Quadratic roots, narrow range

Percentage Accurate: 55.5% → 91.9%

Time: 23.2s

Alternatives: 16

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: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{t\_0 - {b}^{2}} \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \left(a \cdot \frac{20 \cdot {c}^{4}}{{b}^{7}}\right) \cdot -0.25\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -1.2)
     (/ 1.0 (* (/ (* a 2.0) (- t_0 (pow b 2.0))) (+ b (sqrt t_0))))
     (-
      (*
       a
       (-
        (*
         a
         (fma
          -2.0
          (/ (pow c 3.0) (pow b 5.0))
          (* (* a (/ (* 20.0 (pow c 4.0)) (pow b 7.0))) -0.25)))
        (/ (pow c 2.0) (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -1.2) {
		tmp = 1.0 / (((a * 2.0) / (t_0 - pow(b, 2.0))) * (b + sqrt(t_0)));
	} else {
		tmp = (a * ((a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), ((a * ((20.0 * pow(c, 4.0)) / pow(b, 7.0))) * -0.25))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -1.2)
		tmp = Float64(1.0 / Float64(Float64(Float64(a * 2.0) / Float64(t_0 - (b ^ 2.0))) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(a * Float64(Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(Float64(a * Float64(Float64(20.0 * (c ^ 4.0)) / (b ^ 7.0))) * -0.25))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -1.2], N[(1.0 / N[(N[(N[(a * 2.0), $MachinePrecision] / N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(N[(20.0 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.19999999999999996

   1. Initial program 83.3%

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

      1. +-commutative 83.3%

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

      2. sqr-neg 83.3%

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

      3. unsub-neg 83.3%

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

      4. sqr-neg 83.3%

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

      5. sub-neg 83.3%

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

      6. +-commutative 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-*r* 83.3%

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

      9. distribute-rgt-neg-in 83.3%

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

      10. fma-define 83.4%

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

      11. *-commutative 83.4%

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

      12. distribute-rgt-neg-in 83.4%

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

      13. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

      1. add-sqr-sqrt 82.5%

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

      2. pow2 82.5%

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

      3. pow1/2 82.5%

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

      4. sqrt-pow1 82.6%

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

      5. pow2 82.6%

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

      6. metadata-eval 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. pow-pow 83.4%

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

      2. metadata-eval 83.4%

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

      3. pow1/2 83.4%

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

      4. flip-- 84.5%

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

      5. add-sqr-sqrt 86.2%

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

      6. unpow2 86.2%

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

   8. Applied egg-rr 86.2%

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

      1. clear-num 86.1%

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

      2. inv-pow 86.1%

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

      3. +-commutative 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. unpow-1 86.1%

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

       2. associate-/r/ 86.2%

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

   12. Simplified 86.2%

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

   if -1.19999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.6%

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

      1. *-commutative 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in a around 0 92.7%

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

      1. +-commutative 92.7%

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

      2. mul-1-neg 92.7%

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

      3. unsub-neg 92.7%

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

   7. Simplified 92.7%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \left(a \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{b} \cdot 20\right)\right) \cdot -0.25\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

