The quadratic formula (r2)

Percentage Accurate: 51.6% → 89.4%

Time: 14.5s

Alternatives: 11

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: 51.6% 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: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -4\right)\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -2.75 \cdot 10^{-304}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{\mathsf{fma}\left(b, b, t_0\right)}}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+114}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{t_0 + b \cdot b}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c -4.0))))
   (if (<= b -3.5e+43)
     (/ (- c) b)
     (if (<= b -2.75e-304)
       (* -0.5 (/ (* c 4.0) (- b (sqrt (fma b b t_0)))))
       (if (<= b 4.2e+114)
         (* -0.5 (+ (/ b a) (/ (sqrt (+ t_0 (* b b))) a)))
         (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * -4.0);
	double tmp;
	if (b <= -3.5e+43) {
		tmp = -c / b;
	} else if (b <= -2.75e-304) {
		tmp = -0.5 * ((c * 4.0) / (b - sqrt(fma(b, b, t_0))));
	} else if (b <= 4.2e+114) {
		tmp = -0.5 * ((b / a) + (sqrt((t_0 + (b * b))) / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -4.0))
	tmp = 0.0
	if (b <= -3.5e+43)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -2.75e-304)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b - sqrt(fma(b, b, t_0)))));
	elseif (b <= 4.2e+114)
		tmp = Float64(-0.5 * Float64(Float64(b / a) + Float64(sqrt(Float64(t_0 + Float64(b * b))) / a)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.5e+43], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -2.75e-304], N[(-0.5 * N[(N[(c * 4.0), $MachinePrecision] / N[(b - N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.2e+114], N[(-0.5 * N[(N[(b / a), $MachinePrecision] + N[(N[Sqrt[N[(t$95$0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.5000000000000001e43

   1. Initial program 4.2%

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

   2. Taylor expanded in b around -inf 95.3%

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

      1. associate-*r/ 95.3%

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

      2. neg-mul-1 95.3%

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

   4. Simplified 95.3%

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

   if -3.5000000000000001e43 < b < -2.75000000000000017e-304

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

      1. fma-udef 57.7%

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

      2. associate-*r* 57.7%

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

      3. metadata-eval 57.7%

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

      4. distribute-rgt-neg-in 57.7%

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

      5. *-commutative 57.7%

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

      6. +-commutative 57.7%

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

      7. sub-neg 57.7%

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

      8. *-commutative 57.7%

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

      9. associate-*l* 57.7%

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

   5. Applied egg-rr 57.7%

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

      1. div-inv 57.6%

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

   7. Applied egg-rr 57.6%

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

      1. *-commutative 57.6%

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

      2. flip-+ 57.2%

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

      3. add-sqr-sqrt 57.1%

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

      4. frac-times 46.1%

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

   9. Applied egg-rr 63.6%

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

       1. times-frac 83.9%

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

       2. *-inverses 83.9%

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

       3. *-lft-identity 83.9%

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

   11. Simplified 83.9%

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

   if -2.75000000000000017e-304 < b < 4.2000000000000001e114

   1. Initial program 90.4%

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

   3. Simplified 90.4%

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

      1. fma-udef 90.4%

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

      2. associate-*r* 90.4%

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

      3. metadata-eval 90.4%

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

      4. distribute-rgt-neg-in 90.4%

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

      5. *-commutative 90.4%

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

      6. +-commutative 90.4%

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

      7. sub-neg 90.4%

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

      8. *-commutative 90.4%

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

      9. associate-*l* 90.4%

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

   5. Applied egg-rr 90.4%

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

      1. div-inv 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. *-commutative 90.2%

