quadm (p42, negative)

Percentage Accurate: 51.4% → 75.4%

Time: 11.4s

Alternatives: 7

Speedup: 1.0×

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: 51.4% 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: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\\ \mathbf{if}\;b \leq -7 \cdot 10^{-81}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\left(b - t\_0\right) \cdot \left(a \cdot -2\right)}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{t\_0}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* c (* a -4.0))))))
   (if (<= b -7e-81)
     (/ (* 4.0 (* a c)) (* (- b t_0) (* a -2.0)))
     (if (<= b 1.1e+45) (fma b (/ -0.5 a) (/ t_0 (* a -2.0))) (- (/ b a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (c * (a * -4.0))));
	double tmp;
	if (b <= -7e-81) {
		tmp = (4.0 * (a * c)) / ((b - t_0) * (a * -2.0));
	} else if (b <= 1.1e+45) {
		tmp = fma(b, (-0.5 / a), (t_0 / (a * -2.0)));
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))
	tmp = 0.0
	if (b <= -7e-81)
		tmp = Float64(Float64(4.0 * Float64(a * c)) / Float64(Float64(b - t_0) * Float64(a * -2.0)));
	elseif (b <= 1.1e+45)
		tmp = fma(b, Float64(-0.5 / a), Float64(t_0 / Float64(a * -2.0)));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -7e-81], N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b - t$95$0), $MachinePrecision] * N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e+45], N[(b * N[(-0.5 / a), $MachinePrecision] + N[(t$95$0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.99999999999999973e-81

   1. Initial program 16.5%

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

   4. Applied egg-rr 7.9%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 63.2

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

   7. Simplified 63.2%

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

   if -6.99999999999999973e-81 < b < 1.1e45

   1. Initial program 87.6%

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

   4. Applied egg-rr 87.7%

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

   if 1.1e45 < b

   1. Initial program 60.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. rgt-mult-inverse N/A

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

      16. distribute-lft-out N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 40.6

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

   5. Simplified 40.6%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 48.5

         \[\leadsto \frac{\color{blue}{-0.5 \cdot b}}{a} \]

   8. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot b}{a}} \]

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 95.3

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

   11. Simplified 95.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-81}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \left(a \cdot -2\right)}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{0.5}{\frac{a}{0}}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.7e+88)
   (/ 0.5 (/ a 0.0))
   (if (<= b 1.1e+45)
     (fma (/ b a) -0.5 (/ (sqrt (fma b b (* c (* a -4.0)))) (* a -2.0)))
     (- (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.7e+88) {
		tmp = 0.5 / (a / 0.0);
	} else if (b <= 1.1e+45) {
		tmp = fma((b / a), -0.5, (sqrt(fma(b, b, (c * (a * -4.0)))) / (a * -2.0)));
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.7e+88)
		tmp = Float64(0.5 / Float64(a / 0.0));
	elseif (b <= 1.1e+45)
		tmp = fma(Float64(b / a), -0.5, Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0)));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.7e+88], N[(0.5 / N[(a / 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e+45], N[(N[(b / a), $MachinePrecision] * -0.5 + N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.70000000000000002e88

   1. Initial program 7.4%

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

   4. Applied egg-rr 1.9%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{{b}^{3}}}} \]
   6. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 1.7

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

   7. Simplified 1.7%

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

   8. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\color{blue}{0}}} \]
   9. Step-by-step derivation

   10. Simplified 35.5%

       \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{0}}} \]

   if -1.70000000000000002e88 < b < 1.1e45

   1. Initial program 73.0%

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

   4. Applied egg-rr 73.0%

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

   if 1.1e45 < b

   1. Initial program 60.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. rgt-mult-inverse N/A

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

      16. distribute-lft-out N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 40.6

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

   5. Simplified 40.6%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 48.5

         \[\leadsto \frac{\color{blue}{-0.5 \cdot b}}{a} \]

   8. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot b}{a}} \]

