quadm (p42, negative)

Percentage Accurate: 51.1% → 87.4%

Time: 13.4s

Alternatives: 13

Speedup: 2.5×

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: 51.1% 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: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{t\_0}{a \cdot 2}}{b - \sqrt{\mathsf{fma}\left(b, b, t\_0\right)}}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -4.0 (* c a))))
   (if (<= b -3.2e-8)
     (/ (- c) b)
     (if (<= b -5e-197)
       (/ (/ t_0 (* a 2.0)) (- b (sqrt (fma b b t_0))))
       (if (<= b 1.5e+116)
         (fma b (/ -0.5 a) (/ (sqrt (fma b b (* c (* a -4.0)))) (* a -2.0)))
         (/ (fma a (/ c b) (- b)) a))))))

C

double code(double a, double b, double c) {
	double t_0 = -4.0 * (c * a);
	double tmp;
	if (b <= -3.2e-8) {
		tmp = -c / b;
	} else if (b <= -5e-197) {
		tmp = (t_0 / (a * 2.0)) / (b - sqrt(fma(b, b, t_0)));
	} else if (b <= 1.5e+116) {
		tmp = fma(b, (-0.5 / a), (sqrt(fma(b, b, (c * (a * -4.0)))) / (a * -2.0)));
	} else {
		tmp = fma(a, (c / b), -b) / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(-4.0 * Float64(c * a))
	tmp = 0.0
	if (b <= -3.2e-8)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -5e-197)
		tmp = Float64(Float64(t_0 / Float64(a * 2.0)) / Float64(b - sqrt(fma(b, b, t_0))));
	elseif (b <= 1.5e+116)
		tmp = fma(b, Float64(-0.5 / a), Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0)));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.2e-8], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -5e-197], N[(N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b - N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.5e+116], N[(b * N[(-0.5 / a), $MachinePrecision] + N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.2000000000000002e-8

   1. Initial program 14.6%

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

   4. Applied rewrites 13.7%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if -3.2000000000000002e-8 < b < -5.0000000000000002e-197

   1. Initial program 60.6%

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 89.6%

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

   if -5.0000000000000002e-197 < b < 1.4999999999999999e116

   1. Initial program 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 79.7%

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

   if 1.4999999999999999e116 < b

   1. Initial program 56.6%

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\mathsf{neg}\left(b\right)}\right)}{a} \]

      7. lower-neg.f64 99.4

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

   10. Applied rewrites 99.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(c \cdot a\right)}{a \cdot 2}}{b - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.14e-72)
   (/ (- c) b)
   (if (<= b 1.5e+116)
     (fma b (/ -0.5 a) (/ (sqrt (fma b b (* c (* a -4.0)))) (* a -2.0)))
     (/ (fma a (/ c b) (- b)) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.14e-72) {
		tmp = -c / b;
	} else if (b <= 1.5e+116) {
		tmp = fma(b, (-0.5 / a), (sqrt(fma(b, b, (c * (a * -4.0)))) / (a * -2.0)));
	} else {
		tmp = fma(a, (c / b), -b) / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.14e-72)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.5e+116)
		tmp = fma(b, Float64(-0.5 / a), Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0)));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.14e-72], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.5e+116], N[(b * N[(-0.5 / a), $MachinePrecision] + N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14e-72

   1. Initial program 18.7%

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if -1.14e-72 < b < 1.4999999999999999e116

   1. Initial program 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 80.3%

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

   if 1.4999999999999999e116 < b

   1. Initial program 56.6%

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\mathsf{neg}\left(b\right)}\right)}{a} \]