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

      2. distribute-rgt-in 90.3%

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

      3. div-inv 90.3%

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

      4. fma-neg 90.3%

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

      5. distribute-rgt-neg-in 90.3%

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

      6. distribute-rgt-neg-in 90.3%

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

      7. metadata-eval 90.3%

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

   9. Applied egg-rr 90.3%

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

       1. associate-*r/ 90.4%

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

       2. *-rgt-identity 90.4%

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

   11. Simplified 90.4%

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

       1. fma-udef 90.4%

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

   13. Applied egg-rr 90.4%

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

   if 4.2000000000000001e114 < b

   1. Initial program 61.9%

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

   2. Taylor expanded in b around inf 94.4%

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

      1. associate-*r/ 94.4%

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

      2. mul-1-neg 94.4%

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

   4. Simplified 94.4%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -2.75 \cdot 10^{-304}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+114}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot 4\right) \cdot a\\ \mathbf{if}\;b \leq -1 \cdot 10^{-26}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-100}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{t_0}{b - \sqrt{b \cdot b - t_0}}}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+114}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c 4.0) a)))
   (if (<= b -1e-26)
     (/ (- c) b)
     (if (<= b -9.6e-100)
       (* -0.5 (/ (/ t_0 (- b (sqrt (- (* b b) t_0)))) a))
       (if (<= b 8.5e+114)
         (* -0.5 (+ (/ b a) (/ (sqrt (+ (* a (* c -4.0)) (* b b))) a)))
         (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * 4.0) * a;
	double tmp;
	if (b <= -1e-26) {
		tmp = -c / b;
	} else if (b <= -9.6e-100) {
		tmp = -0.5 * ((t_0 / (b - sqrt(((b * b) - t_0)))) / a);
	} else if (b <= 8.5e+114) {
		tmp = -0.5 * ((b / a) + (sqrt(((a * (c * -4.0)) + (b * b))) / 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) :: t_0
    real(8) :: tmp
    t_0 = (c * 4.0d0) * a
    if (b <= (-1d-26)) then
        tmp = -c / b
    else if (b <= (-9.6d-100)) then
        tmp = (-0.5d0) * ((t_0 / (b - sqrt(((b * b) - t_0)))) / a)
    else if (b <= 8.5d+114) then
        tmp = (-0.5d0) * ((b / a) + (sqrt(((a * (c * (-4.0d0))) + (b * b))) / a))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * 4.0) * a;
	double tmp;
	if (b <= -1e-26) {
		tmp = -c / b;
	} else if (b <= -9.6e-100) {
		tmp = -0.5 * ((t_0 / (b - Math.sqrt(((b * b) - t_0)))) / a);
	} else if (b <= 8.5e+114) {
		tmp = -0.5 * ((b / a) + (Math.sqrt(((a * (c * -4.0)) + (b * b))) / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c * 4.0) * a
	tmp = 0
	if b <= -1e-26:
		tmp = -c / b
	elif b <= -9.6e-100:
		tmp = -0.5 * ((t_0 / (b - math.sqrt(((b * b) - t_0)))) / a)
	elif b <= 8.5e+114:
		tmp = -0.5 * ((b / a) + (math.sqrt(((a * (c * -4.0)) + (b * b))) / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * 4.0) * a)
	tmp = 0.0
	if (b <= -1e-26)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -9.6e-100)
		tmp = Float64(-0.5 * Float64(Float64(t_0 / Float64(b - sqrt(Float64(Float64(b * b) - t_0)))) / a));
	elseif (b <= 8.5e+114)
		tmp = Float64(-0.5 * Float64(Float64(b / a) + Float64(sqrt(Float64(Float64(a * Float64(c * -4.0)) + Float64(b * b))) / a)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c * 4.0) * a;
	tmp = 0.0;
	if (b <= -1e-26)
		tmp = -c / b;
	elseif (b <= -9.6e-100)
		tmp = -0.5 * ((t_0 / (b - sqrt(((b * b) - t_0)))) / a);
	elseif (b <= 8.5e+114)
		tmp = -0.5 * ((b / a) + (sqrt(((a * (c * -4.0)) + (b * b))) / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, -1e-26], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -9.6e-100], N[(-0.5 * N[(N[(t$95$0 / N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e+114], N[(-0.5 * N[(N[(b / a), $MachinePrecision] + N[(N[Sqrt[N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1e-26

