The quadratic formula (r2)

Percentage Accurate: 52.4% → 80.5%

Time: 12.2s

Alternatives: 8

Speedup: 19.1×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -150000000:\\ \;\;\;\;\frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{b}\right)}^{3} \cdot a\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -150000000.0)
   (- (/ (- c) b) (* (pow (/ (pow (cbrt c) 2.0) b) 3.0) a))
   (if (<= b 2e-90)
     (* -0.5 (/ 1.0 (* a (/ 1.0 (+ b (hypot b (sqrt (* c (* a -4.0)))))))))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -150000000.0) {
		tmp = (-c / b) - (pow((pow(cbrt(c), 2.0) / b), 3.0) * a);
	} else if (b <= 2e-90) {
		tmp = -0.5 * (1.0 / (a * (1.0 / (b + hypot(b, sqrt((c * (a * -4.0))))))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -150000000.0) {
		tmp = (-c / b) - (Math.pow((Math.pow(Math.cbrt(c), 2.0) / b), 3.0) * a);
	} else if (b <= 2e-90) {
		tmp = -0.5 * (1.0 / (a * (1.0 / (b + Math.hypot(b, Math.sqrt((c * (a * -4.0))))))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -150000000.0)
		tmp = Float64(Float64(Float64(-c) / b) - Float64((Float64((cbrt(c) ^ 2.0) / b) ^ 3.0) * a));
	elseif (b <= 2e-90)
		tmp = Float64(-0.5 * Float64(1.0 / Float64(a * Float64(1.0 / Float64(b + hypot(b, sqrt(Float64(c * Float64(a * -4.0)))))))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -150000000.0], N[(N[((-c) / b), $MachinePrecision] - N[(N[Power[N[(N[Power[N[Power[c, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / b), $MachinePrecision], 3.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-90], N[(-0.5 * N[(1.0 / N[(a * N[(1.0 / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e8

   1. Initial program 13.9%

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

   2. Taylor expanded in b around -inf 79.3%

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

      1. +-commutative 79.3%

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

      2. mul-1-neg 79.3%

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

      3. unsub-neg 79.3%

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

      4. associate-*r/ 79.3%

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

      5. neg-mul-1 79.3%

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

      6. associate-/l* 81.8%

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

      7. unpow2 81.8%

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

   4. Simplified 81.8%

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

      1. add-cube-cbrt 81.8%

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

      2. pow2 81.8%

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

      3. cbrt-div 81.8%

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

      4. cbrt-prod 81.8%

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

      5. pow2 81.8%

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

      6. cbrt-div 81.8%

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

      7. rem-cbrt-cube 81.8%

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

      8. cbrt-div 81.8%

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

      9. cbrt-prod 94.0%

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

      10. pow2 94.0%

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

      11. cbrt-div 94.0%

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

      12. rem-cbrt-cube 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. pow-plus 94.0%

