Quadratic roots, medium range

Percentage Accurate: 31.5% → 94.1%

Time: 21.9s

Alternatives: 10

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.5% 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: 94.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{-4 \cdot \left(c \cdot a\right) + {b}^{2}} - b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -50000000.0)
   (* (/ 0.5 a) (- (sqrt (+ (* -4.0 (* c a)) (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) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0) {
		tmp = (0.5 / a) * (sqrt(((-4.0 * (c * a)) + 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;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-50000000.0d0)) then
        tmp = (0.5d0 / a) * (sqrt((((-4.0d0) * (c * a)) + (b ** 2.0d0))) - b)
    else
        tmp = (a * (((-2.0d0) * (a * ((c ** 3.0d0) / (b ** 5.0d0)))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0) {
		tmp = (0.5 / a) * (Math.sqrt(((-4.0 * (c * a)) + Math.pow(b, 2.0))) - b);
	} else {
		tmp = (a * ((-2.0 * (a * (Math.pow(c, 3.0) / Math.pow(b, 5.0)))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0:
		tmp = (0.5 / a) * (math.sqrt(((-4.0 * (c * a)) + math.pow(b, 2.0))) - b)
	else:
		tmp = (a * ((-2.0 * (a * (math.pow(c, 3.0) / math.pow(b, 5.0)))) - (math.pow(c, 2.0) / math.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(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -50000000.0)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(-4.0 * Float64(c * a)) + (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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0)
		tmp = (0.5 / a) * (sqrt(((-4.0 * (c * a)) + (b ^ 2.0))) - b);
	else
		tmp = (a * ((-2.0 * (a * ((c ^ 3.0) / (b ^ 5.0)))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -50000000.0], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * N[(c * a), $MachinePrecision]), $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)) < -5e7

   1. Initial program 95.1%

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

      1. *-commutative 95.1%

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

      2. +-commutative 95.1%

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

      3. sqr-neg 95.1%

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

      4. unsub-neg 95.1%

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

      5. sqr-neg 95.1%

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

      6. fma-neg 95.1%

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

      7. distribute-lft-neg-in 95.1%

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

      8. *-commutative 95.1%

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

      9. *-commutative 95.1%

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

      10. distribute-rgt-neg-in 95.1%

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

      11. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

      1. *-commutative 95.1%

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

      2. metadata-eval 95.1%

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

      3. distribute-lft-neg-in 95.1%

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

      4. distribute-rgt-neg-in 95.1%

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

      5. *-commutative 95.1%

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

      6. fma-neg 95.1%

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

      7. flip-- 94.8%

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

      8. div-sub 94.2%

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

      9. pow2 94.2%

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

      10. pow2 94.2%

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

      11. pow-prod-up 94.2%

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

      12. metadata-eval 94.2%

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

      13. fma-define 94.4%

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

      14. associate-*l* 94.4%

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

      15. pow2 94.4%

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

      16. associate-*l* 94.4%

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

      17. fma-define 94.4%

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

      18. associate-*l* 94.4%

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

   6. Applied egg-rr 94.4%

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

      1. div-sub 93.4%

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

      2. sub-div 93.6%

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

      3. unpow-prod-down 93.6%

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

      4. metadata-eval 93.6%

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

      5. *-commutative 93.6%

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

      6. *-commutative 93.6%

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

      7. *-commutative 93.6%

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

      8. *-commutative 93.6%

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

   8. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]
   9. Step-by-step derivation

      1. div-sub 94.8%

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

      2. *-lft-identity 94.8%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right)}}{2 \cdot a} \]

      3. *-commutative 94.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot 1}}{2 \cdot a} \]

      4. associate-*r/ 94.8%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot \frac{1}{2 \cdot a}} \]

      5. *-commutative 94.8%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right)} \]

      6. associate-/r* 94.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \]

      7. metadata-eval 94.8%

         \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \]

   10. Simplified 94.8%

       \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4} + -16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} - b\right)} \]