   1. Initial program 7.1%

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

   2. Taylor expanded in b around -inf 92.1%

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

      1. associate-*r/ 92.1%

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

      2. neg-mul-1 92.1%

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

   4. Simplified 92.1%

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

   if -1e-26 < b < -9.600000000000001e-100

   1. Initial program 45.9%

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

   3. Simplified 45.9%

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

      1. fma-udef 45.9%

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

      2. associate-*r* 45.9%

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

      3. metadata-eval 45.9%

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

      4. distribute-rgt-neg-in 45.9%

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

      5. *-commutative 45.9%

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

      6. +-commutative 45.9%

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

      7. sub-neg 45.9%

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

      8. *-commutative 45.9%

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

      9. associate-*l* 45.9%

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

   5. Applied egg-rr 45.9%

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

      1. flip-+ 45.1%

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

      2. add-sqr-sqrt 45.1%

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

   7. Applied egg-rr 45.1%

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

      1. add-log-exp 2.9%

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

      2. associate--r- 2.9%

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

      3. exp-sum 2.9%

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

      4. +-inverses 2.9%

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

      5. 1-exp 2.9%

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

      6. *-un-lft-identity 2.9%

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

      7. add-log-exp 75.0%

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

      8. *-commutative 75.0%

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

   9. Applied egg-rr 75.0%

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

   if -9.600000000000001e-100 < b < 8.5000000000000001e114

   1. Initial program 87.1%

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

   3. Simplified 87.1%

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

      1. fma-udef 87.1%

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

      2. associate-*r* 87.1%

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

      3. metadata-eval 87.1%

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

      4. distribute-rgt-neg-in 87.1%

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

      5. *-commutative 87.1%

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

      6. +-commutative 87.1%

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

      7. sub-neg 87.1%

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

      8. *-commutative 87.1%

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

      9. associate-*l* 87.1%

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

   5. Applied egg-rr 87.1%

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

      1. div-inv 86.9%

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

   7. Applied egg-rr 86.9%

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

      1. *-commutative 86.9%

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

      2. distribute-rgt-in 87.0%

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

      3. div-inv 87.0%

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

      4. fma-neg 87.0%

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

      5. distribute-rgt-neg-in 87.0%

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

      6. distribute-rgt-neg-in 87.0%

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

      7. metadata-eval 87.0%

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

   9. Applied egg-rr 87.0%

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

       1. associate-*r/ 87.1%

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

       2. *-rgt-identity 87.1%

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

   11. Simplified 87.1%

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

       1. fma-udef 87.1%

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

   13. Applied egg-rr 87.1%

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

   if 8.5000000000000001e114 < b

   1. Initial program 61.9%

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

   2. Taylor expanded in b around inf 94.4%

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

      1. associate-*r/ 94.4%

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

      2. mul-1-neg 94.4%

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

   4. Simplified 94.4%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-26}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-100}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{\left(c \cdot 4\right) \cdot a}{b - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+114}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+113}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e-63)
   (/ (- c) b)
   (if (<= b 9.5e+113)
     (* -0.5 (+ (/ b a) (/ (sqrt (+ (* a (* c -4.0)) (* b b))) a)))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-63) {
		tmp = -c / b;
	} else if (b <= 9.5e+113) {
		tmp = -0.5 * ((b / a) + (sqrt(((a * (c * -4.0)) + (b * b))) / 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 <= (-7.8d-63)) then
        tmp = -c / b
    else if (b <= 9.5d+113) then
        tmp = (-0.5d0) * ((b / a) + (sqrt(((a * (c * (-4.0d0))) + (b * b))) / 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 <= -7.8e-63) {
		tmp = -c / b;
	} else if (b <= 9.5e+113) {
		tmp = -0.5 * ((b / a) + (Math.sqrt(((a * (c * -4.0)) + (b * b))) / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.8e-63:
		tmp = -c / b
	elif b <= 9.5e+113:
		tmp = -0.5 * ((b / a) + (math.sqrt(((a * (c * -4.0)) + (b * b))) / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e-63)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 9.5e+113)
		tmp = Float64(-0.5 * Float64(Float64(b / a) + Float64(sqrt(Float64(Float64(a * Float64(c * -4.0)) + Float64(b * b))) / a)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.8e-63)
		tmp = -c / b;
	elseif (b <= 9.5e+113)
		tmp = -0.5 * ((b / a) + (sqrt(((a * (c * -4.0)) + (b * b))) / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e-63], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 9.5e+113], N[(-0.5 * N[(N[(b / a), $MachinePrecision] + N[(N[Sqrt[N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.80000000000000044e-63

