%

         \[\leadsto \left(\sqrt{2 \cdot \left(\color{blue}{\left(-4 \cdot c\right)} \cdot a\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      8. associate-*r* 82.9%

         \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(-4 \cdot \left(c \cdot a\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      9. *-rgt-identity 82.9%

         \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]

      10. fma-neg 82.9%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

      11. *-commutative 82.9%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{\left(c \cdot -4\right) \cdot a}\right)} - b\right) \cdot \frac{0.5}{a} \]

      12. *-commutative 82.9%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{\left(-4 \cdot c\right)} \cdot a\right)} - b\right) \cdot \frac{0.5}{a} \]

      13. associate-*r* 82.9%

          \[\leadsto \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]

   7. Simplified 82.9%

      \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]
   8. Step-by-step derivation

      1. flip-- 82.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} \cdot \sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}} \cdot \frac{0.5}{a} \]

      2. add-sqr-sqrt 83.8%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right)} - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]

      3. associate-*r* 83.8%

         \[\leadsto \frac{\left(\color{blue}{\left(2 \cdot -4\right) \cdot \left(c \cdot a\right)} + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]

      4. metadata-eval 83.8%

         \[\leadsto \frac{\left(\color{blue}{-8} \cdot \left(c \cdot a\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]

      5. cancel-sign-sub-inv 83.8%

         \[\leadsto \frac{\left(-8 \cdot \left(c \cdot a\right) + \color{blue}{\left(b \cdot b + \left(--4\right) \cdot \left(c \cdot a\right)\right)}\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]

      6. metadata-eval 83.8%

         \[\leadsto \frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + \color{blue}{4} \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]

   9. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}} \cdot \frac{0.5}{a} \]

   if -0.043999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.2%

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

      1. *-commutative 46.2%

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

      2. +-commutative 46.2%

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

      3. unsub-neg 46.2%

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

      4. fma-neg 46.3%

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

      5. associate-*l* 46.3%

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

      6. *-commutative 46.3%

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

      7. distribute-rgt-neg-in 46.3%

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

      8. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in b around inf 36.3%

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

      1. flip-- 36.1%

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

      2. associate-/l* 36.1%

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

      3. associate-/l* 36.1%

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

      4. associate-/l* 36.1%

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

   6. Applied egg-rr 36.1%

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

   7. Taylor expanded in b around inf 90.9%

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

      1. *-commutative 90.9%

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

      2. associate-*r* 90.9%

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

   9. Simplified 90.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\left(-8 \cdot \left(a \cdot c\right) + \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)\right) - b \cdot b}{b + \sqrt{-8 \cdot \left(a \cdot c\right) + \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 6: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.044:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.044)
     t_0
     (/ (/ (* c (* a -4.0)) (+ b (+ b (* -2.0 (/ c (/ b a)))))) (* a 2.0)))))

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 <= -0.044) {
		tmp = t_0;
	} else {
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	}
	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 <= (-0.044d0)) then
        tmp = t_0
    else
        tmp = ((c * (a * (-4.0d0))) / (b + (b + ((-2.0d0) * (c / (b / a)))))) / (a * 2.0d0)
    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 <= -0.044) {
		tmp = t_0;
	} else {
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	}
	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 <= -0.044:
		tmp = t_0
	else:
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0)
	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 <= -0.044)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c * Float64(a * -4.0)) / Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a)))))) / Float64(a * 2.0));
	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 <= -0.044)
		tmp = t_0;
	else
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	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, -0.044], t$95$0, N[(N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $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)) < -0.043999999999999997

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. +-commutative 83.0%

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

      3. unsub-neg 83.0%

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

      4. fma-neg 83.1%

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

      5. associate-*l* 83.1%

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

      6. *-commutative 83.1%

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

      7. distribute-rgt-neg-in 83.1%

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

      8. metadata-eval 83.1%

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

   3. Simplified 83.1%

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

      1. fma-udef 83.0%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b\right) \cdot \frac{0.5}{a} \]

      2. *-commutative 83.0%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      3. metadata-eval 83.0%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(-4\right)} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a} \]

      4. cancel-sign-sub-inv 83.0%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      5. associate-*l* 83.0%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}} - b\right) \cdot \frac{0.5}{a} \]