   8. Taylor expanded in c around 0 92.7%

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

      1. associate-*r/ 92.7%

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

   10. Simplified 92.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \left(a \cdot \frac{20 \cdot {c}^{4}}{{b}^{7}}\right) \cdot -0.25\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{t\_0 - {b}^{2}} \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{c \cdot \frac{20 \cdot {a}^{4}}{{b}^{6}}}{b}\right)\right)\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -1.2)
     (/ 1.0 (* (/ (* a 2.0) (- t_0 (pow b 2.0))) (+ b (sqrt t_0))))
     (/
      (*
       c
       (+
        (* -2.0 (/ a b))
        (*
         c
         (+
          (* -2.0 (/ (pow a 2.0) (pow b 3.0)))
          (*
           c
           (+
            (* -4.0 (/ (pow a 3.0) (pow b 5.0)))
            (* -0.5 (/ (* c (/ (* 20.0 (pow a 4.0)) (pow b 6.0))) b))))))))
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -1.2) {
		tmp = 1.0 / (((a * 2.0) / (t_0 - pow(b, 2.0))) * (b + sqrt(t_0)));
	} else {
		tmp = (c * ((-2.0 * (a / b)) + (c * ((-2.0 * (pow(a, 2.0) / pow(b, 3.0))) + (c * ((-4.0 * (pow(a, 3.0) / pow(b, 5.0))) + (-0.5 * ((c * ((20.0 * pow(a, 4.0)) / pow(b, 6.0))) / b)))))))) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -1.2)
		tmp = Float64(1.0 / Float64(Float64(Float64(a * 2.0) / Float64(t_0 - (b ^ 2.0))) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(c * Float64(Float64(-2.0 * Float64(a / b)) + Float64(c * Float64(Float64(-2.0 * Float64((a ^ 2.0) / (b ^ 3.0))) + Float64(c * Float64(Float64(-4.0 * Float64((a ^ 3.0) / (b ^ 5.0))) + Float64(-0.5 * Float64(Float64(c * Float64(Float64(20.0 * (a ^ 4.0)) / (b ^ 6.0))) / b)))))))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -1.2], N[(1.0 / N[(N[(N[(a * 2.0), $MachinePrecision] / N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(-2.0 * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-4.0 * N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(c * N[(N[(20.0 * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $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 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.19999999999999996

   1. Initial program 83.3%

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

      1. +-commutative 83.3%

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

      2. sqr-neg 83.3%

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

      3. unsub-neg 83.3%

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

      4. sqr-neg 83.3%

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

      5. sub-neg 83.3%

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

      6. +-commutative 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-*r* 83.3%

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

      9. distribute-rgt-neg-in 83.3%

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

      10. fma-define 83.4%

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

      11. *-commutative 83.4%

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

      12. distribute-rgt-neg-in 83.4%

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

      13. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

      1. add-sqr-sqrt 82.5%

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

      2. pow2 82.5%

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

      3. pow1/2 82.5%

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

      4. sqrt-pow1 82.6%

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

      5. pow2 82.6%

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

      6. metadata-eval 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. pow-pow 83.4%

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

      2. metadata-eval 83.4%

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

      3. pow1/2 83.4%

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

      4. flip-- 84.5%

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

      5. add-sqr-sqrt 86.2%

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

      6. unpow2 86.2%

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

   8. Applied egg-rr 86.2%

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

      1. clear-num 86.1%

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

      2. inv-pow 86.1%

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

      3. +-commutative 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. unpow-1 86.1%

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

       2. associate-/r/ 86.2%

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

   12. Simplified 86.2%

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

   if -1.19999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.6%

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

      1. *-commutative 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in c around 0 92.5%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)}}{a \cdot 2} \]

   6. Taylor expanded in b around 0 92.5%

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{\color{blue}{\frac{c \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{{b}^{6}}}}{b}\right)\right)\right)}{a \cdot 2} \]
   7. Step-by-step derivation

      1. associate-/l* 92.5%

         \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{\color{blue}{c \cdot \frac{4 \cdot {a}^{4} + 16 \cdot {a}^{4}}{{b}^{6}}}}{b}\right)\right)\right)}{a \cdot 2} \]

      2. distribute-rgt-out 92.5%

         \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{c \cdot \frac{\color{blue}{{a}^{4} \cdot \left(4 + 16\right)}}{{b}^{6}}}{b}\right)\right)\right)}{a \cdot 2} \]

      3. metadata-eval 92.5%

         \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{c \cdot \frac{{a}^{4} \cdot \color{blue}{20}}{{b}^{6}}}{b}\right)\right)\right)}{a \cdot 2} \]

   8. Simplified 92.5%

      \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{\color{blue}{c \cdot \frac{{a}^{4} \cdot 20}{{b}^{6}}}}{b}\right)\right)\right)}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{c \cdot \frac{20 \cdot {a}^{4}}{{b}^{6}}}{b}\right)\right)\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{t\_0 - {b}^{2}} \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.098)
     (/ 1.0 (* (/ (* a 2.0) (- t_0 (pow b 2.0))) (+ b (sqrt t_0))))
     (-
      (*
       a
       (-
        (* -2.0 (* a (/ (pow c 3.0) (pow b 5.0))))
        (/ (pow c 2.0) (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.098) {
		tmp = 1.0 / (((a * 2.0) / (t_0 - pow(b, 2.0))) * (b + sqrt(t_0)));
	} else {
		tmp = (a * ((-2.0 * (a * (pow(c, 3.0) / pow(b, 5.0)))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.098)
		tmp = Float64(1.0 / Float64(Float64(Float64(a * 2.0) / Float64(t_0 - (b ^ 2.0))) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(a * Float64((c ^ 3.0) / (b ^ 5.0)))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.098], N[(1.0 / N[(N[(N[(a * 2.0), $MachinePrecision] / N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-2.0 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.098000000000000004