   1. Initial program 11.5%

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

   2. Taylor expanded in b around -inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   4. Simplified 87.2%

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

   if -7.80000000000000044e-63 < b < 9.5000000000000001e113

   1. Initial program 84.7%

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

   3. Simplified 84.7%

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

      1. fma-udef 84.7%

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

      2. associate-*r* 84.7%

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

      3. metadata-eval 84.7%

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

      4. distribute-rgt-neg-in 84.7%

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

      5. *-commutative 84.7%

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

      6. +-commutative 84.7%

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

      7. sub-neg 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*l* 84.7%

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

   5. Applied egg-rr 84.7%

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

      1. div-inv 84.6%

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

   7. Applied egg-rr 84.6%

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

      1. *-commutative 84.6%

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

      2. distribute-rgt-in 84.6%

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

      3. div-inv 84.6%

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

      4. fma-neg 84.6%

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

      5. distribute-rgt-neg-in 84.6%

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

      6. distribute-rgt-neg-in 84.6%

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

      7. metadata-eval 84.6%

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

   9. Applied egg-rr 84.6%

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

       1. associate-*r/ 84.8%

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

       2. *-rgt-identity 84.8%

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

   11. Simplified 84.8%

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

       1. fma-udef 84.8%

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

   13. Applied egg-rr 84.8%

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

   if 9.5000000000000001e113 < b

   1. Initial program 61.9%

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

   2. Taylor expanded in b around inf 94.4%

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

      1. associate-*r/ 94.4%

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

      2. mul-1-neg 94.4%

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

   4. Simplified 94.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+113}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+114}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.2e-62)
   (/ (- c) b)
   (if (<= b 2e+114)
     (* -0.5 (/ (+ b (sqrt (- (* b b) (* (* c 4.0) a)))) a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.2e-62) {
		tmp = -c / b;
	} else if (b <= 2e+114) {
		tmp = -0.5 * ((b + sqrt(((b * b) - ((c * 4.0) * a)))) / 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 <= (-9.2d-62)) then
        tmp = -c / b
    else if (b <= 2d+114) then
        tmp = (-0.5d0) * ((b + sqrt(((b * b) - ((c * 4.0d0) * a)))) / 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 <= -9.2e-62) {
		tmp = -c / b;
	} else if (b <= 2e+114) {
		tmp = -0.5 * ((b + Math.sqrt(((b * b) - ((c * 4.0) * a)))) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.2e-62:
		tmp = -c / b
	elif b <= 2e+114:
		tmp = -0.5 * ((b + math.sqrt(((b * b) - ((c * 4.0) * a)))) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.2e-62)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2e+114)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a)))) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.2e-62)
		tmp = -c / b;
	elseif (b <= 2e+114)
		tmp = -0.5 * ((b + sqrt(((b * b) - ((c * 4.0) * a)))) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.2e-62], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2e+114], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.20000000000000002e-62

