quadm (p42, negative)

Percentage Accurate: 51.7% → 90.9%

Time: 6.6s

Alternatives: 8

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.7% 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: 90.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-308}:\\ \;\;\;\;\frac{-2 \cdot c}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a} + \frac{b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e+81)
   (/ c (- b))
   (if (<= b 2.7e-308)
     (/ (* -2.0 c) (- b (sqrt (fma (* a c) -4.0 (* b b)))))
     (if (<= b 5.2e+121)
       (+ (/ (sqrt (fma -4.0 (* c a) (* b b))) (* -2.0 a)) (/ b (* -2.0 a)))
       (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e+81) {
		tmp = c / -b;
	} else if (b <= 2.7e-308) {
		tmp = (-2.0 * c) / (b - sqrt(fma((a * c), -4.0, (b * b))));
	} else if (b <= 5.2e+121) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) / (-2.0 * a)) + (b / (-2.0 * a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e+81)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 2.7e-308)
		tmp = Float64(Float64(-2.0 * c) / Float64(b - sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))));
	elseif (b <= 5.2e+121)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) / Float64(-2.0 * a)) + Float64(b / Float64(-2.0 * a)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e+81], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 2.7e-308], N[(N[(-2.0 * c), $MachinePrecision] / N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e+121], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision] + N[(b / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -7.599999999999999e81

   1. Initial program 8.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -7.599999999999999e81 < b < 2.70000000000000015e-308

   1. Initial program 59.0%

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

   4. Applied rewrites 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 58.9

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

      7. lift--.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate--l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. +-inverses 81.4

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{0}\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   6. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(4, a \cdot c, 0\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 86.9

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

   9. Applied rewrites 86.9%

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

   if 2.70000000000000015e-308 < b < 5.1999999999999998e121

   1. Initial program 93.0%

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

   4. Applied rewrites 93.0%

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

   if 5.1999999999999998e121 < b

   1. Initial program 41.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 2: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-308}:\\ \;\;\;\;\frac{-2 \cdot c}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e+81)
   (/ c (- b))
   (if (<= b 2.7e-308)
     (/ (* -2.0 c) (- b (sqrt (fma (* a c) -4.0 (* b b)))))
     (if (<= b 5.2e+121)
       (/ (+ (sqrt (fma -4.0 (* c a) (* b b))) b) (* -2.0 a))
       (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e+81) {
		tmp = c / -b;
	} else if (b <= 2.7e-308) {
		tmp = (-2.0 * c) / (b - sqrt(fma((a * c), -4.0, (b * b))));
	} else if (b <= 5.2e+121) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e+81)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 2.7e-308)
		tmp = Float64(Float64(-2.0 * c) / Float64(b - sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))));
	elseif (b <= 5.2e+121)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e+81], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 2.7e-308], N[(N[(-2.0 * c), $MachinePrecision] / N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e+121], 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[((-b) / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -7.599999999999999e81

   1. Initial program 8.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -7.599999999999999e81 < b < 2.70000000000000015e-308

   1. Initial program 59.0%

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

   4. Applied rewrites 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 58.9

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

      7. lift--.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate--l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. +-inverses 81.4

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{0}\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   6. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(4, a \cdot c, 0\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 86.9

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

   9. Applied rewrites 86.9%

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

   if 2.70000000000000015e-308 < b < 5.1999999999999998e121

   1. Initial program 93.0%

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

   4. Applied rewrites 93.0%

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

   if 5.1999999999999998e121 < b

   1. Initial program 41.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 3: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{-2 \cdot c}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e+81)
   (/ c (- b))
   (if (<= b 8.4e-36)
     (/ (* -2.0 c) (- b (sqrt (fma (* a c) -4.0 (* b b)))))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e+81) {
		tmp = c / -b;
	} else if (b <= 8.4e-36) {
		tmp = (-2.0 * c) / (b - sqrt(fma((a * c), -4.0, (b * b))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e+81)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 8.4e-36)
		tmp = Float64(Float64(-2.0 * c) / Float64(b - sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e+81], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 8.4e-36], N[(N[(-2.0 * c), $MachinePrecision] / N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.599999999999999e81

   1. Initial program 8.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -7.599999999999999e81 < b < 8.39999999999999964e-36

   1. Initial program 69.3%

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

   4. Applied rewrites 57.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 64.8

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

      7. lift--.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate--l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. +-inverses 80.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{0}\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \]

   6. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(4, a \cdot c, 0\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 83.9

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

   9. Applied rewrites 83.9%

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

   if 8.39999999999999964e-36 < b

   1. Initial program 60.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 91.6

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

   5. Applied rewrites 91.6%

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

Alternative 4: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e-109)
   (/ c (- b))
   (if (<= b 1.9e-35)
     (/ (+ (sqrt (* -4.0 (* a c))) b) (* -2.0 a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-109) {
		tmp = c / -b;
	} else if (b <= 1.9e-35) {
		tmp = (sqrt((-4.0 * (a * c))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.8d-109)) then
        tmp = c / -b
    else if (b <= 1.9d-35) then
        tmp = (sqrt(((-4.0d0) * (a * c))) + b) / ((-2.0d0) * a)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-109) {
		tmp = c / -b;
	} else if (b <= 1.9e-35) {
		tmp = (Math.sqrt((-4.0 * (a * c))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.8e-109:
		tmp = c / -b
	elif b <= 1.9e-35:
		tmp = (math.sqrt((-4.0 * (a * c))) + b) / (-2.0 * a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e-109)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.9e-35)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.8e-109)
		tmp = c / -b;
	elseif (b <= 1.9e-35)
		tmp = (sqrt((-4.0 * (a * c))) + b) / (-2.0 * a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e-109], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.9e-35], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.80000000000000046e-109

   1. Initial program 19.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -7.80000000000000046e-109 < b < 1.9000000000000001e-35

   1. Initial program 85.5%

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 79.5

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

   7. Applied rewrites 79.5%

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

   if 1.9000000000000001e-35 < b

   1. Initial program 60.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 91.6

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

   5. Applied rewrites 91.6%

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

Alternative 5: 68.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left|b\right|}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-307) (/ c (- b)) (/ (- (fabs b)) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-307) {
		tmp = c / -b;
	} else {
		tmp = -fabs(b) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.8d-307)) then
        tmp = c / -b
    else
        tmp = -abs(b) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-307) {
		tmp = c / -b;
	} else {
		tmp = -Math.abs(b) / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e-307:
		tmp = c / -b
	else:
		tmp = -math.fabs(b) / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-307)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-abs(b)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e-307)
		tmp = c / -b;
	else
		tmp = -abs(b) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-307], N[(c / (-b)), $MachinePrecision], N[((-N[Abs[b], $MachinePrecision]) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.8e-307

   1. Initial program 37.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 63.0

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

   5. Applied rewrites 63.0%

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

   if -2.8e-307 < b

   1. Initial program 69.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.3

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

   5. Applied rewrites 67.3%

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

   7. Applied rewrites 67.3%

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

Alternative 6: 68.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 37.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 70.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.9

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

   5. Applied rewrites 67.9%

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

Alternative 7: 35.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 35.5

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

5. Applied rewrites 35.5%

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

Alternative 8: 10.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 30.9

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

5. Applied rewrites 30.9%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 10.0%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024339 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

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

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