   11. Taylor expanded in b around 0 95.2%

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

   if -5e7 < (/.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 29.3%

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

      1. *-commutative 29.3%

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

      2. +-commutative 29.3%

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

      3. sqr-neg 29.3%

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

      4. unsub-neg 29.3%

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

      5. sqr-neg 29.3%

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

      6. fma-neg 29.3%

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

      7. distribute-lft-neg-in 29.3%

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

      8. *-commutative 29.3%

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

      9. *-commutative 29.3%

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

      10. distribute-rgt-neg-in 29.3%

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

      11. metadata-eval 29.3%

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

   3. Simplified 29.3%

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

      1. *-commutative 29.3%

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

      2. metadata-eval 29.3%

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

      3. distribute-lft-neg-in 29.3%

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

      4. distribute-rgt-neg-in 29.3%

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

      5. *-commutative 29.3%

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

      6. fma-neg 29.3%

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

      7. flip-- 29.3%

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

      8. div-sub 29.2%

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

      9. pow2 29.2%

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

      10. pow2 29.2%

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

      11. pow-prod-up 29.2%

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

      12. metadata-eval 29.2%

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

      13. fma-define 29.4%

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

      14. associate-*l* 29.4%

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

      15. pow2 29.4%

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

      16. associate-*l* 29.4%

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

      17. fma-define 29.4%

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

      18. associate-*l* 29.4%

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

   6. Applied egg-rr 29.4%

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

      1. div-sub 28.8%

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

      2. sub-div 28.8%

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

      3. unpow-prod-down 28.8%

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

      4. metadata-eval 28.8%

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

      5. *-commutative 28.8%

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

      6. *-commutative 28.8%

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

      7. *-commutative 28.8%

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

      8. *-commutative 28.8%

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

   8. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]
   9. Step-by-step derivation

      1. div-sub 29.3%

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

      2. *-lft-identity 29.3%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right)}}{2 \cdot a} \]

      3. *-commutative 29.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot 1}}{2 \cdot a} \]

      4. associate-*r/ 29.3%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot \frac{1}{2 \cdot a}} \]

      5. *-commutative 29.3%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right)} \]

      6. associate-/r* 29.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \]

      7. metadata-eval 29.3%

         \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \]

   10. Simplified 29.3%

       \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4} + -16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} - b\right)} \]

   11. Taylor expanded in a around 0 94.4%

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

       1. neg-mul-1 94.4%

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

       2. +-commutative 94.4%

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

       3. unsub-neg 94.4%

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

       4. mul-1-neg 94.4%

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

       5. unsub-neg 94.4%

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

       6. associate-/l* 94.4%

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

   13. Simplified 94.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{-4 \cdot \left(c \cdot a\right) + {b}^{2}} - b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

5. Taylor expanded in a around 0 94.1%

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

6. Taylor expanded in c around 0 94.1%

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

   1. *-commutative 94.1%

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

8. Simplified 94.1%

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

   1. associate-*r/ 94.1%

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

10. Applied egg-rr 94.1%

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

    1. mul-1-neg 94.1%

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

12. Simplified 94.1%

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

    1. unpow2 94.1%

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

14. Applied egg-rr 94.1%

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

15. Final simplification 94.1%

    \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{a \cdot {c}^{4}}{{b}^{7}} \cdot -5\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b} \]
16. Add Preprocessing

Alternative 3: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{-4 \cdot \left(c \cdot a\right) + {b}^{2}} - b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -50000000.0)
   (* (/ 0.5 a) (- (sqrt (+ (* -4.0 (* c a)) (pow b 2.0))) b))
   (*
    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 (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0) {
		tmp = (0.5 / a) * (sqrt(((-4.0 * (c * a)) + pow(b, 2.0))) - b);
	} 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;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-50000000.0d0)) then
        tmp = (0.5d0 / a) * (sqrt((((-4.0d0) * (c * a)) + (b ** 2.0d0))) - b)
    else
        tmp = c * ((c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0) {
		tmp = (0.5 / a) * (Math.sqrt(((-4.0 * (c * a)) + Math.pow(b, 2.0))) - b);
	} else {
		tmp = c * ((c * ((-2.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0:
		tmp = (0.5 / a) * (math.sqrt(((-4.0 * (c * a)) + math.pow(b, 2.0))) - b)
	else:
		tmp = c * ((c * ((-2.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -50000000.0)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(-4.0 * Float64(c * a)) + (b ^ 2.0))) - b));
	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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0)
		tmp = (0.5 / a) * (sqrt(((-4.0 * (c * a)) + (b ^ 2.0))) - b);
	else
		tmp = c * ((c * ((-2.0 * ((c * (a ^ 2.0)) / (b ^ 5.0))) - (a / (b ^ 3.0)))) + (-1.0 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -50000000.0], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $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 (/.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)) < -5e7