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

      2. metadata-eval 94.0%

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

      3. associate-/r/ 94.0%

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

      4. cube-prod 94.1%

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

      5. rem-cube-cbrt 94.1%

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

   8. Simplified 94.1%

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

   if -1.5e8 < b < 1.99999999999999999e-90

   1. Initial program 67.7%

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

      1. /-rgt-identity 67.7%

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

      2. metadata-eval 67.7%

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

      3. associate-/l* 67.6%

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

      4. associate-/r/ 67.6%

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

      5. *-commutative 67.6%

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

      6. metadata-eval 67.6%

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

      7. metadata-eval 67.6%

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

      8. associate-*l/ 67.6%

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

      9. associate-/r/ 67.6%

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

      10. times-frac 67.7%

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

      11. *-commutative 67.7%

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

      12. times-frac 67.7%

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

      13. metadata-eval 67.7%

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

      14. associate-/r/ 67.7%

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

      15. *-commutative 67.7%

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

      16. div-sub 67.7%

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

   3. Simplified 67.7%

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

      1. clear-num 67.6%

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

      2. inv-pow 67.6%

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

   5. Applied egg-rr 67.6%

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

      1. unpow-1 67.6%

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

      2. fma-udef 67.6%

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

      3. *-commutative 67.6%

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

      4. associate-*l* 67.6%

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

      5. +-commutative 67.6%

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

      6. fma-def 67.6%

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

      7. *-commutative 67.6%

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

   7. Simplified 67.6%

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

      1. div-inv 67.6%

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

      2. fma-udef 67.6%

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

      3. add-sqr-sqrt 67.6%

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

      4. hypot-def 69.2%

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

   9. Applied egg-rr 69.2%

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

   if 1.99999999999999999e-90 < b

   1. Initial program 59.1%

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

   2. Taylor expanded in b around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. mul-1-neg 88.9%

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

   4. Simplified 88.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -150000000:\\ \;\;\;\;\frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{b}\right)}^{3} \cdot a\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -64000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -64000.0)
   (/ (- c) b)
   (if (<= b 2.2e-90)
     (* -0.5 (/ 1.0 (* a (/ 1.0 (+ b (hypot b (sqrt (* c (* a -4.0)))))))))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -64000.0) {
		tmp = -c / b;
	} else if (b <= 2.2e-90) {
		tmp = -0.5 * (1.0 / (a * (1.0 / (b + hypot(b, sqrt((c * (a * -4.0))))))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -64000.0) {
		tmp = -c / b;
	} else if (b <= 2.2e-90) {
		tmp = -0.5 * (1.0 / (a * (1.0 / (b + Math.hypot(b, Math.sqrt((c * (a * -4.0))))))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -64000.0:
		tmp = -c / b
	elif b <= 2.2e-90:
		tmp = -0.5 * (1.0 / (a * (1.0 / (b + math.hypot(b, math.sqrt((c * (a * -4.0))))))))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -64000.0)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2.2e-90)
		tmp = Float64(-0.5 * Float64(1.0 / Float64(a * Float64(1.0 / Float64(b + hypot(b, sqrt(Float64(c * Float64(a * -4.0)))))))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -64000.0)
		tmp = -c / b;
	elseif (b <= 2.2e-90)
		tmp = -0.5 * (1.0 / (a * (1.0 / (b + hypot(b, sqrt((c * (a * -4.0))))))));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -64000.0], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2.2e-90], N[(-0.5 * N[(1.0 / N[(a * N[(1.0 / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -64000

   1. Initial program 13.8%

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

   2. Taylor expanded in b around -inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. neg-mul-1 93.0%

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

   4. Simplified 93.0%

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

   if -64000 < b < 2.19999999999999986e-90

   1. Initial program 68.4%

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

      1. /-rgt-identity 68.4%

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

      2. metadata-eval 68.4%

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

      3. associate-/l* 68.3%

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

      4. associate-/r/ 68.3%

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

      5. *-commutative 68.3%

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

      6. metadata-eval 68.3%

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

      7. metadata-eval 68.3%

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

      8. associate-*l/ 68.3%

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

      9. associate-/r/ 68.3%

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

      10. times-frac 68.4%

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

      11. *-commutative 68.4%

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

      12. times-frac 68.4%

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

      13. metadata-eval 68.4%

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

      14. associate-/r/ 68.4%

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

      15. *-commutative 68.4%

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

      16. div-sub 68.5%

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

   3. Simplified 68.4%

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

      1. clear-num 68.3%

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

      2. inv-pow 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. unpow-1 68.3%

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

      2. fma-udef 68.3%

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

      3. *-commutative 68.3%

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

      4. associate-*l* 68.3%

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

      5. +-commutative 68.3%

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

      6. fma-def 68.3%

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

      7. *-commutative 68.3%

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

   7. Simplified 68.3%

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

      1. div-inv 68.3%

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

      2. fma-udef 68.3%

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

      3. add-sqr-sqrt 68.3%

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

      4. hypot-def 70.0%

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

   9. Applied egg-rr 70.0%

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

   if 2.19999999999999986e-90 < b

   1. Initial program 59.1%

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

   2. Taylor expanded in b around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. mul-1-neg 88.9%

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

   4. Simplified 88.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -64000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -440000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -440000.0)
   (/ (- c) b)
   (if (<= b 4e-40)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* 2.0 a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -440000.0) {
		tmp = -c / b;
	} else if (b <= 4e-40) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-440000.0d0)) then
        tmp = -c / b
    else if (b <= 4d-40) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (2.0d0 * a)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -440000.0) {
		tmp = -c / b;
	} else if (b <= 4e-40) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -440000.0:
		tmp = -c / b
	elif b <= 4e-40:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -440000.0)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 4e-40)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -440000.0)
		tmp = -c / b;
	elseif (b <= 4e-40)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -440000.0], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 4e-40], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4e5

   1. Initial program 13.8%

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

   2. Taylor expanded in b around -inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. neg-mul-1 93.0%

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

   4. Simplified 93.0%

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

   if -4.4e5 < b < 3.9999999999999997e-40

   1. Initial program 71.6%

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

   if 3.9999999999999997e-40 < b

   1. Initial program 53.6%

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

   2. Taylor expanded in b around inf 88.7%

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

      1. associate-*r/ 88.7%

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

      2. mul-1-neg 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -440000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -85000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-96}:\\ \;\;\;\;-\frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -85000.0)
   (/ (- c) b)
   (if (<= b 5.2e-96)
     (- (/ (+ b (sqrt (* -4.0 (* c a)))) (* 2.0 a)))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -85000.0) {
		tmp = -c / b;
	} else if (b <= 5.2e-96) {
		tmp = -((b + sqrt((-4.0 * (c * a)))) / (2.0 * a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-85000.0d0)) then
        tmp = -c / b
    else if (b <= 5.2d-96) then
        tmp = -((b + sqrt(((-4.0d0) * (c * a)))) / (2.0d0 * a))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -85000.0) {
		tmp = -c / b;
	} else if (b <= 5.2e-96) {
		tmp = -((b + Math.sqrt((-4.0 * (c * a)))) / (2.0 * a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -85000.0:
		tmp = -c / b
	elif b <= 5.2e-96:
		tmp = -((b + math.sqrt((-4.0 * (c * a)))) / (2.0 * a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -85000.0)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 5.2e-96)
		tmp = Float64(-Float64(Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))) / Float64(2.0 * a)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -85000.0)
		tmp = -c / b;
	elseif (b <= 5.2e-96)
		tmp = -((b + sqrt((-4.0 * (c * a)))) / (2.0 * a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -85000.0], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 5.2e-96], (-N[(N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -85000