      6. *-un-lft-identity 83.0%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      7. prod-diff 83.1%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 82.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-rgt-identity 82.9%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      2. fma-neg 82.9%

         \[\leadsto \left(\sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      3. fma-udef 82.9%

         \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

      4. *-rgt-identity 82.9%

         \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      5. *-rgt-identity 82.9%

         \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]

      6. associate--r- 83.0%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -4\right) - \left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

      7. associate--r+ 83.0%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right) - a \cdot \left(c \cdot -4\right)\right) - a \cdot \left(c \cdot -4\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

      8. +-inverses 83.0%

         \[\leadsto \left(\sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a} \]

      9. neg-sub0 83.0%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -4\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

      10. associate-*r* 83.0%

          \[\leadsto \left(\sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -4}\right)} - b\right) \cdot \frac{0.5}{a} \]

      11. distribute-rgt-neg-in 83.0%

          \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot \left(--4\right)}} - b\right) \cdot \frac{0.5}{a} \]

      12. metadata-eval 83.0%

          \[\leadsto \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot \color{blue}{4}} - b\right) \cdot \frac{0.5}{a} \]

      13. *-commutative 83.0%

          \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(c \cdot a\right)} \cdot 4} - b\right) \cdot \frac{0.5}{a} \]

      14. associate-*r* 83.0%

          \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{c \cdot \left(a \cdot 4\right)}} - b\right) \cdot \frac{0.5}{a} \]

   7. Simplified 83.0%

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

   if -0.043999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.2%

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

      1. *-commutative 46.2%

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

      2. +-commutative 46.2%

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

      3. unsub-neg 46.2%

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

      4. fma-neg 46.3%

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

      5. associate-*l* 46.3%

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

      6. *-commutative 46.3%

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

      7. distribute-rgt-neg-in 46.3%

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

      8. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in b around inf 36.3%

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

      1. flip-- 36.1%

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

      2. associate-/l* 36.1%

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

      3. associate-/l* 36.1%

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

      4. associate-/l* 36.1%

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

   6. Applied egg-rr 36.1%

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

   7. Taylor expanded in b around inf 90.9%

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

      1. *-commutative 90.9%

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

      2. associate-*r* 90.9%

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

   9. Simplified 90.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.044:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 7: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.075)
   (* (/ 0.5 a) (- (sqrt (- (* b b) (* (* 4.0 a) c))) b))
   (/ (/ (* c (* a -4.0)) (+ b (+ b (* -2.0 (/ c (/ b a)))))) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.075) {
		tmp = (0.5 / a) * (sqrt(((b * b) - ((4.0 * a) * c))) - b);
	} else {
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	}
	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) :: tmp
    if (b <= 0.075d0) then
        tmp = (0.5d0 / a) * (sqrt(((b * b) - ((4.0d0 * a) * c))) - b)
    else
        tmp = ((c * (a * (-4.0d0))) / (b + (b + ((-2.0d0) * (c / (b / a)))))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.075) {
		tmp = (0.5 / a) * (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b);
	} else {
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.075:
		tmp = (0.5 / a) * (math.sqrt(((b * b) - ((4.0 * a) * c))) - b)
	else:
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.075)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b));
	else
		tmp = Float64(Float64(Float64(c * Float64(a * -4.0)) / Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a)))))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.075)
		tmp = (0.5 / a) * (sqrt(((b * b) - ((4.0 * a) * c))) - b);
	else
		tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (c / (b / a)))))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.075], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0749999999999999972

   1. Initial program 85.3%

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

      1. /-rgt-identity 85.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval 85.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l* 85.3%

         \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

      4. associate-*r/ 85.2%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

      5. +-commutative 85.2%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

      6. unsub-neg 85.2%

         \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

      7. fma-neg 85.4%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      8. associate-*l* 85.4%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      9. *-commutative 85.4%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      10. distribute-rgt-neg-in 85.4%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      11. metadata-eval 85.4%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

      12. associate-/r* 85.4%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

      13. metadata-eval 85.4%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

      14. metadata-eval 85.4%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Simplified 85.4%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 85.2%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b\right) \cdot \frac{0.5}{a} \]

      2. *-commutative 85.2%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      3. metadata-eval 85.2%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(-4\right)} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a} \]