   1. Initial program 11.5%

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

   2. Taylor expanded in b around -inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   4. Simplified 87.2%

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

   if -9.20000000000000002e-62 < b < 2e114

   1. Initial program 84.7%

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

   3. Simplified 84.7%

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

      1. fma-udef 84.7%

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

      2. associate-*r* 84.7%

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

      3. metadata-eval 84.7%

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

      4. distribute-rgt-neg-in 84.7%

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

      5. *-commutative 84.7%

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

      6. +-commutative 84.7%

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

      7. sub-neg 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*l* 84.7%

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

   5. Applied egg-rr 84.7%

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

   if 2e114 < b

   1. Initial program 61.9%

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

   2. Taylor expanded in b around inf 94.4%

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

      1. associate-*r/ 94.4%

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

      2. mul-1-neg 94.4%

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

   4. Simplified 94.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+114}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-130}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e-70)
   (/ (- c) b)
   (if (<= b 2.25e-130)
     (* -0.5 (/ (+ b (sqrt (* c (* a -4.0)))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-70) {
		tmp = -c / b;
	} else if (b <= 2.25e-130) {
		tmp = -0.5 * ((b + sqrt((c * (a * -4.0)))) / a);
	} 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 <= (-4.4d-70)) then
        tmp = -c / b
    else if (b <= 2.25d-130) then
        tmp = (-0.5d0) * ((b + sqrt((c * (a * (-4.0d0))))) / a)
    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 <= -4.4e-70) {
		tmp = -c / b;
	} else if (b <= 2.25e-130) {
		tmp = -0.5 * ((b + Math.sqrt((c * (a * -4.0)))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e-70:
		tmp = -c / b
	elif b <= 2.25e-130:
		tmp = -0.5 * ((b + math.sqrt((c * (a * -4.0)))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e-70)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2.25e-130)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / a));
	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 <= -4.4e-70)
		tmp = -c / b;
	elseif (b <= 2.25e-130)
		tmp = -0.5 * ((b + sqrt((c * (a * -4.0)))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e-70], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2.25e-130], N[(-0.5 * N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.3999999999999998e-70

   1. Initial program 12.3%

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

   2. Taylor expanded in b around -inf 85.3%

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

      1. associate-*r/ 85.3%

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

      2. neg-mul-1 85.3%

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

   4. Simplified 85.3%

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

   if -4.3999999999999998e-70 < b < 2.25e-130

   1. Initial program 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in a around inf 76.5%

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

      1. *-commutative 76.5%

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

      2. associate-*r* 76.5%

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

   6. Simplified 76.5%

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

   if 2.25e-130 < b

   1. Initial program 79.6%

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

   2. Taylor expanded in b around inf 84.8%

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

      1. mul-1-neg 84.8%

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

      2. unsub-neg 84.8%

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

   4. Simplified 84.8%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-130}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 68.2% 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 31.9%

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

   2. Taylor expanded in b around -inf 63.1%

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

      1. associate-*r/ 63.1%

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

      2. neg-mul-1 63.1%

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

   4. Simplified 63.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 79.7%

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

   2. Taylor expanded in b around inf 70.9%

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

      1. mul-1-neg 70.9%

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

      2. unsub-neg 70.9%

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

   4. Simplified 70.9%

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

4. Final simplification 67.2%

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

Alternative 7: 43.7% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-292}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -7.5e-292) 0.0 (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-292) {
		tmp = 0.0;
	} 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 <= (-7.5d-292)) then
        tmp = 0.0d0
    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 <= -7.5e-292) {
		tmp = 0.0;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e-292:
		tmp = 0.0
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e-292)
		tmp = 0.0;
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e-292)
		tmp = 0.0;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e-292], 0.0, N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.5000000000000002e-292

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in a around 0 4.1%

      \[\leadsto -0.5 \cdot \frac{b + \sqrt{\color{blue}{{b}^{2}}}}{a} \]
   5. Step-by-step derivation

      1. unpow2 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in b around -inf 12.6%