   1. Initial program 81.6%

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

      1. +-commutative 81.6%

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

      2. sqr-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. sqr-neg 81.6%

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

      5. sub-neg 81.6%

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

      6. +-commutative 81.6%

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

      7. *-commutative 81.6%

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

      8. associate-*r* 81.6%

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

      9. distribute-rgt-neg-in 81.6%

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

      10. fma-define 81.6%

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

      11. *-commutative 81.6%

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

      12. distribute-rgt-neg-in 81.6%

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

      13. metadata-eval 81.6%

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

   3. Simplified 81.6%

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

      1. add-sqr-sqrt 79.9%

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

      2. pow2 79.9%

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

      3. pow1/2 79.9%

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

      4. sqrt-pow1 79.9%

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

      5. pow2 79.9%

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

      6. metadata-eval 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. pow-pow 81.6%

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

      2. metadata-eval 81.6%

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

      3. pow1/2 81.6%

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

      4. flip-- 81.9%

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

      5. add-sqr-sqrt 83.6%

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

      6. unpow2 83.6%

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

   8. Applied egg-rr 83.6%

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

      1. clear-num 83.6%

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

      2. inv-pow 83.6%

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

      3. +-commutative 83.6%

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

   10. Applied egg-rr 83.6%

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

       1. unpow-1 83.6%

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

       2. associate-/r/ 83.6%

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

   12. Simplified 83.6%

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

   if -0.098000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 91.4%

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

      1. +-commutative 91.4%

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

      2. mul-1-neg 91.4%

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

      3. unsub-neg 91.4%

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

      4. mul-1-neg 91.4%

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

      5. unsub-neg 91.4%

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

      6. associate-/l* 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.098)
   (/
    (/
     (- (fma a (* c -4.0) (pow b 2.0)) (pow b 2.0))
     (+ b (sqrt (* a (+ (* c -4.0) (/ (pow b 2.0) a))))))
    (* a 2.0))
   (-
    (*
     a
     (-
      (* -2.0 (* a (/ (pow c 3.0) (pow b 5.0))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.098) {
		tmp = ((fma(a, (c * -4.0), pow(b, 2.0)) - pow(b, 2.0)) / (b + sqrt((a * ((c * -4.0) + (pow(b, 2.0) / a)))))) / (a * 2.0);
	} else {
		tmp = (a * ((-2.0 * (a * (pow(c, 3.0) / pow(b, 5.0)))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.098)
		tmp = Float64(Float64(Float64(fma(a, Float64(c * -4.0), (b ^ 2.0)) - (b ^ 2.0)) / Float64(b + sqrt(Float64(a * Float64(Float64(c * -4.0) + Float64((b ^ 2.0) / a)))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(a * Float64((c ^ 3.0) / (b ^ 5.0)))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.098], N[(N[(N[(N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(N[(c * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-2.0 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.098000000000000004

   1. Initial program 81.6%

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

      1. +-commutative 81.6%

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

      2. sqr-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. sqr-neg 81.6%

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

      5. sub-neg 81.6%

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

      6. +-commutative 81.6%

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

      7. *-commutative 81.6%

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

      8. associate-*r* 81.6%

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

      9. distribute-rgt-neg-in 81.6%

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

      10. fma-define 81.6%

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

      11. *-commutative 81.6%

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

      12. distribute-rgt-neg-in 81.6%

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

      13. metadata-eval 81.6%

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

   3. Simplified 81.6%

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

      1. add-sqr-sqrt 79.9%

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

      2. pow2 79.9%

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

      3. pow1/2 79.9%

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

      4. sqrt-pow1 79.9%

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

      5. pow2 79.9%

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

      6. metadata-eval 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. pow-pow 81.6%

