quad2m (problem 3.2.1, negative)

Percentage Accurate: 53.3% → 88.9%

Time: 9.9s

Alternatives: 10

Speedup: 15.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 53.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \frac{a}{b_2}\right)\right)}\\ \mathbf{elif}\;b_2 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{a}{a} \cdot \frac{c}{\mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right) - b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+127}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e+32)
   (* (/ a a) (/ c (fma b_2 -2.0 (* c (* 0.5 (/ a b_2))))))
   (if (<= b_2 -2e-309)
     (* (/ a a) (/ c (- (hypot b_2 (sqrt (* a (- c)))) b_2)))
     (if (<= b_2 1e+127)
       (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
       (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e+32) {
		tmp = (a / a) * (c / fma(b_2, -2.0, (c * (0.5 * (a / b_2)))));
	} else if (b_2 <= -2e-309) {
		tmp = (a / a) * (c / (hypot(b_2, sqrt((a * -c))) - b_2));
	} else if (b_2 <= 1e+127) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e+32)
		tmp = Float64(Float64(a / a) * Float64(c / fma(b_2, -2.0, Float64(c * Float64(0.5 * Float64(a / b_2))))));
	elseif (b_2 <= -2e-309)
		tmp = Float64(Float64(a / a) * Float64(c / Float64(hypot(b_2, sqrt(Float64(a * Float64(-c)))) - b_2)));
	elseif (b_2 <= 1e+127)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e+32], N[(N[(a / a), $MachinePrecision] * N[(c / N[(b$95$2 * -2.0 + N[(c * N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -2e-309], N[(N[(a / a), $MachinePrecision] * N[(c / N[(N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1e+127], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.90000000000000003e32

   1. Initial program 8.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fma-neg 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fma-neg 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. *-commutative 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. fma-udef 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. distribute-lft-neg-in 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. *-commutative 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. distribute-rgt-neg-in 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. fma-def 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. *-commutative 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

      15. fma-udef 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}\right)}}{a} \]

      16. distribute-lft-neg-in 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)\right)}}{a} \]

      17. *-commutative 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)\right)}}{a} \]

      18. distribute-rgt-neg-in 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)\right)}}{a} \]

      19. fma-def 8.2%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}\right)}}{a} \]

   3. Applied egg-rr 8.2%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      2. count-2 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

      3. *-commutative 8.2%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)}}{a} \]

   5. Simplified 8.2%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
   6. Step-by-step derivation

      1. flip-- 7.1%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)} \cdot \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}}{a} \]

      2. add-sqr-sqrt 7.1%

         \[\leadsto \frac{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \color{blue}{\left(\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)\right)}}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

      3. *-commutative 7.1%

         \[\leadsto \frac{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \left(\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2}\right)}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

      4. *-commutative 7.1%

         \[\leadsto \frac{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \left(\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2\right)}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2}}}}{a} \]

   7. Applied egg-rr 7.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \left(\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2\right)}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2}}}}{a} \]

   9. Simplified 59.9%

      \[\leadsto \frac{\color{blue}{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{a} \]
   10. Step-by-step derivation

       1. div-inv 59.9%

          \[\leadsto \color{blue}{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \cdot \frac{1}{a}} \]

       2. sub-neg 59.9%

          \[\leadsto \frac{c \cdot a}{\sqrt{\color{blue}{b_2 \cdot b_2 + \left(-c \cdot a\right)}} - b_2} \cdot \frac{1}{a} \]

       3. distribute-rgt-neg-out 59.9%

          \[\leadsto \frac{c \cdot a}{\sqrt{b_2 \cdot b_2 + \color{blue}{c \cdot \left(-a\right)}} - b_2} \cdot \frac{1}{a} \]

       4. add-sqr-sqrt 35.2%

          \[\leadsto \frac{c \cdot a}{\sqrt{b_2 \cdot b_2 + \color{blue}{\sqrt{c \cdot \left(-a\right)} \cdot \sqrt{c \cdot \left(-a\right)}}} - b_2} \cdot \frac{1}{a} \]

       5. hypot-udef 51.1%

          \[\leadsto \frac{c \cdot a}{\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)} - b_2} \cdot \frac{1}{a} \]

   11. Applied egg-rr 51.1%

       \[\leadsto \color{blue}{\frac{c \cdot a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2} \cdot \frac{1}{a}} \]
   12. Step-by-step derivation

       1. times-frac 48.9%

          \[\leadsto \color{blue}{\frac{\left(c \cdot a\right) \cdot 1}{\left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right) \cdot a}} \]

       2. *-rgt-identity 48.9%

          \[\leadsto \frac{\color{blue}{c \cdot a}}{\left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right) \cdot a} \]

