The quadratic formula (r2)

Percentage Accurate: 51.9% → 87.0%

Time: 8.8s

Alternatives: 10

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: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(4, a \cdot c, 0\right)}{-2 \cdot a}}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b}, c, \frac{-b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+63)
   (/ c (- b))
   (if (<= b -1.35e-147)
     (/
      (/ (fma 4.0 (* a c) 0.0) (* -2.0 a))
      (- b (sqrt (fma (* a c) -4.0 (* b b)))))
     (if (<= b 5e+92)
       (/ (+ (sqrt (fma -4.0 (* c a) (* b b))) b) (* -2.0 a))
       (fma (/ (fma (/ a b) (/ c b) 1.0) b) c (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+63) {
		tmp = c / -b;
	} else if (b <= -1.35e-147) {
		tmp = (fma(4.0, (a * c), 0.0) / (-2.0 * a)) / (b - sqrt(fma((a * c), -4.0, (b * b))));
	} else if (b <= 5e+92) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = fma((fma((a / b), (c / b), 1.0) / b), c, (-b / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+63)
		tmp = Float64(c / Float64(-b));
	elseif (b <= -1.35e-147)
		tmp = Float64(Float64(fma(4.0, Float64(a * c), 0.0) / Float64(-2.0 * a)) / Float64(b - sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))));
	elseif (b <= 5e+92)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(fma(Float64(a / b), Float64(c / b), 1.0) / b), c, Float64(Float64(-b) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+63], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, -1.35e-147], N[(N[(N[(4.0 * N[(a * c), $MachinePrecision] + 0.0), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision] / N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+92], 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[(N[(N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision] * c + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.50000000000000005e63

   1. Initial program 3.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.50000000000000005e63 < b < -1.35e-147

   1. Initial program 50.0%

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, 4, b \cdot b\right) - b \cdot b}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(-2 \cdot a\right)}} \]
   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 50.1

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

      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. lower-*.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 79.7

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

   6. Applied rewrites 79.7%

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

   if -1.35e-147 < b < 5.00000000000000022e92

   1. Initial program 77.0%

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

   4. Applied rewrites 77.0%

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

   if 5.00000000000000022e92 < b

   1. Initial program 46.1%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 96.9

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

   5. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b}, c, \frac{-b}{a}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.04 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{\left(b - t\_0\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+92}:\\ \;\;\;\;\frac{t\_0 + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b}, c, \frac{-b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma -4.0 (* c a) (* b b)))))
   (if (<= b -1.6e+63)
     (/ c (- b))
     (if (<= b -1.04e-69)
       (/ (* (* a c) 4.0) (* (- b t_0) (* -2.0 a)))
       (if (<= b 5e+92)
         (/ (+ t_0 b) (* -2.0 a))
         (fma (/ (fma (/ a b) (/ c b) 1.0) b) c (/ (- b) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(-4.0, (c * a), (b * b)));
	double tmp;
	if (b <= -1.6e+63) {
		tmp = c / -b;
	} else if (b <= -1.04e-69) {
		tmp = ((a * c) * 4.0) / ((b - t_0) * (-2.0 * a));
	} else if (b <= 5e+92) {
		tmp = (t_0 + b) / (-2.0 * a);
	} else {
		tmp = fma((fma((a / b), (c / b), 1.0) / b), c, (-b / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))
	tmp = 0.0
	if (b <= -1.6e+63)
		tmp = Float64(c / Float64(-b));
	elseif (b <= -1.04e-69)
		tmp = Float64(Float64(Float64(a * c) * 4.0) / Float64(Float64(b - t_0) * Float64(-2.0 * a)));
	elseif (b <= 5e+92)
		tmp = Float64(Float64(t_0 + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(fma(Float64(a / b), Float64(c / b), 1.0) / b), c, Float64(Float64(-b) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.6e+63], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, -1.04e-69], N[(N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision] / N[(N[(b - t$95$0), $MachinePrecision] * N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+92], N[(N[(t$95$0 + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision] * c + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.60000000000000006e63

   1. Initial program 3.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.60000000000000006e63 < b < -1.04000000000000001e-69