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

      2. metadata-eval 81.6%

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

      3. pow1/2 81.6%

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

      4. flip-- 81.9%

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

      5. add-sqr-sqrt 83.6%

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

      6. unpow2 83.6%

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

   8. Applied egg-rr 83.6%

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

   9. Taylor expanded in a around inf 83.6%

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

   if -0.098000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 91.4%

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

      1. +-commutative 91.4%

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

      2. mul-1-neg 91.4%

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

      3. unsub-neg 91.4%

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

      4. mul-1-neg 91.4%

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

      5. unsub-neg 91.4%

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

      6. associate-/l* 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.098)
   (/
    (/
     (- (fma a (* c -4.0) (pow b 2.0)) (pow b 2.0))
     (+ b (sqrt (* c (+ (* a -4.0) (/ (pow b 2.0) c))))))
    (* a 2.0))
   (-
    (*
     a
     (-
      (* -2.0 (* a (/ (pow c 3.0) (pow b 5.0))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.098) {
		tmp = ((fma(a, (c * -4.0), pow(b, 2.0)) - pow(b, 2.0)) / (b + sqrt((c * ((a * -4.0) + (pow(b, 2.0) / c)))))) / (a * 2.0);
	} else {
		tmp = (a * ((-2.0 * (a * (pow(c, 3.0) / pow(b, 5.0)))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.098)
		tmp = Float64(Float64(Float64(fma(a, Float64(c * -4.0), (b ^ 2.0)) - (b ^ 2.0)) / Float64(b + sqrt(Float64(c * Float64(Float64(a * -4.0) + Float64((b ^ 2.0) / c)))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(a * Float64((c ^ 3.0) / (b ^ 5.0)))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.098], N[(N[(N[(N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(N[(a * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-2.0 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.098000000000000004

   1. Initial program 81.6%

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

      1. +-commutative 81.6%

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

      2. sqr-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. sqr-neg 81.6%

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

      5. sub-neg 81.6%

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

      6. +-commutative 81.6%

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

      7. *-commutative 81.6%

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

      8. associate-*r* 81.6%

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

      9. distribute-rgt-neg-in 81.6%

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

      10. fma-define 81.6%

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

      11. *-commutative 81.6%

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

      12. distribute-rgt-neg-in 81.6%

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

      13. metadata-eval 81.6%

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

   3. Simplified 81.6%

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

      1. add-sqr-sqrt 79.9%

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

      2. pow2 79.9%

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

      3. pow1/2 79.9%

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

      4. sqrt-pow1 79.9%

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

      5. pow2 79.9%

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

      6. metadata-eval 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. pow-pow 81.6%

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

      2. metadata-eval 81.6%

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

      3. pow1/2 81.6%

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

      4. flip-- 81.9%

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

      5. add-sqr-sqrt 83.6%

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

      6. unpow2 83.6%

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

   8. Applied egg-rr 83.6%

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

   9. Taylor expanded in c around inf 83.6%

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

   if -0.098000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 91.4%

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

      1. +-commutative 91.4%

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

      2. mul-1-neg 91.4%

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

      3. unsub-neg 91.4%

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

      4. mul-1-neg 91.4%

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

      5. unsub-neg 91.4%

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

      6. associate-/l* 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{{b}^{2} + a \cdot \left(c \cdot -4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.098)
   (/
    (/
     (- (fma a (* c -4.0) (pow b 2.0)) (pow b 2.0))
     (+ b (sqrt (+ (pow b 2.0) (* a (* c -4.0))))))
    (* a 2.0))
   (-
    (*
     a
     (-
      (* -2.0 (* a (/ (pow c 3.0) (pow b 5.0))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.098) {
		tmp = ((fma(a, (c * -4.0), pow(b, 2.0)) - pow(b, 2.0)) / (b + sqrt((pow(b, 2.0) + (a * (c * -4.0)))))) / (a * 2.0);
	} else {
		tmp = (a * ((-2.0 * (a * (pow(c, 3.0) / pow(b, 5.0)))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.098)
		tmp = Float64(Float64(Float64(fma(a, Float64(c * -4.0), (b ^ 2.0)) - (b ^ 2.0)) / Float64(b + sqrt(Float64((b ^ 2.0) + Float64(a * Float64(c * -4.0)))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(a * Float64((c ^ 3.0) / (b ^ 5.0)))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.098], N[(N[(N[(N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-2.0 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.098000000000000004

