quadp (p42, positive)

Percentage Accurate: 51.9% → 85.5%

Time: 8.9s

Alternatives: 11

Speedup: 2.5×

Specification

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.9% 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: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+125}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a} - \frac{b}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(-c\right) \cdot \mathsf{fma}\left(\frac{c}{{b}^{3}}, a, \frac{1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e+125)
   (/ (- b) a)
   (if (<= b 6.6e-47)
     (- (/ (sqrt (fma b b (* -4.0 (* c a)))) (* 2.0 a)) (* (/ b a) 0.5))
     (* (- c) (fma (/ c (pow b 3.0)) a (/ 1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+125) {
		tmp = -b / a;
	} else if (b <= 6.6e-47) {
		tmp = (sqrt(fma(b, b, (-4.0 * (c * a)))) / (2.0 * a)) - ((b / a) * 0.5);
	} else {
		tmp = -c * fma((c / pow(b, 3.0)), a, (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e+125)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.6e-47)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) / Float64(2.0 * a)) - Float64(Float64(b / a) * 0.5));
	else
		tmp = Float64(Float64(-c) * fma(Float64(c / (b ^ 3.0)), a, Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e+125], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.6e-47], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[(b / a), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[((-c) * N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.00000000000000023e125

   1. Initial program 51.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -7.00000000000000023e125 < b < 6.60000000000000007e-47

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 80.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. neg-mul-1 N/A

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

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

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 80.8%

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

   if 6.60000000000000007e-47 < b

   1. Initial program 10.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot c}{{b}^{3}}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right) \]

      3. distribute-neg-out N/A

         \[\leadsto c \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)\right)\right)} \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(c \cdot \left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)\right)} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-c \cdot \left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto -\color{blue}{c \cdot \left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)} \]

      7. associate-/l* N/A

         \[\leadsto -c \cdot \left(\color{blue}{a \cdot \frac{c}{{b}^{3}}} + \frac{1}{b}\right) \]

      8. *-commutative N/A

         \[\leadsto -c \cdot \left(\color{blue}{\frac{c}{{b}^{3}} \cdot a} + \frac{1}{b}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto -c \cdot \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{3}}, a, \frac{1}{b}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto -c \cdot \mathsf{fma}\left(\color{blue}{\frac{c}{{b}^{3}}}, a, \frac{1}{b}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto -c \cdot \mathsf{fma}\left(\frac{c}{\color{blue}{{b}^{3}}}, a, \frac{1}{b}\right) \]

      12. lower-/.f64 90.9

          \[\leadsto -c \cdot \mathsf{fma}\left(\frac{c}{{b}^{3}}, a, \color{blue}{\frac{1}{b}}\right) \]

   5. Applied rewrites 90.9%

      \[\leadsto \color{blue}{-c \cdot \mathsf{fma}\left(\frac{c}{{b}^{3}}, a, \frac{1}{b}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+125}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a} - \frac{b}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(-c\right) \cdot \mathsf{fma}\left(\frac{c}{{b}^{3}}, a, \frac{1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+125}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a} - \frac{b}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e+125)
   (/ (- b) a)
   (if (<= b 6.6e-47)
     (- (/ (sqrt (fma b b (* -4.0 (* c a)))) (* 2.0 a)) (* (/ b a) 0.5))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+125) {
		tmp = -b / a;
	} else if (b <= 6.6e-47) {
		tmp = (sqrt(fma(b, b, (-4.0 * (c * a)))) / (2.0 * a)) - ((b / a) * 0.5);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e+125)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.6e-47)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) / Float64(2.0 * a)) - Float64(Float64(b / a) * 0.5));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e+125], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.6e-47], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[(b / a), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.00000000000000023e125

   1. Initial program 51.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -7.00000000000000023e125 < b < 6.60000000000000007e-47

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 80.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. neg-mul-1 N/A

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

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

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 80.8%

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

   if 6.60000000000000007e-47 < b

   1. Initial program 10.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+125}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a} - \frac{b}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+125}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e+125)
   (/ (- b) a)
   (if (<= b 6.6e-47)
     (/ (- (sqrt (fma b b (* -4.0 (* c a)))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+125) {
		tmp = -b / a;
	} else if (b <= 6.6e-47) {
		tmp = (sqrt(fma(b, b, (-4.0 * (c * a)))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e+125)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.6e-47)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e+125], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.6e-47], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.00000000000000023e125

   1. Initial program 51.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -7.00000000000000023e125 < b < 6.60000000000000007e-47

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 80.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift--.f64 80.7

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

   6. Applied rewrites 80.8%

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

   if 6.60000000000000007e-47 < b

   1. Initial program 10.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+125}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+125}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e+125)
   (/ (- b) a)
   (if (<= b 6.6e-47)
     (/ (- (sqrt (fma -4.0 (* c a) (* b b))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+125) {
		tmp = -b / a;
	} else if (b <= 6.6e-47) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e+125)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.6e-47)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e+125], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.6e-47], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.00000000000000023e125

   1. Initial program 51.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -7.00000000000000023e125 < b < 6.60000000000000007e-47

   1. Initial program 80.7%

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

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 80.7

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval 80.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 80.7