      7. lower-neg.f64 99.4

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

   10. Applied rewrites 99.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 9.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2} \leq -1 \cdot 10^{-247}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0)) -1e-247)
   (* b (- b))
   (* b b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)) <= -1e-247) {
		tmp = b * -b;
	} else {
		tmp = b * b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)) <= (-1d-247)) then
        tmp = b * -b
    else
        tmp = b * b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)) <= -1e-247) {
		tmp = b * -b;
	} else {
		tmp = b * b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)) <= -1e-247:
		tmp = b * -b
	else:
		tmp = b * b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0)) <= -1e-247)
		tmp = Float64(b * Float64(-b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)) <= -1e-247)
		tmp = b * -b;
	else
		tmp = b * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -1e-247], N[(b * (-b)), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a)) < -1e-247

   1. Initial program 57.0%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot {b}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({b}^{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{b \cdot b}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \]

      7. lower-neg.f64 10.9

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

   5. Applied rewrites 10.9%

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

   if -1e-247 < (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 #s(literal 4 binary64) (*.f64 a c))))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 50.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({b}^{2} \cdot \left(1 + \frac{1}{b}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot b\right)} \cdot \left(1 + \frac{1}{b}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{b \cdot \left(b \cdot \left(1 + \frac{1}{b}\right)\right)}\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(b \cdot \left(1 + \frac{1}{b}\right)\right)\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot 1 + b \cdot \frac{1}{b}\right)}\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(\color{blue}{b} + b \cdot \frac{1}{b}\right)\right)\right) \]

      7. rgt-mult-inverse N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(b + \color{blue}{1}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + b\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(1 + b\right)\right)\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(b\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto b \cdot \left(\color{blue}{-1} + \left(\mathsf{neg}\left(b\right)\right)\right) \]

      12. unsub-neg N/A

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

      13. lower--.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto {b}^{\color{blue}{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.8%

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

4. Final simplification 10.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2} \leq -1 \cdot 10^{-247}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.14e-72)
   (/ (- c) b)
   (if (<= b 1.5e+116)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (/ (fma a (/ c b) (- b)) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.14e-72) {
		tmp = -c / b;
	} else if (b <= 1.5e+116) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = fma(a, (c / b), -b) / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.14e-72)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.5e+116)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.14e-72], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.5e+116], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14e-72

   1. Initial program 18.7%

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if -1.14e-72 < b < 1.4999999999999999e116

   1. Initial program 80.2%

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

   if 1.4999999999999999e116 < b

   1. Initial program 56.6%

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\mathsf{neg}\left(b\right)}\right)}{a} \]

      7. lower-neg.f64 99.4

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

   10. Applied rewrites 99.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.14e-72)
   (/ (- c) b)
   (if (<= b 1.5e+116)
     (* (/ -0.5 a) (+ b (sqrt (fma b b (* c (* a -4.0))))))
     (/ (fma a (/ c b) (- b)) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.14e-72) {
		tmp = -c / b;
	} else if (b <= 1.5e+116) {
		tmp = (-0.5 / a) * (b + sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = fma(a, (c / b), -b) / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.14e-72)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.5e+116)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.14e-72], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.5e+116], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14e-72

   1. Initial program 18.7%

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if -1.14e-72 < b < 1.4999999999999999e116

   1. Initial program 80.2%

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

   4. Applied rewrites 80.1%

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

   if 1.4999999999999999e116 < b

   1. Initial program 56.6%

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\mathsf{neg}\left(b\right)}\right)}{a} \]

      7. lower-neg.f64 99.4

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

   10. Applied rewrites 99.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.14e-72)
   (/ (- c) b)
   (if (<= b 7e-18)
     (/ (- (- b) (sqrt (* c (* a -4.0)))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.14e-72) {
		tmp = -c / b;
	} else if (b <= 7e-18) {
		tmp = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
	} 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 <= (-1.14d-72)) then
        tmp = -c / b
    else if (b <= 7d-18) then
        tmp = (-b - sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
    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 <= -1.14e-72) {
		tmp = -c / b;
	} else if (b <= 7e-18) {
		tmp = (-b - Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.14e-72:
		tmp = -c / b
	elif b <= 7e-18:
		tmp = (-b - math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.14e-72)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 7e-18)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
	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 <= -1.14e-72)
		tmp = -c / b;
	elseif (b <= 7e-18)
		tmp = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.14e-72], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 7e-18], N[(N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14e-72