   1. Initial program 81.6%

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

      1. +-commutative 81.6%

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

      2. sqr-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. sqr-neg 81.6%

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

      5. sub-neg 81.6%

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

      6. +-commutative 81.6%

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

      7. *-commutative 81.6%

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

      8. associate-*r* 81.6%

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

      9. distribute-rgt-neg-in 81.6%

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

      10. fma-define 81.6%

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

      11. *-commutative 81.6%

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

      12. distribute-rgt-neg-in 81.6%

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

      13. metadata-eval 81.6%

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

   3. Simplified 81.6%

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

      1. add-sqr-sqrt 79.9%

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

      2. pow2 79.9%

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

      3. pow1/2 79.9%

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

      4. sqrt-pow1 79.9%

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

      5. pow2 79.9%

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

      6. metadata-eval 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. pow-pow 81.6%

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

      2. metadata-eval 81.6%

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

      3. pow1/2 81.6%

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

      4. flip-- 81.9%

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

      5. add-sqr-sqrt 83.6%

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

      6. unpow2 83.6%

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

   8. Applied egg-rr 83.6%

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

      1. fma-undefine 83.6%

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

   10. Applied egg-rr 83.6%

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

   if -0.098000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 91.4%

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

      1. +-commutative 91.4%

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

      2. mul-1-neg 91.4%

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

      3. unsub-neg 91.4%

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

      4. mul-1-neg 91.4%

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

      5. unsub-neg 91.4%

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

      6. associate-/l* 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{{b}^{2} + a \cdot \left(c \cdot -4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.8:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -1.8)
   (* (/ 0.5 a) (- (sqrt (fma a (* c -4.0) (pow b 2.0))) b))
   (-
    (*
     a
     (-
      (* -2.0 (* a (/ (pow c 3.0) (pow b 5.0))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -1.8) {
		tmp = (0.5 / a) * (sqrt(fma(a, (c * -4.0), pow(b, 2.0))) - b);
	} else {
		tmp = (a * ((-2.0 * (a * (pow(c, 3.0) / pow(b, 5.0)))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -1.8)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(a, Float64(c * -4.0), (b ^ 2.0))) - b));
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(a * Float64((c ^ 3.0) / (b ^ 5.0)))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -1.8], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-2.0 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.80000000000000004

   1. Initial program 84.2%

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

      1. +-commutative 84.2%

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

      2. sqr-neg 84.2%

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

      3. unsub-neg 84.2%

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

      4. sqr-neg 84.2%

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

      5. sub-neg 84.2%

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

      6. +-commutative 84.2%

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

      7. *-commutative 84.2%

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

      8. associate-*r* 84.2%

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

      9. distribute-rgt-neg-in 84.2%

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

      10. fma-define 84.2%

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

      11. *-commutative 84.2%

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

      12. distribute-rgt-neg-in 84.2%

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

      13. metadata-eval 84.2%

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

   3. Simplified 84.2%

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

      1. div-sub 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. *-commutative 84.0%

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

      4. times-frac 84.0%

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

      5. metadata-eval 84.0%

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

      6. pow2 84.0%

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

      7. *-un-lft-identity 84.0%

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

      8. *-commutative 84.0%

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

      9. times-frac 84.0%

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

      10. metadata-eval 84.0%

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

   6. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - 0.5 \cdot \frac{b}{a}} \]
   7. Step-by-step derivation

