The quadratic formula (r2)

Percentage Accurate: 53.0% → 85.6%

Time: 10.8s

Alternatives: 9

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: 53.0% 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: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right) \cdot c}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-86)
   (/ (* (fma (/ a b) (/ c b) 1.0) c) (- b))
   (if (<= b 5e+94)
     (/ (+ (sqrt (fma (* c a) -4.0 (* b b))) b) (* -2.0 a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-86) {
		tmp = (fma((a / b), (c / b), 1.0) * c) / -b;
	} else if (b <= 5e+94) {
		tmp = (sqrt(fma((c * a), -4.0, (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e-86)
		tmp = Float64(Float64(fma(Float64(a / b), Float64(c / b), 1.0) * c) / Float64(-b));
	elseif (b <= 5e+94)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e-86], N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 1.0), $MachinePrecision] * c), $MachinePrecision] / (-b)), $MachinePrecision], If[LessEqual[b, 5e+94], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.59999999999999992e-86

   1. Initial program 20.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 20.3

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 20.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 20.3

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

   4. Applied rewrites 20.3%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. 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 2.8

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 86.1

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

   10. Applied rewrites 86.1%

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

   if -4.59999999999999992e-86 < b < 5.0000000000000001e94

   1. Initial program 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. div-inv N/A

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

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

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 82.5%

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

   6. Applied rewrites 82.5%

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

   8. Applied rewrites 82.5%

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

   if 5.0000000000000001e94 < b

   1. Initial program 55.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, \frac{c}{b}, c\right)}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-86)
   (/ (fma (* (/ c b) a) (/ c b) c) (- b))
   (if (<= b 5e+94)
     (/ (+ (sqrt (fma (* c a) -4.0 (* b b))) b) (* -2.0 a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-86) {
		tmp = fma(((c / b) * a), (c / b), c) / -b;
	} else if (b <= 5e+94) {
		tmp = (sqrt(fma((c * a), -4.0, (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e-86)
		tmp = Float64(fma(Float64(Float64(c / b) * a), Float64(c / b), c) / Float64(-b));
	elseif (b <= 5e+94)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e-86], N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * N[(c / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision], If[LessEqual[b, 5e+94], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.59999999999999992e-86

   1. Initial program 20.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -4.59999999999999992e-86 < b < 5.0000000000000001e94

   1. Initial program 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. div-inv N/A

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

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

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 82.5%

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

   6. Applied rewrites 82.5%

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

   8. Applied rewrites 82.5%

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

   if 5.0000000000000001e94 < b

   1. Initial program 55.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-86)
   (/ c (- b))
   (if (<= b 5e+94)
     (/ (+ (sqrt (fma (* c a) -4.0 (* b b))) b) (* -2.0 a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-86) {
		tmp = c / -b;
	} else if (b <= 5e+94) {
		tmp = (sqrt(fma((c * a), -4.0, (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-86)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5e+94)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-86], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5e+94], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.8e-86

   1. Initial program 20.3%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 85.8

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

   5. Applied rewrites 85.8%

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

   if -3.8e-86 < b < 5.0000000000000001e94

   1. Initial program 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. div-inv N/A

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

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

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 82.5%

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

   6. Applied rewrites 82.5%

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

   8. Applied rewrites 82.5%

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

   if 5.0000000000000001e94 < b

   1. Initial program 55.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} + b}{\left(-a\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-86)
   (/ c (- b))
   (if (<= b 6e-70)
     (/ (+ (sqrt (* (* c a) -4.0)) b) (* (- a) 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-86) {
		tmp = c / -b;
	} else if (b <= 6e-70) {
		tmp = (sqrt(((c * a) * -4.0)) + b) / (-a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.8d-86)) then
        tmp = c / -b
    else if (b <= 6d-70) then
        tmp = (sqrt(((c * a) * (-4.0d0))) + b) / (-a * 2.0d0)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-86) {
		tmp = c / -b;
	} else if (b <= 6e-70) {
		tmp = (Math.sqrt(((c * a) * -4.0)) + b) / (-a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-86:
		tmp = c / -b
	elif b <= 6e-70:
		tmp = (math.sqrt(((c * a) * -4.0)) + b) / (-a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-86)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 6e-70)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) + b) / Float64(Float64(-a) * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-86)
		tmp = c / -b;
	elseif (b <= 6e-70)
		tmp = (sqrt(((c * a) * -4.0)) + b) / (-a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-86], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 6e-70], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[((-a) * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.8e-86

   1. Initial program 20.3%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 85.8

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

   5. Applied rewrites 85.8%

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

   if -3.8e-86 < b < 6.0000000000000003e-70

   1. Initial program 79.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   if 6.0000000000000003e-70 < b

   1. Initial program 67.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 92.9

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

   5. Applied rewrites 92.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} + b}{\left(-a\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 36.2%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 67.1

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

   5. Applied rewrites 67.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 72.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   4. 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 66.1

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

   5. Applied rewrites 66.1%

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

Alternative 6: 67.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-308) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-308) {
		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 <= (-9.5d-308)) 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 <= -9.5e-308) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-308:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-308)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e-308)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-308], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.49999999999999963e-308

   1. Initial program 36.2%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 67.1

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

   5. Applied rewrites 67.1%

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

   if -9.49999999999999963e-308 < b

   1. Initial program 72.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 65.7

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

   5. Applied rewrites 65.7%

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

Alternative 7: 35.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
4. 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 36.4

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

5. Applied rewrites 36.4%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Alternative 8: 11.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 N/A

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

   7. lower-neg.f64 53.3

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 53.3

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 53.3

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

4. Applied rewrites 53.3%

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

5. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 32.3

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

7. Applied rewrites 32.3%

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

8. Taylor expanded in c around inf

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

10. Applied rewrites 8.6%

    \[\leadsto \frac{c}{\color{blue}{b}} \]
11. Add Preprocessing

Alternative 9: 2.5% accurate, 4.2× 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 / 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 53.3%

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

4. Applied rewrites 35.4%

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

5. Taylor expanded in b around -inf

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

   1. lower-/.f64 2.7

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

7. Applied rewrites 2.7%

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

Developer Target 1: 71.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024273 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

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