   1. Initial program 95.1%

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

      1. *-commutative 95.1%

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

      2. +-commutative 95.1%

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

      3. sqr-neg 95.1%

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

      4. unsub-neg 95.1%

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

      5. sqr-neg 95.1%

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

      6. fma-neg 95.1%

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

      7. distribute-lft-neg-in 95.1%

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

      8. *-commutative 95.1%

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

      9. *-commutative 95.1%

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

      10. distribute-rgt-neg-in 95.1%

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

      11. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

      1. *-commutative 95.1%

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

      2. metadata-eval 95.1%

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

      3. distribute-lft-neg-in 95.1%

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

      4. distribute-rgt-neg-in 95.1%

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

      5. *-commutative 95.1%

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

      6. fma-neg 95.1%

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

      7. flip-- 94.8%

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

      8. div-sub 94.2%

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

      9. pow2 94.2%

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

      10. pow2 94.2%

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

      11. pow-prod-up 94.2%

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

      12. metadata-eval 94.2%

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

      13. fma-define 94.4%

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

      14. associate-*l* 94.4%

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

      15. pow2 94.4%

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

      16. associate-*l* 94.4%

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

      17. fma-define 94.4%

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

      18. associate-*l* 94.4%

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

   6. Applied egg-rr 94.4%

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

      1. div-sub 93.4%

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

      2. sub-div 93.6%

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

      3. unpow-prod-down 93.6%

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

      4. metadata-eval 93.6%

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

      5. *-commutative 93.6%

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

      6. *-commutative 93.6%

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

      7. *-commutative 93.6%

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

      8. *-commutative 93.6%

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

   8. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]
   9. Step-by-step derivation

      1. div-sub 94.8%

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

      2. *-lft-identity 94.8%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right)}}{2 \cdot a} \]

      3. *-commutative 94.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot 1}}{2 \cdot a} \]

      4. associate-*r/ 94.8%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot \frac{1}{2 \cdot a}} \]

      5. *-commutative 94.8%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right)} \]

      6. associate-/r* 94.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \]

      7. metadata-eval 94.8%

         \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \]

   10. Simplified 94.8%

       \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4} + -16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} - b\right)} \]

   11. Taylor expanded in b around 0 95.2%

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

   if -5e7 < (/.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 29.3%

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

      1. *-commutative 29.3%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in c around 0 94.2%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{-4 \cdot \left(c \cdot a\right) + {b}^{2}} - b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{-4 \cdot \left(c \cdot a\right) + {b}^{2}} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -50000000.0)
   (* (/ 0.5 a) (- (sqrt (+ (* -4.0 (* c a)) (pow b 2.0))) b))
   (- (/ c (- b)) (* a (/ (pow c 2.0) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0) {
		tmp = (0.5 / a) * (sqrt(((-4.0 * (c * a)) + pow(b, 2.0))) - b);
	} else {
		tmp = (c / -b) - (a * (pow(c, 2.0) / 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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-50000000.0d0)) then
        tmp = (0.5d0 / a) * (sqrt((((-4.0d0) * (c * a)) + (b ** 2.0d0))) - b)
    else
        tmp = (c / -b) - (a * ((c ** 2.0d0) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0) {
		tmp = (0.5 / a) * (Math.sqrt(((-4.0 * (c * a)) + Math.pow(b, 2.0))) - b);
	} else {
		tmp = (c / -b) - (a * (Math.pow(c, 2.0) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0:
		tmp = (0.5 / a) * (math.sqrt(((-4.0 * (c * a)) + math.pow(b, 2.0))) - b)
	else:
		tmp = (c / -b) - (a * (math.pow(c, 2.0) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -50000000.0)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(-4.0 * Float64(c * a)) + (b ^ 2.0))) - b));
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0)
		tmp = (0.5 / a) * (sqrt(((-4.0 * (c * a)) + (b ^ 2.0))) - b);
	else
		tmp = (c / -b) - (a * ((c ^ 2.0) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -50000000.0], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $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)) < -5e7