      1. sub-neg 84.0%

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

      2. associate-*r/ 84.0%

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

   8. Applied egg-rr 84.0%

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

      1. unsub-neg 84.0%

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

      2. associate-*r/ 84.0%

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

      3. distribute-lft-out-- 84.0%

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

      4. div-sub 84.2%

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

      5. associate-*r/ 84.2%

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

      6. associate-*l/ 84.3%

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

   10. Simplified 84.3%

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

   if -1.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.0%

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

      1. *-commutative 52.0%

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

   3. Simplified 52.0%

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

   5. Taylor expanded in a around 0 89.9%

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

      1. +-commutative 89.9%

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

      2. mul-1-neg 89.9%

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

      3. unsub-neg 89.9%

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

      4. mul-1-neg 89.9%

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

      5. unsub-neg 89.9%

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

      6. associate-/l* 89.9%

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

   7. Simplified 89.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.8:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - {\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.098)
   (* (/ 0.5 a) (- (sqrt (fma a (* c -4.0) (pow b 2.0))) b))
   (/ (- (- c) (pow (/ (* c (sqrt a)) b) 2.0)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.098) {
		tmp = (0.5 / a) * (sqrt(fma(a, (c * -4.0), pow(b, 2.0))) - b);
	} else {
		tmp = (-c - pow(((c * sqrt(a)) / b), 2.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.098)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(a, Float64(c * -4.0), (b ^ 2.0))) - b));
	else
		tmp = Float64(Float64(Float64(-c) - (Float64(Float64(c * sqrt(a)) / b) ^ 2.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.098], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[Power[N[(N[(c * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.098000000000000004

   1. Initial program 81.6%

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

      1. +-commutative 81.6%

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

      2. sqr-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. sqr-neg 81.6%

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

      5. sub-neg 81.6%

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

      6. +-commutative 81.6%

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

      7. *-commutative 81.6%

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

      8. associate-*r* 81.6%

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

      9. distribute-rgt-neg-in 81.6%

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

      10. fma-define 81.6%

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

      11. *-commutative 81.6%

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

      12. distribute-rgt-neg-in 81.6%

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

      13. metadata-eval 81.6%

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

   3. Simplified 81.6%

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

      1. div-sub 81.0%

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

      2. *-un-lft-identity 81.0%

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

      3. *-commutative 81.0%

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

      4. times-frac 81.0%

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

      5. metadata-eval 81.0%

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

      6. pow2 81.0%

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

      7. *-un-lft-identity 81.0%

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

      8. *-commutative 81.0%

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

      9. times-frac 81.0%

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

      10. metadata-eval 81.0%

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

   6. Applied egg-rr 81.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - 0.5 \cdot \frac{b}{a}} \]
   7. Step-by-step derivation

      1. sub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. unsub-neg 81.0%

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

      2. associate-*r/ 81.0%

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

      3. distribute-lft-out-- 81.0%

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

      4. div-sub 81.6%

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

      5. associate-*r/ 81.6%

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

      6. associate-*l/ 81.7%

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

   10. Simplified 81.7%

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

   if -0.098000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in b around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. unsub-neg 86.5%

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

      3. mul-1-neg 86.5%

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

   7. Simplified 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. add-sqr-sqrt 86.5%

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

      3. pow2 86.5%

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

      4. sqrt-div 86.5%

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

      5. *-commutative 86.5%

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

      6. sqrt-prod 86.5%

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

      7. sqrt-pow1 86.5%

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

      8. metadata-eval 86.5%

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

      9. pow1 86.5%

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

      10. sqrt-pow1 86.5%

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

      11. metadata-eval 86.5%

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

      12. pow1 86.5%

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

   9. Applied egg-rr 86.5%

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

       1. *-lft-identity 86.5%

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

   11. Simplified 86.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - {\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - {\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.098)
   (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
   (/ (- (- c) (pow (/ (* c (sqrt a)) b) 2.0)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.098) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = (-c - pow(((c * sqrt(a)) / b), 2.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.098)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - (Float64(Float64(c * sqrt(a)) / b) ^ 2.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.098], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[Power[N[(N[(c * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.098000000000000004

   1. Initial program 81.6%

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

      1. +-commutative 81.6%

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

      2. sqr-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. sqr-neg 81.6%

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

      5. sub-neg 81.6%

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

      6. +-commutative 81.6%

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

      7. *-commutative 81.6%

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

      8. associate-*r* 81.6%

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

      9. distribute-rgt-neg-in 81.6%

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

      10. fma-define 81.6%

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

      11. *-commutative 81.6%

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

      12. distribute-rgt-neg-in 81.6%

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

      13. metadata-eval 81.6%

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

   3. Simplified 81.6%

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

   if -0.098000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in b around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. unsub-neg 86.5%