       3. *-commutative 48.9%

          \[\leadsto \frac{\color{blue}{a \cdot c}}{\left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right) \cdot a} \]

       4. *-commutative 48.9%

          \[\leadsto \frac{a \cdot c}{\color{blue}{a \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right)}} \]

       5. times-frac 54.8%

          \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{c}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}} \]

   13. Simplified 54.8%

       \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{c}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}} \]

   14. Taylor expanded in b_2 around -inf 0.0%

       \[\leadsto \frac{a}{a} \cdot \frac{c}{\color{blue}{-2 \cdot b_2 + -0.5 \cdot \frac{c \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot a\right)}{b_2}}} \]
   15. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\color{blue}{b_2 \cdot -2} + -0.5 \cdot \frac{c \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot a\right)}{b_2}} \]

       2. fma-def 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(b_2, -2, -0.5 \cdot \frac{c \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot a\right)}{b_2}\right)}} \]

       3. associate-*r/ 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{\frac{-0.5 \cdot \left(c \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot a\right)\right)}{b_2}}\right)} \]

       4. associate-*r* 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{-0.5 \cdot \color{blue}{\left(\left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot a\right)}}{b_2}\right)} \]

       5. associate-*r* 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\color{blue}{\left(-0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot a}}{b_2}\right)} \]

       6. unpow2 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\left(-0.5 \cdot \left(c \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right)\right) \cdot a}{b_2}\right)} \]

       7. rem-square-sqrt 96.1%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\left(-0.5 \cdot \left(c \cdot \color{blue}{-1}\right)\right) \cdot a}{b_2}\right)} \]

       8. associate-*l* 96.1%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\color{blue}{\left(\left(-0.5 \cdot c\right) \cdot -1\right)} \cdot a}{b_2}\right)} \]

       9. *-commutative 96.1%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\left(\color{blue}{\left(c \cdot -0.5\right)} \cdot -1\right) \cdot a}{b_2}\right)} \]

       10. associate-*l* 96.1%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\color{blue}{\left(c \cdot \left(-0.5 \cdot -1\right)\right)} \cdot a}{b_2}\right)} \]

       11. metadata-eval 96.1%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\left(c \cdot \color{blue}{0.5}\right) \cdot a}{b_2}\right)} \]

       12. associate-/l* 98.7%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{\frac{c \cdot 0.5}{\frac{b_2}{a}}}\right)} \]

       13. *-rgt-identity 98.7%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\color{blue}{\left(c \cdot 0.5\right) \cdot 1}}{\frac{b_2}{a}}\right)} \]

       14. associate-*r/ 98.7%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{\left(c \cdot 0.5\right) \cdot \frac{1}{\frac{b_2}{a}}}\right)} \]

       15. associate-*l* 98.7%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{c \cdot \left(0.5 \cdot \frac{1}{\frac{b_2}{a}}\right)}\right)} \]

       16. associate-/r/ 98.7%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \color{blue}{\left(\frac{1}{b_2} \cdot a\right)}\right)\right)} \]

       17. *-commutative 98.7%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \color{blue}{\left(a \cdot \frac{1}{b_2}\right)}\right)\right)} \]

       18. associate-*r/ 98.7%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \color{blue}{\frac{a \cdot 1}{b_2}}\right)\right)} \]

       19. *-rgt-identity 98.7%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \frac{\color{blue}{a}}{b_2}\right)\right)} \]

   16. Simplified 98.7%

       \[\leadsto \frac{a}{a} \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \frac{a}{b_2}\right)\right)}} \]

   if -2.90000000000000003e32 < b_2 < -1.9999999999999988e-309

   1. Initial program 59.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 59.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 59.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fma-neg 59.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 59.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 59.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fma-neg 59.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 59.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. *-commutative 59.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. fma-udef 59.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. distribute-lft-neg-in 59.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. *-commutative 59.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. distribute-rgt-neg-in 59.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. fma-def 59.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. *-commutative 59.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

      15. fma-udef 59.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}\right)}}{a} \]

      16. distribute-lft-neg-in 59.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)\right)}}{a} \]

      17. *-commutative 59.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)\right)}}{a} \]

      18. distribute-rgt-neg-in 59.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)\right)}}{a} \]

      19. fma-def 59.0%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}\right)}}{a} \]

   3. Applied egg-rr 59.0%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 59.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      2. count-2 59.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

      3. *-commutative 59.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)}}{a} \]

   5. Simplified 59.0%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
   6. Step-by-step derivation

      1. flip-- 58.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)} \cdot \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}}{a} \]

      2. add-sqr-sqrt 59.0%

         \[\leadsto \frac{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \color{blue}{\left(\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)\right)}}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