   1. Initial program 95.1%

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

      1. *-commutative 95.1%

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

      2. +-commutative 95.1%

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

      3. sqr-neg 95.1%

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

      4. unsub-neg 95.1%

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

      5. sqr-neg 95.1%

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

      6. fma-neg 95.1%

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

      7. distribute-lft-neg-in 95.1%

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

      8. *-commutative 95.1%

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

      9. *-commutative 95.1%

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

      10. distribute-rgt-neg-in 95.1%

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

      11. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

      1. *-commutative 95.1%

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

      2. metadata-eval 95.1%

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

      3. distribute-lft-neg-in 95.1%

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

      4. distribute-rgt-neg-in 95.1%

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

      5. *-commutative 95.1%

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

      6. fma-neg 95.1%

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

      7. flip-- 94.8%

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

      8. div-sub 94.2%

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

      9. pow2 94.2%

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

      10. pow2 94.2%

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

      11. pow-prod-up 94.2%

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

      12. metadata-eval 94.2%

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

      13. fma-define 94.4%

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

      14. associate-*l* 94.4%

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

      15. pow2 94.4%

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

      16. associate-*l* 94.4%

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

      17. fma-define 94.4%

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

      18. associate-*l* 94.4%

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

   6. Applied egg-rr 94.4%

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

      1. div-sub 93.4%

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

      2. sub-div 93.6%

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

      3. unpow-prod-down 93.6%

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

      4. metadata-eval 93.6%

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

      5. *-commutative 93.6%

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

      6. *-commutative 93.6%

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

      7. *-commutative 93.6%

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

      8. *-commutative 93.6%

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

   8. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]
   9. Step-by-step derivation

      1. div-sub 94.8%

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

      2. *-lft-identity 94.8%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right)}}{2 \cdot a} \]

      3. *-commutative 94.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot 1}}{2 \cdot a} \]

      4. associate-*r/ 94.8%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot \frac{1}{2 \cdot a}} \]

      5. *-commutative 94.8%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right)} \]

      6. associate-/r* 94.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \]

      7. metadata-eval 94.8%

         \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{\frac{{b}^{4} - 16 \cdot {\left(c \cdot a\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}} - b\right) \]

   10. Simplified 94.8%

       \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4} + -16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} - b\right)} \]

   11. Taylor expanded in b around 0 95.2%

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

   if -5e7 < (/.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 29.3%

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

      1. *-commutative 29.3%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in a around 0 92.0%

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

      1. mul-1-neg 92.0%

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

      2. unsub-neg 92.0%

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

      3. mul-1-neg 92.0%

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

      4. distribute-neg-frac2 92.0%

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

      5. associate-/l* 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{-4 \cdot \left(c \cdot a\right) + {b}^{2}} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -50000000.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ c (- b)) (* a (/ (pow c 2.0) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000000.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c / -b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -50000000.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -50000000.0], 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 / (-b)), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $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)) < -5e7

   1. Initial program 95.1%

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

      1. *-commutative 95.1%

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

      2. +-commutative 95.1%

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

      3. sqr-neg 95.1%

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

      4. unsub-neg 95.1%

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

      5. sqr-neg 95.1%

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

      6. fma-neg 95.1%

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

      7. distribute-lft-neg-in 95.1%

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

      8. *-commutative 95.1%

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

      9. *-commutative 95.1%

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

      10. distribute-rgt-neg-in 95.1%

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

      11. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

   if -5e7 < (/.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 29.3%

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

      1. *-commutative 29.3%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in a around 0 92.0%

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

      1. mul-1-neg 92.0%

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

      2. unsub-neg 92.0%

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

      3. mul-1-neg 92.0%

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

      4. distribute-neg-frac2 92.0%

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

      5. associate-/l* 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -50000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -50000000.0)
     t_0
     (- (/ c (- b)) (* a (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -50000000.0) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * (pow(c, 2.0) / 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) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-50000000.0d0)) then
        tmp = t_0
    else
        tmp = (c / -b) - (a * ((c ** 2.0d0) / (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) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -50000000.0) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * (Math.pow(c, 2.0) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -50000000.0:
		tmp = t_0
	else:
		tmp = (c / -b) - (a * (math.pow(c, 2.0) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -50000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -50000000.0)
		tmp = t_0;
	else
		tmp = (c / -b) - (a * ((c ^ 2.0) / (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[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000.0], t$95$0, N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $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)) < -5e7

   1. Initial program 95.1%

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

   if -5e7 < (/.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 29.3%

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

      1. *-commutative 29.3%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in a around 0 92.0%