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

      3. mul-1-neg 86.5%

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

   7. Simplified 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. add-sqr-sqrt 86.5%

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

      3. pow2 86.5%

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

      4. sqrt-div 86.5%

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

      5. *-commutative 86.5%

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

      6. sqrt-prod 86.5%

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

      7. sqrt-pow1 86.5%

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

      8. metadata-eval 86.5%

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

      9. pow1 86.5%

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

      10. sqrt-pow1 86.5%

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

      11. metadata-eval 86.5%

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

      12. pow1 86.5%

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

   9. Applied egg-rr 86.5%

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

       1. *-lft-identity 86.5%

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

   11. Simplified 86.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - {\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - {\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.098)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (- (- c) (pow (/ (* c (sqrt a)) b) 2.0)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.098) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - pow(((c * sqrt(a)) / b), 2.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.098)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - (Float64(Float64(c * sqrt(a)) / b) ^ 2.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.098], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[Power[N[(N[(c * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.098000000000000004

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. +-commutative 81.6%

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

      3. sqr-neg 81.6%

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

      4. unsub-neg 81.6%

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

      5. sqr-neg 81.6%

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

      6. fma-neg 81.7%

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

      7. distribute-lft-neg-in 81.7%

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

      8. *-commutative 81.7%

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

      9. *-commutative 81.7%

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

      10. distribute-rgt-neg-in 81.7%

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

      11. metadata-eval 81.7%

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

   3. Simplified 81.7%

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

   if -0.098000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in b around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. unsub-neg 86.5%

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

      3. mul-1-neg 86.5%

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

   7. Simplified 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. add-sqr-sqrt 86.5%

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

      3. pow2 86.5%

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

      4. sqrt-div 86.5%

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

      5. *-commutative 86.5%

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

      6. sqrt-prod 86.5%

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

      7. sqrt-pow1 86.5%

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

      8. metadata-eval 86.5%

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

      9. pow1 86.5%

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

      10. sqrt-pow1 86.5%

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

      11. metadata-eval 86.5%

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

      12. pow1 86.5%

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

   9. Applied egg-rr 86.5%

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

       1. *-lft-identity 86.5%

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

   11. Simplified 86.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - {\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.098:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - {\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.098) t_0 (/ (- (- c) (pow (/ (* c (sqrt a)) b) 2.0)) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.098) {
		tmp = t_0;
	} else {
		tmp = (-c - pow(((c * sqrt(a)) / b), 2.0)) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.098d0)) then
        tmp = t_0
    else
        tmp = (-c - (((c * sqrt(a)) / b) ** 2.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.098) {
		tmp = t_0;
	} else {
		tmp = (-c - Math.pow(((c * Math.sqrt(a)) / b), 2.0)) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.098:
		tmp = t_0
	else:
		tmp = (-c - math.pow(((c * math.sqrt(a)) / b), 2.0)) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.098)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-c) - (Float64(Float64(c * sqrt(a)) / b) ^ 2.0)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.098)
		tmp = t_0;
	else
		tmp = (-c - (((c * sqrt(a)) / b) ^ 2.0)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.098], t$95$0, N[(N[((-c) - N[Power[N[(N[(c * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.098000000000000004

   1. Initial program 81.6%

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

   if -0.098000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in b around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. unsub-neg 86.5%

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

      3. mul-1-neg 86.5%

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

   7. Simplified 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. add-sqr-sqrt 86.5%

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

      3. pow2 86.5%

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

      4. sqrt-div 86.5%

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

      5. *-commutative 86.5%

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

      6. sqrt-prod 86.5%

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

      7. sqrt-pow1 86.5%

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

      8. metadata-eval 86.5%

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

      9. pow1 86.5%

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

      10. sqrt-pow1 86.5%

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

      11. metadata-eval 86.5%

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

      12. pow1 86.5%

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

   9. Applied egg-rr 86.5%

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

       1. *-lft-identity 86.5%

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

   11. Simplified 86.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - {\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (*
    c
    (+
     (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.39999999999999991