   1. Initial program 18.7%

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if -1.14e-72 < b < 6.9999999999999997e-18

   1. Initial program 76.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if 6.9999999999999997e-18 < b

   1. Initial program 70.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.14e-72)
   (/ (- c) b)
   (if (<= b 7e-18)
     (* (/ -0.5 a) (+ b (sqrt (* -4.0 (* c a)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.14e-72) {
		tmp = -c / b;
	} else if (b <= 7e-18) {
		tmp = (-0.5 / a) * (b + sqrt((-4.0 * (c * 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 <= (-1.14d-72)) then
        tmp = -c / b
    else if (b <= 7d-18) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((-4.0d0) * (c * 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 <= -1.14e-72) {
		tmp = -c / b;
	} else if (b <= 7e-18) {
		tmp = (-0.5 / a) * (b + Math.sqrt((-4.0 * (c * a))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.14e-72:
		tmp = -c / b
	elif b <= 7e-18:
		tmp = (-0.5 / a) * (b + math.sqrt((-4.0 * (c * a))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.14e-72)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 7e-18)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(-4.0 * Float64(c * 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 <= -1.14e-72)
		tmp = -c / b;
	elseif (b <= 7e-18)
		tmp = (-0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.14e-72], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 7e-18], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14e-72

   1. Initial program 18.7%

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if -1.14e-72 < b < 6.9999999999999997e-18

   1. Initial program 76.1%

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 68.4

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

   7. Applied rewrites 68.4%

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

   if 6.9999999999999997e-18 < b

   1. Initial program 70.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-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 <= (-1d-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 <= -1e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-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 <= -1e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-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 < -9.999999999999969e-311

   1. Initial program 29.2%

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 72.4

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

   7. Applied rewrites 72.4%

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

   if -9.999999999999969e-311 < b

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 67.0

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

   7. Applied rewrites 67.0%

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

4. Final simplification 69.5%

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

Alternative 9: 67.8% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (- c) b) (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		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 <= (-1d-310)) 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 <= -1e-310) {
		tmp = -c / b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -c / b
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = -c / b;
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[((-c) / b), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 29.2%

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 72.4

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

   7. Applied rewrites 72.4%

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

   if -9.999999999999969e-311 < b

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      6. lower-neg.f64 66.6

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

   7. Applied rewrites 66.6%

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

4. Final simplification 69.3%

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

Alternative 10: 9.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-310}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(b, b, b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= a -1e-310) (* b b) (- (fma b b b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (a <= -1e-310) {
		tmp = b * b;
	} else {
		tmp = -fma(b, b, b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (a <= -1e-310)
		tmp = Float64(b * b);
	else
		tmp = Float64(-fma(b, b, b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[a, -1e-310], N[(b * b), $MachinePrecision], (-N[(b * b + b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if a < -9.999999999999969e-311

   1. Initial program 56.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({b}^{2} \cdot \left(1 + \frac{1}{b}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot b\right)} \cdot \left(1 + \frac{1}{b}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{b \cdot \left(b \cdot \left(1 + \frac{1}{b}\right)\right)}\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(b \cdot \left(1 + \frac{1}{b}\right)\right)\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot 1 + b \cdot \frac{1}{b}\right)}\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(\color{blue}{b} + b \cdot \frac{1}{b}\right)\right)\right) \]

      7. rgt-mult-inverse N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(b + \color{blue}{1}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + b\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(1 + b\right)\right)\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(b\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto b \cdot \left(\color{blue}{-1} + \left(\mathsf{neg}\left(b\right)\right)\right) \]

