Quadratic roots, full range

Percentage Accurate: 51.5% → 90.1%

Time: 10.9s

Alternatives: 6

Speedup: 2.5×

Specification

Math

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

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

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

Alternative 1: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq -2.8 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e+159)
   (/ b (- a))
   (if (<= b -2.8e-257)
     (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
     (if (<= b 5.8e+77)
       (/ (* c -2.0) (+ b (sqrt (fma c (* a -4.0) (* b b)))))
       (/ c (- b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e+159) {
		tmp = b / -a;
	} else if (b <= -2.8e-257) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else if (b <= 5.8e+77) {
		tmp = (c * -2.0) / (b + sqrt(fma(c, (a * -4.0), (b * b))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e+159)
		tmp = Float64(b / Float64(-a));
	elseif (b <= -2.8e-257)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	elseif (b <= 5.8e+77)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e+159], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, -2.8e-257], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e+77], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.5499999999999999e159

   1. Initial program 47.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if -1.5499999999999999e159 < b < -2.80000000000000001e-257

   1. Initial program 85.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 85.1

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 85.1

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

   4. Applied rewrites 85.1%

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

   if -2.80000000000000001e-257 < b < 5.8000000000000003e77

   1. Initial program 51.0%

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

   4. Applied rewrites 42.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 70.6

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

   7. Applied rewrites 70.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 78.7%

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

   10. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 85.7

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

   12. Applied rewrites 85.7%

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

   if 5.8000000000000003e77 < b

   1. Initial program 10.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq -2.8 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{-a}\right)\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.8e-82)
   (fma b (/ c (* b b)) (/ b (- a)))
   (if (<= b 5.8e+77)
     (/ (* c -2.0) (+ b (sqrt (fma c (* a -4.0) (* b b)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e-82) {
		tmp = fma(b, (c / (b * b)), (b / -a));
	} else if (b <= 5.8e+77) {
		tmp = (c * -2.0) / (b + sqrt(fma(c, (a * -4.0), (b * b))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.8e-82)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(b / Float64(-a)));
	elseif (b <= 5.8e+77)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.8e-82], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e+77], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.7999999999999995e-82

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   4. 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. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -6.7999999999999995e-82 < b < 5.8000000000000003e77

   1. Initial program 59.6%

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 70.5

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

   7. Applied rewrites 70.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 78.3%

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

   10. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 83.5

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

   12. Applied rewrites 83.5%

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

   if 5.8000000000000003e77 < b

   1. Initial program 10.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 96.6

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

   5. Applied rewrites 96.6%

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

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{-a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-92)
   (fma b (/ c (* b b)) (/ b (- a)))
   (if (<= b 1.25e-100)
     (/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-92) {
		tmp = fma(b, (c / (b * b)), (b / -a));
	} else if (b <= 1.25e-100) {
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-92)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(b / Float64(-a)));
	elseif (b <= 1.25e-100)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-92], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-100], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.99999999999999995e-92

   1. Initial program 69.5%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   4. 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. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if -3.99999999999999995e-92 < b < 1.25e-100

   1. Initial program 80.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 80.6

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 79.4

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

   7. Applied rewrites 79.4%

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

   if 1.25e-100 < b

   1. Initial program 15.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{-a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{-a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-100}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-92)
   (fma b (/ c (* b b)) (/ b (- a)))
   (if (<= b 1.25e-100)
     (* (/ 0.5 a) (- (sqrt (* a (* c -4.0))) b))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-92) {
		tmp = fma(b, (c / (b * b)), (b / -a));
	} else if (b <= 1.25e-100) {
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-92)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(b / Float64(-a)));
	elseif (b <= 1.25e-100)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(a * Float64(c * -4.0))) - b));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-92], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-100], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.99999999999999995e-92

   1. Initial program 69.5%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   4. 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. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if -3.99999999999999995e-92 < b < 1.25e-100

   1. Initial program 80.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 80.6

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b}} - b}{2 \cdot a} \]

      2. lower-*.f64 5.0

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b}} - b}{2 \cdot a} \]

   7. Applied rewrites 5.0%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b}} - b}{2 \cdot a} \]

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 79.4

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

   10. Applied rewrites 79.4%

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

       1. lift-/.f64 N/A

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

       2. div-inv N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. lift-*.f64 N/A

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

       7. associate-/r* N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f64 79.2

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

   12. Applied rewrites 79.2%

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

   if 1.25e-100 < b

   1. Initial program 15.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 5: 67.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.5e-301) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.5e-301) {
		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 <= 4.5d-301) 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 <= 4.5e-301) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.5e-301:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.5e-301)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.5e-301)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.5e-301], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5000000000000002e-301

   1. Initial program 73.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if 4.5000000000000002e-301 < b

   1. Initial program 29.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 68.1

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

   5. Applied rewrites 68.1%

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

Alternative 6: 34.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 35.7

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

5. Applied rewrites 35.7%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))