      3. *-commutative 59.0%

         \[\leadsto \frac{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \left(\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2}\right)}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

      4. *-commutative 59.0%

         \[\leadsto \frac{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \left(\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2\right)}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2}}}}{a} \]

   7. Applied egg-rr 59.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \left(\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2\right)}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2}}}}{a} \]

   9. Simplified 71.7%

      \[\leadsto \frac{\color{blue}{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{a} \]
   10. Step-by-step derivation

       1. div-inv 71.5%

          \[\leadsto \color{blue}{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \cdot \frac{1}{a}} \]

       2. sub-neg 71.5%

          \[\leadsto \frac{c \cdot a}{\sqrt{\color{blue}{b_2 \cdot b_2 + \left(-c \cdot a\right)}} - b_2} \cdot \frac{1}{a} \]

       3. distribute-rgt-neg-out 71.5%

          \[\leadsto \frac{c \cdot a}{\sqrt{b_2 \cdot b_2 + \color{blue}{c \cdot \left(-a\right)}} - b_2} \cdot \frac{1}{a} \]

       4. add-sqr-sqrt 69.8%

          \[\leadsto \frac{c \cdot a}{\sqrt{b_2 \cdot b_2 + \color{blue}{\sqrt{c \cdot \left(-a\right)} \cdot \sqrt{c \cdot \left(-a\right)}}} - b_2} \cdot \frac{1}{a} \]

       5. hypot-udef 69.8%

          \[\leadsto \frac{c \cdot a}{\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)} - b_2} \cdot \frac{1}{a} \]

   11. Applied egg-rr 69.8%

       \[\leadsto \color{blue}{\frac{c \cdot a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2} \cdot \frac{1}{a}} \]
   12. Step-by-step derivation

       1. times-frac 63.8%

          \[\leadsto \color{blue}{\frac{\left(c \cdot a\right) \cdot 1}{\left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right) \cdot a}} \]

       2. *-rgt-identity 63.8%

          \[\leadsto \frac{\color{blue}{c \cdot a}}{\left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right) \cdot a} \]

       3. *-commutative 63.8%

          \[\leadsto \frac{\color{blue}{a \cdot c}}{\left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right) \cdot a} \]

       4. *-commutative 63.8%

          \[\leadsto \frac{a \cdot c}{\color{blue}{a \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right)}} \]

       5. times-frac 85.4%

          \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{c}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}} \]

   13. Simplified 85.4%

       \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{c}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}} \]

   if -1.9999999999999988e-309 < b_2 < 9.99999999999999955e126

   1. Initial program 85.8%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 9.99999999999999955e126 < b_2

   1. Initial program 45.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 97.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \frac{a}{b_2}\right)\right)}\\ \mathbf{elif}\;b_2 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{a}{a} \cdot \frac{c}{\mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right) - b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+127}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -9.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \frac{a}{b_2}\right)\right)}\\ \mathbf{elif}\;b_2 \leq 7.7 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.8e-79)
   (* (/ a a) (/ c (fma b_2 -2.0 (* c (* 0.5 (/ a b_2))))))
   (if (<= b_2 7.7e+127)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.8e-79) {
		tmp = (a / a) * (c / fma(b_2, -2.0, (c * (0.5 * (a / b_2)))));
	} else if (b_2 <= 7.7e+127) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9.8e-79)
		tmp = Float64(Float64(a / a) * Float64(c / fma(b_2, -2.0, Float64(c * Float64(0.5 * Float64(a / b_2))))));
	elseif (b_2 <= 7.7e+127)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9.8e-79], N[(N[(a / a), $MachinePrecision] * N[(c / N[(b$95$2 * -2.0 + N[(c * N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.7e+127], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.8000000000000001e-79

   1. Initial program 16.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fma-neg 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fma-neg 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. *-commutative 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. fma-udef 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. distribute-lft-neg-in 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. *-commutative 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. distribute-rgt-neg-in 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. fma-def 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. *-commutative 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

      15. fma-udef 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}\right)}}{a} \]

      16. distribute-lft-neg-in 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)\right)}}{a} \]

      17. *-commutative 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)\right)}}{a} \]

      18. distribute-rgt-neg-in 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)\right)}}{a} \]

      19. fma-def 16.1%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}\right)}}{a} \]

   3. Applied egg-rr 16.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      2. count-2 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

      3. *-commutative 16.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)}}{a} \]

   5. Simplified 16.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
   6. Step-by-step derivation

      1. flip-- 15.3%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)} \cdot \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}}{a} \]

      2. add-sqr-sqrt 15.3%

         \[\leadsto \frac{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \color{blue}{\left(\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)\right)}}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