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

      1. mul-1-neg 92.0%

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

      2. unsub-neg 92.0%

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

      3. mul-1-neg 92.0%

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

      4. distribute-neg-frac2 92.0%

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

      5. associate-/l* 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -50000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -50000000.0) t_0 (/ (- (- c) (* a (pow (/ c (- b)) 2.0))) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -50000000.0) {
		tmp = t_0;
	} else {
		tmp = (-c - (a * pow((c / -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) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-50000000.0d0)) then
        tmp = t_0
    else
        tmp = (-c - (a * ((c / -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) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -50000000.0) {
		tmp = t_0;
	} else {
		tmp = (-c - (a * Math.pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -50000000.0:
		tmp = t_0
	else:
		tmp = (-c - (a * math.pow((c / -b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -50000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -50000000.0)
		tmp = t_0;
	else
		tmp = (-c - (a * ((c / -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[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000.0], t$95$0, N[(N[((-c) - N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $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)) < -5e7

   1. Initial program 95.1%

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

   if -5e7 < (/.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 29.3%

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

      1. *-commutative 29.3%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in a around 0 92.0%

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

      1. mul-1-neg 92.0%

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

      2. unsub-neg 92.0%

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

      3. mul-1-neg 92.0%

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

      4. distribute-neg-frac2 92.0%

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

      5. associate-/l* 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in b around inf 91.9%

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

      1. neg-mul-1 91.9%

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

      2. mul-1-neg 91.9%

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

      3. unsub-neg 91.9%

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

      4. associate-/l* 91.9%

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

      5. unpow2 91.9%

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

      6. unpow2 91.9%

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

      7. times-frac 91.9%

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

      8. sqr-neg 91.9%

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

      9. distribute-frac-neg2 91.9%

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

      10. distribute-frac-neg2 91.9%

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

      11. unpow2 91.9%

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

   10. Simplified 91.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-c - (a * pow((c / -b), 2.0))) / 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 - (a * ((c / -b) ** 2.0d0))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

5. Taylor expanded in a around 0 90.1%

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

   1. mul-1-neg 90.1%

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

   2. unsub-neg 90.1%

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

   3. mul-1-neg 90.1%

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

   4. distribute-neg-frac2 90.1%

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

   5. associate-/l* 90.1%

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

7. Simplified 90.1%

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

8. Taylor expanded in b around inf 90.1%

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

   1. neg-mul-1 90.1%

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

   2. mul-1-neg 90.1%

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

   3. unsub-neg 90.1%

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

   4. associate-/l* 90.1%

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

   5. unpow2 90.1%

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

   6. unpow2 90.1%

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

   7. times-frac 90.1%

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

   8. sqr-neg 90.1%

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

   9. distribute-frac-neg2 90.1%

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

   10. distribute-frac-neg2 90.1%

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

   11. unpow2 90.1%

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

10. Simplified 90.1%

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

Alternative 9: 81.1% 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 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

5. Taylor expanded in b around inf 81.4%

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

   1. associate-*r/ 81.4%

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

   2. mul-1-neg 81.4%

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

7. Simplified 81.4%

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

8. Final simplification 81.4%

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

Alternative 10: 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 31.4%

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

   1. *-commutative 31.4%

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

   2. +-commutative 31.4%

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

   3. sqr-neg 31.4%

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

   4. unsub-neg 31.4%

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

   5. sqr-neg 31.4%

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

   6. fma-neg 31.4%

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

   7. distribute-lft-neg-in 31.4%

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

   8. *-commutative 31.4%

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

   9. *-commutative 31.4%

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

   10. distribute-rgt-neg-in 31.4%

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

   11. metadata-eval 31.4%

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

3. Simplified 31.4%

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

   1. *-commutative 31.4%

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

   2. metadata-eval 31.4%

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

   3. distribute-lft-neg-in 31.4%

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

   4. distribute-rgt-neg-in 31.4%

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

   5. *-commutative 31.4%

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

   6. fma-neg 31.4%

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

   7. flip-- 31.3%

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

   8. div-sub 31.3%

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

   9. pow2 31.3%

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

   10. pow2 31.2%

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

   11. pow-prod-up 31.2%

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

   12. metadata-eval 31.2%

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

   13. fma-define 31.4%

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

   14. associate-*l* 31.4%

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

   15. pow2 31.4%

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

   16. associate-*l* 31.4%

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

   17. fma-define 31.4%

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

   18. associate-*l* 31.4%

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

6. Applied egg-rr 31.4%

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

   1. *-un-lft-identity 31.4%

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

   2. add-sqr-sqrt 31.4%

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

   3. prod-diff 32.5%

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

8. Applied egg-rr 32.4%

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

9. Taylor expanded in c 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. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))