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 95.3

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

   11. Simplified 95.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{0.5}{\frac{a}{0}}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, a, a\right), a\right)}{b}}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+154)
   (/ 0.5 (/ (fma b (fma b a a) a) b))
   (if (<= b 1.1e+45)
     (/ (+ b (sqrt (fma b b (* c (* a -4.0))))) (* a -2.0))
     (- (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e+154) {
		tmp = 0.5 / (fma(b, fma(b, a, a), a) / b);
	} else if (b <= 1.1e+45) {
		tmp = (b + sqrt(fma(b, b, (c * (a * -4.0))))) / (a * -2.0);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(0.5 / Float64(fma(b, fma(b, a, a), a) / b));
	elseif (b <= 1.1e+45)
		tmp = Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))) / Float64(a * -2.0));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+154], N[(0.5 / N[(N[(b * N[(b * a + a), $MachinePrecision] + a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e+45], N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 1.6%

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

   4. Applied egg-rr 1.6%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{{b}^{3}}}} \]
   6. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 1.7

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

   7. Simplified 1.7%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 41.3

          \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, a, a\right)}, a\right)}{b}} \]

   10. Simplified 41.3%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, a, a\right), a\right)}{b}}} \]

   if -1.35000000000000003e154 < b < 1.1e45

   1. Initial program 66.5%

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

   4. Applied egg-rr 66.5%

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

   if 1.1e45 < b

   1. Initial program 60.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. rgt-mult-inverse N/A

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

      16. distribute-lft-out N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 40.6

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

   5. Simplified 40.6%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 48.5

         \[\leadsto \frac{\color{blue}{-0.5 \cdot b}}{a} \]

   8. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot b}{a}} \]

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 95.3

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

   11. Simplified 95.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, a, a\right), a\right)}{b}}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, a, a\right), a\right)}{b}}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+154)
   (/ 0.5 (/ (fma b (fma b a a) a) b))
   (if (<= b 1.1e+45)
     (* (/ -0.5 a) (+ b (sqrt (fma b b (* c (* a -4.0))))))
     (- (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e+154) {
		tmp = 0.5 / (fma(b, fma(b, a, a), a) / b);
	} else if (b <= 1.1e+45) {
		tmp = (-0.5 / a) * (b + sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(0.5 / Float64(fma(b, fma(b, a, a), a) / b));
	elseif (b <= 1.1e+45)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+154], N[(0.5 / N[(N[(b * N[(b * a + a), $MachinePrecision] + a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e+45], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 1.6%

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

   4. Applied egg-rr 1.6%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{{b}^{3}}}} \]
   6. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 1.7

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

   7. Simplified 1.7%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 41.3

          \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, a, a\right)}, a\right)}{b}} \]

   10. Simplified 41.3%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, a, a\right), a\right)}{b}}} \]

   if -1.35000000000000003e154 < b < 1.1e45

   1. Initial program 66.5%

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

   4. Applied egg-rr 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 66.4

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

   6. Applied egg-rr 66.4%

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

   if 1.1e45 < b

   1. Initial program 60.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. rgt-mult-inverse N/A

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

      16. distribute-lft-out N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 40.6

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

   5. Simplified 40.6%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 48.5

         \[\leadsto \frac{\color{blue}{-0.5 \cdot b}}{a} \]

   8. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot b}{a}} \]

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 95.3

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

   11. Simplified 95.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, a, a\right), a\right)}{b}}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.5}{\frac{a}{0}}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e+32)
   (/ 0.5 (/ a 0.0))
   (if (<= b 1.25e-123)
     (/ (+ b (sqrt (* (* a c) -4.0))) (* a -2.0))
     (- (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e+32) {
		tmp = 0.5 / (a / 0.0);
	} else if (b <= 1.25e-123) {
		tmp = (b + sqrt(((a * c) * -4.0))) / (a * -2.0);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.2d+32)) then
        tmp = 0.5d0 / (a / 0.0d0)
    else if (b <= 1.25d-123) then
        tmp = (b + sqrt(((a * c) * (-4.0d0)))) / (a * (-2.0d0))
    else
        tmp = -(b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e+32) {
		tmp = 0.5 / (a / 0.0);
	} else if (b <= 1.25e-123) {
		tmp = (b + Math.sqrt(((a * c) * -4.0))) / (a * -2.0);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e+32:
		tmp = 0.5 / (a / 0.0)
	elif b <= 1.25e-123:
		tmp = (b + math.sqrt(((a * c) * -4.0))) / (a * -2.0)
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e+32)
		tmp = Float64(0.5 / Float64(a / 0.0));
	elseif (b <= 1.25e-123)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(a * c) * -4.0))) / Float64(a * -2.0));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.2e+32)
		tmp = 0.5 / (a / 0.0);
	elseif (b <= 1.25e-123)
		tmp = (b + sqrt(((a * c) * -4.0))) / (a * -2.0);
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e+32], N[(0.5 / N[(a / 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-123], N[(N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -4.2000000000000001e32