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

   4. Applied rewrites 80.7%

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

   if 6.60000000000000007e-47 < b

   1. Initial program 10.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

Alternative 5: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+116)
   (/ (- b) a)
   (if (<= b 6.6e-47)
     (* (/ 0.5 a) (- (sqrt (fma -4.0 (* c a) (* b b))) b))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+116) {
		tmp = -b / a;
	} else if (b <= 6.6e-47) {
		tmp = (0.5 / a) * (sqrt(fma(-4.0, (c * a), (b * b))) - b);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+116)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.6e-47)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+116], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.6e-47], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.50000000000000016e116

   1. Initial program 53.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if -4.50000000000000016e116 < b < 6.60000000000000007e-47

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 80.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 80.2

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

   4. Applied rewrites 80.2%

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

   if 6.60000000000000007e-47 < b

   1. Initial program 10.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-46)
   (- (/ c b) (/ b a))
   (if (<= b 6.6e-47)
     (/ (- (sqrt (* -4.0 (* c a))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-46) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6e-47) {
		tmp = (sqrt((-4.0 * (c * a))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.15d-46)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.6d-47) then
        tmp = (sqrt(((-4.0d0) * (c * a))) - b) / (2.0d0 * a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-46) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6e-47) {
		tmp = (Math.sqrt((-4.0 * (c * a))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-46:
		tmp = (c / b) - (b / a)
	elif b <= 6.6e-47:
		tmp = (math.sqrt((-4.0 * (c * a))) - b) / (2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-46)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.6e-47)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-46)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.6e-47)
		tmp = (sqrt((-4.0 * (c * a))) - b) / (2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-46], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6e-47], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15e-46

   1. Initial program 70.7%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 85.9

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 86.1%

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

   if -1.15e-46 < b < 6.60000000000000007e-47

   1. Initial program 73.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 73.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift--.f64 73.7

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

   6. Applied rewrites 73.7%

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

   7. Taylor expanded in c around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}} - b}{2 \cdot a} \]
   8. 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 66.4

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

   9. Applied rewrites 66.4%

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

   if 6.60000000000000007e-47 < b

   1. Initial program 10.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-46)
   (- (/ c b) (/ b a))
   (if (<= b 6.6e-47)
     (* (- (sqrt (* -4.0 (* c a))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-46) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6e-47) {
		tmp = (sqrt((-4.0 * (c * a))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.15d-46)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.6d-47) then
        tmp = (sqrt(((-4.0d0) * (c * a))) - b) * (0.5d0 / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-46) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6e-47) {
		tmp = (Math.sqrt((-4.0 * (c * a))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-46:
		tmp = (c / b) - (b / a)
	elif b <= 6.6e-47:
		tmp = (math.sqrt((-4.0 * (c * a))) - b) * (0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-46)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.6e-47)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-46)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.6e-47)
		tmp = (sqrt((-4.0 * (c * a))) - b) * (0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-46], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6e-47], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15e-46

   1. Initial program 70.7%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 85.9

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 86.1%

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

   if -1.15e-46 < b < 6.60000000000000007e-47

   1. Initial program 73.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 73.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 73.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. sub-neg N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift--.f64 73.6

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

   6. Applied rewrites 73.6%

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

   7. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 66.4

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

   9. Applied rewrites 66.4%

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

   if 6.60000000000000007e-47 < b

   1. Initial program 10.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-47}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 73.7%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 65.3

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

   8. Applied rewrites 65.3%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 66.4%

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

   if -3.999999999999988e-310 < b

   1. Initial program 23.9%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 74.5

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

   5. Applied rewrites 74.5%

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

Alternative 9: 67.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-253}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.5e-253) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.5e-253) {
		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 <= 8.5d-253) 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 <= 8.5e-253) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.5e-253:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.5e-253)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.5e-253)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.5e-253], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.4999999999999999e-253

   1. Initial program 73.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 63.6

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

   5. Applied rewrites 63.6%

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

   if 8.4999999999999999e-253 < b

   1. Initial program 21.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 77.3

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

   5. Applied rewrites 77.3%

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

Alternative 10: 43.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 73.7%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if -3.999999999999988e-310 < b

   1. Initial program 23.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 23.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 23.9

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

   4. Applied rewrites 23.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift--.f64 23.9

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

   6. Applied rewrites 23.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. clear-num N/A

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

      18. associate-/r/ N/A

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

      19. lift-/.f64 N/A

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

      20. associate-*l* N/A

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

   8. Applied rewrites 20.6%

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

   9. Taylor expanded in c around 0

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

       1. distribute-rgt-out N/A

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

       2. metadata-eval N/A

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

       3. mul0-rgt 23.8

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

   11. Applied rewrites 23.8%

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

Alternative 11: 10.9% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 48.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 48.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 48.6

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

4. Applied rewrites 48.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. lift--.f64 48.6

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

6. Applied rewrites 48.6%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lift-/.f64 N/A

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

   13. div-inv N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. metadata-eval N/A

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

   17. clear-num N/A

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

   18. associate-/r/ N/A

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

   19. lift-/.f64 N/A

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

   20. associate-*l* N/A

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

8. Applied rewrites 46.9%

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

9. Taylor expanded in c around 0

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

    1. distribute-rgt-out N/A

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

    2. metadata-eval N/A

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

    3. mul0-rgt 13.4

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

11. Applied rewrites 13.4%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024268 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

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

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