      4. cancel-sign-sub-inv 85.2%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      5. associate-*l* 85.2%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}} - b\right) \cdot \frac{0.5}{a} \]

      6. *-un-lft-identity 85.2%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

      7. prod-diff 85.4%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 85.0%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 85.0%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      2. fma-neg 85.2%

         \[\leadsto \left(\sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      3. fma-udef 85.2%

         \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

      4. *-rgt-identity 85.2%

         \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

      5. *-rgt-identity 85.2%

         \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]

      6. associate--r- 85.2%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -4\right) - \left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

      7. associate--r+ 85.2%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right) - a \cdot \left(c \cdot -4\right)\right) - a \cdot \left(c \cdot -4\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

      8. +-inverses 85.2%

         \[\leadsto \left(\sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a} \]

      9. neg-sub0 85.2%

         \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -4\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]

      10. associate-*r* 85.2%

          \[\leadsto \left(\sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -4}\right)} - b\right) \cdot \frac{0.5}{a} \]

      11. distribute-rgt-neg-in 85.2%

          \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot \left(--4\right)}} - b\right) \cdot \frac{0.5}{a} \]

      12. metadata-eval 85.2%

          \[\leadsto \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot \color{blue}{4}} - b\right) \cdot \frac{0.5}{a} \]

      13. *-commutative 85.2%

          \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{\left(c \cdot a\right)} \cdot 4} - b\right) \cdot \frac{0.5}{a} \]

      14. associate-*r* 85.2%

          \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{c \cdot \left(a \cdot 4\right)}} - b\right) \cdot \frac{0.5}{a} \]

   7. Simplified 85.2%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - c \cdot \left(a \cdot 4\right)}} - b\right) \cdot \frac{0.5}{a} \]

   if 0.0749999999999999972 < b

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

      2. +-commutative 51.2%

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

      3. unsub-neg 51.2%

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

      4. fma-neg 51.3%

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

      5. associate-*l* 51.3%

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

      6. *-commutative 51.3%

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

      7. distribute-rgt-neg-in 51.3%

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

      8. metadata-eval 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in b around inf 36.4%

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

      1. flip-- 36.3%

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

      2. associate-/l* 36.3%

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

      3. associate-/l* 36.3%

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

      4. associate-/l* 36.3%

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

   6. Applied egg-rr 36.3%

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

   7. Taylor expanded in b around inf 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*r* 86.4%

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

   9. Simplified 86.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 8: 81.7% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

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 * (a * (-4.0d0))) / (b + (b + ((-2.0d0) * (c / (b / a)))))) / (a * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

   1. *-commutative 54.7%

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

   2. +-commutative 54.7%

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

   3. unsub-neg 54.7%

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

   4. fma-neg 54.8%

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

   5. associate-*l* 54.8%

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

   6. *-commutative 54.8%

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

   7. distribute-rgt-neg-in 54.8%

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

   8. metadata-eval 54.8%

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

3. Simplified 54.8%

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

4. Taylor expanded in b around inf 36.5%

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

   1. flip-- 36.3%

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

   2. associate-/l* 36.3%

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

   3. associate-/l* 36.3%

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

   4. associate-/l* 36.3%

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

6. Applied egg-rr 36.3%

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

7. Taylor expanded in b around inf 83.3%

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

   1. *-commutative 83.3%

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

   2. associate-*r* 83.3%

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

9. Simplified 83.3%

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

10. Final simplification 83.3%

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

Alternative 9: 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 54.7%

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

   1. /-rgt-identity 54.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

   2. metadata-eval 54.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

   3. associate-/l* 54.7%

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

   4. associate-*r/ 54.7%

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

   5. +-commutative 54.7%

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

   6. unsub-neg 54.7%

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

   7. fma-neg 54.8%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   8. associate-*l* 54.8%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   9. *-commutative 54.8%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   10. distribute-rgt-neg-in 54.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   11. metadata-eval 54.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

   12. associate-/r* 54.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

   13. metadata-eval 54.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

   14. metadata-eval 54.8%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

3. Simplified 54.8%

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]

4. Taylor expanded in b around inf 65.3%

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

   1. mul-1-neg 65.3%

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

   2. distribute-neg-frac 65.3%

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

6. Simplified 65.3%

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

7. Final simplification 65.3%

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

Reproduce

herbie shell --seed 2023207 
(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)))