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

   if -7.5000000000000002e-292 < b

   1. Initial program 79.4%

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

   2. Taylor expanded in b around inf 69.3%

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

      1. associate-*r/ 69.3%

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

      2. mul-1-neg 69.3%

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

   4. Simplified 69.3%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-292}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8: 68.0% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-292}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-292) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-292) {
		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 <= (-9d-292)) 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 <= -9e-292) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-292:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-292)
		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 <= -9e-292)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-292], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.99999999999999913e-292

   1. Initial program 31.0%

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

   2. Taylor expanded in b around -inf 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 -8.99999999999999913e-292 < b

   1. Initial program 79.4%

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

   2. Taylor expanded in b around inf 69.3%

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

      1. associate-*r/ 69.3%

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

      2. mul-1-neg 69.3%

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

   4. Simplified 69.3%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-292}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 9: 3.6% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 -1.0)

C

double code(double a, double b, double c) {
	return -1.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -1.0d0
end function

Java

public static double code(double a, double b, double c) {
	return -1.0;
}

Python

def code(a, b, c):
	return -1.0

Julia

function code(a, b, c)
	return -1.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = -1.0;
end

Wolfram

code[a_, b_, c_] := -1.0

Derivation

1. Initial program 57.1%

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

3. Simplified 57.1%

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

4. Taylor expanded in a around 0 32.3%

   \[\leadsto -0.5 \cdot \frac{b + \sqrt{\color{blue}{{b}^{2}}}}{a} \]
5. Step-by-step derivation

   1. unpow2 32.3%

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

6. Simplified 32.3%

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

7. Taylor expanded in b around -inf 7.3%

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

9. Simplified 3.3%

   \[\leadsto \color{blue}{-1} \]

10. Final simplification 3.3%

    \[\leadsto -1 \]

Alternative 10: 3.6% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 -0.5)

C

double code(double a, double b, double c) {
	return -0.5;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -0.5d0
end function

Java

public static double code(double a, double b, double c) {
	return -0.5;
}

Python

def code(a, b, c):
	return -0.5

Julia

function code(a, b, c)
	return -0.5
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5;
end

Wolfram

code[a_, b_, c_] := -0.5

Derivation

1. Initial program 57.1%

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

3. Simplified 57.1%

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

   1. fma-udef 57.1%

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

   2. associate-*r* 57.1%

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

   3. metadata-eval 57.1%

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

   4. distribute-rgt-neg-in 57.1%

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

   5. *-commutative 57.1%

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

   6. +-commutative 57.1%

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

   7. sub-neg 57.1%

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

   8. *-commutative 57.1%

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

   9. associate-*l* 57.1%

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

5. Applied egg-rr 57.1%

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

   1. div-inv 57.0%

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

7. Applied egg-rr 57.0%

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

8. Taylor expanded in b around inf 38.6%

   \[\leadsto -0.5 \cdot \left(\left(b + \color{blue}{b}\right) \cdot \frac{1}{a}\right) \]
9. Step-by-step derivation

   1. expm1-log1p-u 24.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\left(b + b\right) \cdot \frac{1}{a}\right)\right)\right)} \]

   2. expm1-udef 21.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \left(\left(b + b\right) \cdot \frac{1}{a}\right)\right)} - 1} \]

10. Applied egg-rr 0.0%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{a \cdot \frac{0}{0}}\right)} - 1} \]

12. Simplified 3.3%

    \[\leadsto \color{blue}{-0.5} \]

13. Final simplification 3.3%

    \[\leadsto -0.5 \]

Alternative 11: 10.7% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 57.1%

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

3. Simplified 57.1%

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

4. Taylor expanded in a around 0 32.3%

   \[\leadsto -0.5 \cdot \frac{b + \sqrt{\color{blue}{{b}^{2}}}}{a} \]
5. Step-by-step derivation

   1. unpow2 32.3%

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

6. Simplified 32.3%

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

7. Taylor expanded in b around -inf 7.3%

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

8. Final simplification 7.3%

   \[\leadsto 0 \]

Developer target: 69.3% 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 2023257 
(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)))