      12. unsub-neg N/A

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

      13. lower--.f64 2.0

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

   5. Applied rewrites 2.0%

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

   6. Taylor expanded in b around -inf

      \[\leadsto {b}^{\color{blue}{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 10.9%

      \[\leadsto b \cdot \color{blue}{b} \]

   if -9.999999999999969e-311 < a

   1. Initial program 50.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({b}^{2} \cdot \left(1 + \frac{1}{b}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot b\right)} \cdot \left(1 + \frac{1}{b}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{b \cdot \left(b \cdot \left(1 + \frac{1}{b}\right)\right)}\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(b \cdot \left(1 + \frac{1}{b}\right)\right)\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot 1 + b \cdot \frac{1}{b}\right)}\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(\color{blue}{b} + b \cdot \frac{1}{b}\right)\right)\right) \]

      7. rgt-mult-inverse N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(b + \color{blue}{1}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + b\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(1 + b\right)\right)\right)} \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(b\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto b \cdot \left(\color{blue}{-1} + \left(\mathsf{neg}\left(b\right)\right)\right) \]

      12. unsub-neg N/A

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

      13. lower--.f64 9.6

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{\left(-1 \cdot b - 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.6%

      \[\leadsto -\mathsf{fma}\left(b, b, b\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 34.7% accurate, 3.6× 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(b / Float64(-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 53.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. lift-neg.f64 N/A

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

   8. remove-double-neg N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

4. Applied rewrites 50.2%

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

5. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{b}{\mathsf{neg}\left(a\right)}} \]

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

      \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

   6. lower-neg.f64 36.7

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

7. Applied rewrites 36.7%

   \[\leadsto \color{blue}{\frac{b}{-a}} \]
8. Add Preprocessing

Alternative 12: 5.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(b * b)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({b}^{2} \cdot \left(1 + \frac{1}{b}\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot b\right)} \cdot \left(1 + \frac{1}{b}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{b \cdot \left(b \cdot \left(1 + \frac{1}{b}\right)\right)}\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(b \cdot \left(1 + \frac{1}{b}\right)\right)\right)} \]

   5. distribute-lft-in N/A

      \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot 1 + b \cdot \frac{1}{b}\right)}\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(\color{blue}{b} + b \cdot \frac{1}{b}\right)\right)\right) \]

   7. rgt-mult-inverse N/A

      \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(b + \color{blue}{1}\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + b\right)}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(1 + b\right)\right)\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(b\right)\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto b \cdot \left(\color{blue}{-1} + \left(\mathsf{neg}\left(b\right)\right)\right) \]

   12. unsub-neg N/A

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

   13. lower--.f64 5.9

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

5. Applied rewrites 5.9%

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

6. Taylor expanded in b around -inf

   \[\leadsto {b}^{\color{blue}{2}} \]
7. Step-by-step derivation

8. Applied rewrites 6.3%

   \[\leadsto b \cdot \color{blue}{b} \]
9. Add Preprocessing

Alternative 13: 3.3% accurate, 16.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := (-b)

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({b}^{2} \cdot \left(1 + \frac{1}{b}\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot b\right)} \cdot \left(1 + \frac{1}{b}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{b \cdot \left(b \cdot \left(1 + \frac{1}{b}\right)\right)}\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(b \cdot \left(1 + \frac{1}{b}\right)\right)\right)} \]

   5. distribute-lft-in N/A

      \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b \cdot 1 + b \cdot \frac{1}{b}\right)}\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(\color{blue}{b} + b \cdot \frac{1}{b}\right)\right)\right) \]

   7. rgt-mult-inverse N/A

      \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(b + \color{blue}{1}\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + b\right)}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(1 + b\right)\right)\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(b\right)\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto b \cdot \left(\color{blue}{-1} + \left(\mathsf{neg}\left(b\right)\right)\right) \]

   12. unsub-neg N/A

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

   13. lower--.f64 5.9

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

5. Applied rewrites 5.9%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 3.3%

   \[\leadsto -b \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024223 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

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