   1. Initial program 13.8%

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

   2. Taylor expanded in b around -inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. neg-mul-1 93.0%

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

   4. Simplified 93.0%

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

   if -85000 < b < 5.2000000000000003e-96

   1. Initial program 68.4%

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

   2. Taylor expanded in b around 0 65.4%

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

      1. *-commutative 65.4%

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

   4. Simplified 65.4%

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

   if 5.2000000000000003e-96 < b

   1. Initial program 59.1%

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

   2. Taylor expanded in b around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. mul-1-neg 88.9%

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

   4. Simplified 88.9%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -85000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-96}:\\ \;\;\;\;-\frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5: 66.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{1}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (* -0.5 (/ 1.0 (+ (* 0.5 (/ b c)) (* -0.5 (/ a b)))))
   (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.5 * (1.0 / ((0.5 * (b / c)) + (-0.5 * (a / b))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (-0.5d0) * (1.0d0 / ((0.5d0 * (b / c)) + ((-0.5d0) * (a / b))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.5 * (1.0 / ((0.5 * (b / c)) + (-0.5 * (a / b))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -0.5 * (1.0 / ((0.5 * (b / c)) + (-0.5 * (a / b))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(-0.5 * Float64(1.0 / Float64(Float64(0.5 * Float64(b / c)) + Float64(-0.5 * Float64(a / b)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -0.5 * (1.0 / ((0.5 * (b / c)) + (-0.5 * (a / b))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(-0.5 * N[(1.0 / N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 36.7%

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

      1. /-rgt-identity 36.7%

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

      2. metadata-eval 36.7%

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

      3. associate-/l* 36.6%

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

      4. associate-/r/ 36.7%

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

      5. *-commutative 36.7%

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

      6. metadata-eval 36.7%

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

      7. metadata-eval 36.7%

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

      8. associate-*l/ 36.7%

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

      9. associate-/r/ 36.7%

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

      10. times-frac 36.7%

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

      11. *-commutative 36.7%

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

      12. times-frac 36.7%

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

      13. metadata-eval 36.7%

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

      14. associate-/r/ 36.7%

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

      15. *-commutative 36.7%

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

      16. div-sub 36.4%

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

   3. Simplified 36.7%

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

      1. clear-num 36.7%

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

      2. inv-pow 36.7%

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

   5. Applied egg-rr 36.7%

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

      1. unpow-1 36.7%

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

      2. fma-udef 36.7%

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

      3. *-commutative 36.7%

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

      4. associate-*l* 36.7%

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

      5. +-commutative 36.7%

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

      6. fma-def 36.7%

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

      7. *-commutative 36.7%

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

   7. Simplified 36.7%

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

   8. Taylor expanded in b around -inf 64.7%

      \[\leadsto -0.5 \cdot \frac{1}{\color{blue}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 61.7%

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

   2. Taylor expanded in b around inf 71.1%

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

      1. mul-1-neg 71.1%

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

      2. unsub-neg 71.1%

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

   4. Simplified 71.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{1}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 67.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = -c / b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 36.7%

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

   2. Taylor expanded in b around -inf 64.6%

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

      1. associate-*r/ 64.6%

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

      2. neg-mul-1 64.6%

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

   4. Simplified 64.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 61.7%

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

   2. Taylor expanded in b around inf 71.1%

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

      1. mul-1-neg 71.1%

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

      2. unsub-neg 71.1%

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

   4. Simplified 71.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7: 66.8% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.4e-305) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-305) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.4d-305)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-305) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.4e-305:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.4e-305)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.4e-305)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.4e-305], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.3999999999999998e-305

   1. Initial program 36.3%

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

   2. Taylor expanded in b around -inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. neg-mul-1 65.1%

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

   4. Simplified 65.1%

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

   if -5.3999999999999998e-305 < b

   1. Initial program 62.0%

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

   2. Taylor expanded in b around inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. mul-1-neg 70.5%

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

   4. Simplified 70.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8: 34.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

2. Taylor expanded in b around inf 33.1%

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

   1. associate-*r/ 33.1%

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

   2. mul-1-neg 33.1%

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

4. Simplified 33.1%

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

5. Final simplification 33.1%

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

Developer target: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023178 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))