   1. Initial program 40.4%

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

   4. Applied rewrites 40.3%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 79.3

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

   7. Applied rewrites 79.3%

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

   if -1.04000000000000001e-69 < b < 5.00000000000000022e92

   1. Initial program 75.4%

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

   4. Applied rewrites 75.4%

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

   if 5.00000000000000022e92 < b

   1. Initial program 46.1%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 96.9

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

   5. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b}, c, \frac{-b}{a}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-67}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b}, c, \frac{-b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e-67)
   (/ c (- b))
   (if (<= b 5e+92)
     (/ (+ (sqrt (fma -4.0 (* c a) (* b b))) b) (* -2.0 a))
     (fma (/ (fma (/ a b) (/ c b) 1.0) b) c (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e-67) {
		tmp = c / -b;
	} else if (b <= 5e+92) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = fma((fma((a / b), (c / b), 1.0) / b), c, (-b / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e-67)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5e+92)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(fma(Float64(a / b), Float64(c / b), 1.0) / b), c, Float64(Float64(-b) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e-67], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5e+92], 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[(N[(N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision] * c + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.15000000000000013e-67

   1. Initial program 15.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.2

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

   5. Applied rewrites 88.2%

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

   if -2.15000000000000013e-67 < b < 5.00000000000000022e92

   1. Initial program 75.4%

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

   4. Applied rewrites 75.4%

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

   if 5.00000000000000022e92 < b

   1. Initial program 46.1%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 96.9

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

   5. Applied rewrites 96.9%

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

Alternative 4: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-67}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e-67)
   (/ c (- b))
   (if (<= b 5e+92)
     (/ (+ (sqrt (fma -4.0 (* c a) (* b b))) b) (* -2.0 a))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e-67) {
		tmp = c / -b;
	} else if (b <= 5e+92) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e-67)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5e+92)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e-67], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5e+92], 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[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.15000000000000013e-67

   1. Initial program 15.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.2

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

   5. Applied rewrites 88.2%

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

   if -2.15000000000000013e-67 < b < 5.00000000000000022e92

   1. Initial program 75.4%

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

   4. Applied rewrites 75.4%

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

   if 5.00000000000000022e92 < b

   1. Initial program 46.1%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-67}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -4\right) \cdot c} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e-67)
   (/ c (- b))
   (if (<= b 2.5e-46)
     (/ (+ (sqrt (* (* a -4.0) c)) b) (* -2.0 a))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e-67) {
		tmp = c / -b;
	} else if (b <= 2.5e-46) {
		tmp = (sqrt(((a * -4.0) * c)) + b) / (-2.0 * a);
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e-67)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 2.5e-46)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * -4.0) * c)) + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e-67], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 2.5e-46], N[(N[(N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.15000000000000013e-67

   1. Initial program 15.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.2

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

   5. Applied rewrites 88.2%

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

   if -2.15000000000000013e-67 < b < 2.49999999999999996e-46

   1. Initial program 71.8%

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

   4. Applied rewrites 71.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 71.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.8

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

   6. Applied rewrites 71.8%

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

   7. Taylor expanded in a 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 66.1

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

   9. Applied rewrites 66.1%

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

   11. Applied rewrites 66.2%

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

   if 2.49999999999999996e-46 < b

   1. Initial program 58.0%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 6: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-67}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e-67)
   (/ c (- b))
   (if (<= b 2.5e-46)
     (/ (+ (sqrt (* -4.0 (* a c))) b) (* -2.0 a))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e-67) {
		tmp = c / -b;
	} else if (b <= 2.5e-46) {
		tmp = (sqrt((-4.0 * (a * c))) + b) / (-2.0 * a);
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e-67)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 2.5e-46)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e-67], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 2.5e-46], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.15000000000000013e-67

   1. Initial program 15.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.2

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

   5. Applied rewrites 88.2%

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

   if -2.15000000000000013e-67 < b < 2.49999999999999996e-46

   1. Initial program 71.8%

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

   4. Applied rewrites 71.8%

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

   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 66.1

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

   7. Applied rewrites 66.1%

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

   if 2.49999999999999996e-46 < b

   1. Initial program 58.0%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-67}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ c (- b)) (fma (/ b a) -1.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(c / (-b)), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 30.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 67.5

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

   5. Applied rewrites 67.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 66.1%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.9

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

   5. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.8% 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 30.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 67.5

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

   5. Applied rewrites 67.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 66.1%

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

   3. Taylor expanded in 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 61.4

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

   5. Applied rewrites 61.4%

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

Alternative 9: 35.1% 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 49.4%

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

3. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 32.4

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

5. Applied rewrites 32.4%

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

Alternative 10: 11.1% 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 49.4%

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

4. Applied rewrites 49.4%

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

5. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 34.1

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

7. Applied rewrites 34.1%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 10.3%

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

Developer Target 1: 70.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024343 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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