   1. Initial program 8.5%

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

   4. Applied egg-rr 2.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{{b}^{3}}}} \]
   6. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 1.8

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

   7. Simplified 1.8%

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

   8. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\color{blue}{0}}} \]
   9. Step-by-step derivation

   10. Simplified 32.9%

       \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{0}}} \]

   if -4.2000000000000001e32 < b < 1.25000000000000007e-123

   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

   4. Applied egg-rr 70.5%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 66.0

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

   7. Simplified 66.0%

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

   if 1.25000000000000007e-123 < b

   1. Initial program 70.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. rgt-mult-inverse N/A

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

      16. distribute-lft-out N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 34.5

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

   5. Simplified 34.5%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 40.9

         \[\leadsto \frac{\color{blue}{-0.5 \cdot b}}{a} \]

   8. Simplified 40.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot b}{a}} \]

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 90.9

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

   11. Simplified 90.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.5}{\frac{a}{0}}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-305}:\\ \;\;\;\;\frac{0.5}{\frac{a}{0}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e-305) (/ 0.5 (/ a 0.0)) (- (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-305) {
		tmp = 0.5 / (a / 0.0);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.95d-305)) then
        tmp = 0.5d0 / (a / 0.0d0)
    else
        tmp = -(b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-305) {
		tmp = 0.5 / (a / 0.0);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e-305:
		tmp = 0.5 / (a / 0.0)
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e-305)
		tmp = Float64(0.5 / Float64(a / 0.0));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.95e-305)
		tmp = 0.5 / (a / 0.0);
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e-305], N[(0.5 / N[(a / 0.0), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < -1.95000000000000013e-305

   1. Initial program 32.8%

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

   4. Applied egg-rr 27.2%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{{b}^{3}}}} \]
   6. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.1

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

   7. Simplified 3.1%

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

   8. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\color{blue}{0}}} \]
   9. Step-by-step derivation

   10. Simplified 20.1%

       \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{0}}} \]

   if -1.95000000000000013e-305 < b

   1. Initial program 73.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. rgt-mult-inverse N/A

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

      16. distribute-lft-out N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 28.5

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

   5. Simplified 28.5%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 33.6

         \[\leadsto \frac{\color{blue}{-0.5 \cdot b}}{a} \]

   8. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot b}{a}} \]

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 74.6

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

   11. Simplified 74.6%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-305}:\\ \;\;\;\;\frac{0.5}{\frac{a}{0}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. *-lft-identity N/A

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

   5. lft-mult-inverse N/A

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

   6. associate-*r* N/A

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

   7. unpow2 N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. distribute-lft-in N/A

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

   14. *-rgt-identity N/A

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

   15. rgt-mult-inverse N/A

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

   16. distribute-lft-out N/A

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

   17. *-rgt-identity N/A

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

   18. lower-fma.f64 16.2

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

5. Simplified 16.2%

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

6. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 18.9

      \[\leadsto \frac{\color{blue}{-0.5 \cdot b}}{a} \]

8. Simplified 18.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot b}{a}} \]

9. Taylor expanded in b around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 40.4

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

11. Simplified 40.4%

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

12. Final simplification 40.4%

    \[\leadsto -\frac{b}{a} \]
13. 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 2024214 
(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)))