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. +-commutative 81.9%

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

      3. sqr-neg 81.9%

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

      4. unsub-neg 81.9%

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

      5. sqr-neg 81.9%

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

      6. fma-neg 82.0%

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

      7. distribute-lft-neg-in 82.0%

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

      8. *-commutative 82.0%

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

      9. *-commutative 82.0%

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

      10. distribute-rgt-neg-in 82.0%

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

      11. metadata-eval 82.0%

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

   3. Simplified 82.0%

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

   if 2.39999999999999991 < b

   1. Initial program 49.3%

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

      1. *-commutative 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in c around 0 91.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.098:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.098) t_0 (* c (- (/ -1.0 b) (/ (* a c) (pow b 3.0)))))))

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 <= -0.098) {
		tmp = t_0;
	} else {
		tmp = c * ((-1.0 / b) - ((a * c) / pow(b, 3.0)));
	}
	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 <= (-0.098d0)) then
        tmp = t_0
    else
        tmp = c * (((-1.0d0) / b) - ((a * c) / (b ** 3.0d0)))
    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 <= -0.098) {
		tmp = t_0;
	} else {
		tmp = c * ((-1.0 / b) - ((a * c) / Math.pow(b, 3.0)));
	}
	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 <= -0.098:
		tmp = t_0
	else:
		tmp = c * ((-1.0 / b) - ((a * c) / math.pow(b, 3.0)))
	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 <= -0.098)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(a * c) / (b ^ 3.0))));
	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 <= -0.098)
		tmp = t_0;
	else
		tmp = c * ((-1.0 / b) - ((a * c) / (b ^ 3.0)));
	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, -0.098], t$95$0, N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(a * c), $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 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.098000000000000004

   1. Initial program 81.6%

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

   if -0.098000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in c around 0 86.3%

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

      1. associate-*r/ 86.3%

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

      2. neg-mul-1 86.3%

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

      3. distribute-rgt-neg-in 86.3%

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

   7. Simplified 86.3%

      \[\leadsto \color{blue}{c \cdot \left(\frac{a \cdot \left(-c\right)}{{b}^{3}} - \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.098:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (/ (* a c) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((a * c) / pow(b, 3.0)));
}

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 * (((-1.0d0) / b) - ((a * c) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((a * c) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - ((a * c) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(a * c) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - ((a * c) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in c around 0 80.6%

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

   1. associate-*r/ 80.6%

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

   2. neg-mul-1 80.6%

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

   3. distribute-rgt-neg-in 80.6%

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

7. Simplified 80.6%

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

8. Final simplification 80.6%

   \[\leadsto c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right) \]
9. Add Preprocessing

Alternative 15: 64.4% 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(c / Float64(-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 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around inf 63.6%

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

   1. associate-*r/ 63.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. mul-1-neg 63.6%

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

7. Simplified 63.6%

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

8. Final simplification 63.6%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Alternative 16: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. +-commutative 55.9%

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

   2. sqr-neg 55.9%

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

   3. unsub-neg 55.9%

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

   4. sqr-neg 55.9%

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

   5. sub-neg 55.9%

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

   6. +-commutative 55.9%

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

   7. *-commutative 55.9%

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

   8. associate-*r* 55.9%

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

   9. distribute-rgt-neg-in 55.9%

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

   10. fma-define 55.9%

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

   11. *-commutative 55.9%

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

   12. distribute-rgt-neg-in 55.9%

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

   13. metadata-eval 55.9%

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

3. Simplified 55.9%

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

   1. add-sqr-sqrt 54.7%

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

   2. pow2 54.7%

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

   3. pow1/2 54.7%

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

   4. sqrt-pow1 54.9%

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

   5. pow2 54.9%

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

   6. metadata-eval 54.9%

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

6. Applied egg-rr 54.9%

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

   1. unpow2 54.9%

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

   2. add-sqr-sqrt 53.9%

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

   3. prod-diff 54.2%

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

   4. add-sqr-sqrt 54.9%

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

   5. fma-neg 55.3%

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

   6. unpow2 55.3%

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

   7. pow-pow 56.6%

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

   8. metadata-eval 56.6%

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

   9. pow1/2 56.6%

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

   10. add-sqr-sqrt 55.7%

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

8. Applied egg-rr 55.7%

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

9. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

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

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

11. Simplified 3.2%

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

12. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024068 
(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)))