quadp (p42, positive)

Percentage Accurate: 52.3% → 84.8%

Time: 8.6s

Alternatives: 8

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: 52.3% 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: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+84}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{a}, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, \frac{-0.5 \cdot b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+84)
   (- (/ c b) (/ b a))
   (if (<= b 6.8e-105)
     (fma (/ 0.5 a) (sqrt (fma (* a c) -4.0 (* b b))) (/ (* -0.5 b) a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+84) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.8e-105) {
		tmp = fma((0.5 / a), sqrt(fma((a * c), -4.0, (b * b))), ((-0.5 * b) / a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e+84)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.8e-105)
		tmp = fma(Float64(0.5 / a), sqrt(fma(Float64(a * c), -4.0, Float64(b * b))), Float64(Float64(-0.5 * b) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e+84], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-105], N[(N[(0.5 / a), $MachinePrecision] * N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(-0.5 * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999999e84

   1. Initial program 52.4%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 97.7

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 98.0%

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

   if -2.2999999999999999e84 < b < 6.79999999999999984e-105

   1. Initial program 80.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 80.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 80.0

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

   6. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 80.1

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

   8. Applied rewrites 80.1%

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

   if 6.79999999999999984e-105 < b

   1. Initial program 12.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+84}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{a}, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, \frac{-0.5 \cdot b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+149}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+149)
   (/ (- b) a)
   (if (<= b 5e-106)
     (/ (- (sqrt (- (* b b) (* (* a c) 4.0))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+149) {
		tmp = -b / a;
	} else if (b <= 5e-106) {
		tmp = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+149)) then
        tmp = -b / a
    else if (b <= 5d-106) then
        tmp = (sqrt(((b * b) - ((a * c) * 4.0d0))) - b) / (2.0d0 * a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+149) {
		tmp = -b / a;
	} else if (b <= 5e-106) {
		tmp = (Math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+149:
		tmp = -b / a
	elif b <= 5e-106:
		tmp = (math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+149)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 5e-106)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+149)
		tmp = -b / a;
	elseif (b <= 5e-106)
		tmp = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+149], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 5e-106], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000005e149

   1. Initial program 40.8%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -1.00000000000000005e149 < b < 4.99999999999999983e-106

   1. Initial program 81.7%

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

   if 4.99999999999999983e-106 < b

   1. Initial program 12.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+149}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e+74)
   (- (/ c b) (/ b a))
   (if (<= b 5e-106)
     (* (- (sqrt (fma -4.0 (* a c) (* b b))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e+74) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5e-106) {
		tmp = (sqrt(fma(-4.0, (a * c), (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e+74)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5e-106)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(a * c), Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e+74], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-106], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2000000000000001e74

   1. Initial program 54.3%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 97.7

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 98.0%

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

   if -8.2000000000000001e74 < b < 4.99999999999999983e-106

   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

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 79.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if 4.99999999999999983e-106 < b

   1. Initial program 12.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-75}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-106}:\\ \;\;\;\;\left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-75)
   (- (/ c b) (/ b a))
   (if (<= b 1.55e-106)
     (* (- (sqrt (* (* -4.0 a) c)) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-75) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.55e-106) {
		tmp = (sqrt(((-4.0 * a) * c)) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6d-75)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.55d-106) then
        tmp = (sqrt((((-4.0d0) * a) * c)) - b) * (0.5d0 / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-75) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.55e-106) {
		tmp = (Math.sqrt(((-4.0 * a) * c)) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-75:
		tmp = (c / b) - (b / a)
	elif b <= 1.55e-106:
		tmp = (math.sqrt(((-4.0 * a) * c)) - b) * (0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-75)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.55e-106)
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e-75)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.55e-106)
		tmp = (sqrt(((-4.0 * a) * c)) - b) * (0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-75], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-106], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.9999999999999997e-75

   1. Initial program 67.3%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 89.0

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

   8. Applied rewrites 89.0%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 89.2%

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

   if -5.9999999999999997e-75 < b < 1.54999999999999993e-106

   1. Initial program 75.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 75.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 73.7

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

   7. Applied rewrites 73.7%

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

   if 1.54999999999999993e-106 < b

   1. Initial program 12.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-75}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-106}:\\ \;\;\;\;\left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 71.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 64.4

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

   8. Applied rewrites 64.4%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 64.9%

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

   if -3.999999999999988e-310 < b

   1. Initial program 25.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 73.1

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

   5. Applied rewrites 73.1%

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

Alternative 6: 66.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-293}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.7e-293) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.7e-293) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.7d-293) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.7e-293) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.7e-293:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.7e-293)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.7e-293)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.7e-293], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.7e-293

   1. Initial program 71.6%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if 1.7e-293 < b

   1. Initial program 24.1%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 7: 42.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -4e-310) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 71.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if -3.999999999999988e-310 < b

   1. Initial program 25.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 25.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 25.2

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

   4. Applied rewrites 25.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 21.3

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

   6. Applied rewrites 21.3%

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

   7. Taylor expanded in c around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt 18.3

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

   9. Applied rewrites 18.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 11.1% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 45.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 45.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 45.4

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

4. Applied rewrites 45.4%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. lift-neg.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 43.2

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

6. Applied rewrites 43.2%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 11.4

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

9. Applied rewrites 11.4%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

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) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

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 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	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 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	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 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	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(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	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 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	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[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024271 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

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