      3. *-commutative 15.3%

         \[\leadsto \frac{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \left(\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2}\right)}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

      4. *-commutative 15.3%

         \[\leadsto \frac{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \left(\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2\right)}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2}}}}{a} \]

   7. Applied egg-rr 15.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \left(\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2\right)}{\left(-b_2\right) + \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2}}}}{a} \]

   9. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{a} \]
   10. Step-by-step derivation

       1. div-inv 61.6%

          \[\leadsto \color{blue}{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \cdot \frac{1}{a}} \]

       2. sub-neg 61.6%

          \[\leadsto \frac{c \cdot a}{\sqrt{\color{blue}{b_2 \cdot b_2 + \left(-c \cdot a\right)}} - b_2} \cdot \frac{1}{a} \]

       3. distribute-rgt-neg-out 61.6%

          \[\leadsto \frac{c \cdot a}{\sqrt{b_2 \cdot b_2 + \color{blue}{c \cdot \left(-a\right)}} - b_2} \cdot \frac{1}{a} \]

       4. add-sqr-sqrt 42.5%

          \[\leadsto \frac{c \cdot a}{\sqrt{b_2 \cdot b_2 + \color{blue}{\sqrt{c \cdot \left(-a\right)} \cdot \sqrt{c \cdot \left(-a\right)}}} - b_2} \cdot \frac{1}{a} \]

       5. hypot-udef 54.1%

          \[\leadsto \frac{c \cdot a}{\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)} - b_2} \cdot \frac{1}{a} \]

   11. Applied egg-rr 54.1%

       \[\leadsto \color{blue}{\frac{c \cdot a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2} \cdot \frac{1}{a}} \]
   12. Step-by-step derivation

       1. times-frac 49.7%

          \[\leadsto \color{blue}{\frac{\left(c \cdot a\right) \cdot 1}{\left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right) \cdot a}} \]

       2. *-rgt-identity 49.7%

          \[\leadsto \frac{\color{blue}{c \cdot a}}{\left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right) \cdot a} \]

       3. *-commutative 49.7%

          \[\leadsto \frac{\color{blue}{a \cdot c}}{\left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right) \cdot a} \]

       4. *-commutative 49.7%

          \[\leadsto \frac{a \cdot c}{\color{blue}{a \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right)}} \]

       5. times-frac 64.0%

          \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{c}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}} \]

   13. Simplified 64.0%

       \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{c}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}} \]

   14. Taylor expanded in b_2 around -inf 0.0%

       \[\leadsto \frac{a}{a} \cdot \frac{c}{\color{blue}{-2 \cdot b_2 + -0.5 \cdot \frac{c \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot a\right)}{b_2}}} \]
   15. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\color{blue}{b_2 \cdot -2} + -0.5 \cdot \frac{c \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot a\right)}{b_2}} \]

       2. fma-def 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(b_2, -2, -0.5 \cdot \frac{c \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot a\right)}{b_2}\right)}} \]

       3. associate-*r/ 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{\frac{-0.5 \cdot \left(c \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot a\right)\right)}{b_2}}\right)} \]

       4. associate-*r* 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{-0.5 \cdot \color{blue}{\left(\left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot a\right)}}{b_2}\right)} \]

       5. associate-*r* 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\color{blue}{\left(-0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot a}}{b_2}\right)} \]

       6. unpow2 0.0%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\left(-0.5 \cdot \left(c \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right)\right) \cdot a}{b_2}\right)} \]

       7. rem-square-sqrt 88.4%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\left(-0.5 \cdot \left(c \cdot \color{blue}{-1}\right)\right) \cdot a}{b_2}\right)} \]

       8. associate-*l* 88.4%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\color{blue}{\left(\left(-0.5 \cdot c\right) \cdot -1\right)} \cdot a}{b_2}\right)} \]

       9. *-commutative 88.4%

          \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\left(\color{blue}{\left(c \cdot -0.5\right)} \cdot -1\right) \cdot a}{b_2}\right)} \]

       10. associate-*l* 88.4%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\color{blue}{\left(c \cdot \left(-0.5 \cdot -1\right)\right)} \cdot a}{b_2}\right)} \]

       11. metadata-eval 88.4%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\left(c \cdot \color{blue}{0.5}\right) \cdot a}{b_2}\right)} \]

       12. associate-/l* 90.3%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{\frac{c \cdot 0.5}{\frac{b_2}{a}}}\right)} \]

       13. *-rgt-identity 90.3%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \frac{\color{blue}{\left(c \cdot 0.5\right) \cdot 1}}{\frac{b_2}{a}}\right)} \]

       14. associate-*r/ 90.3%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{\left(c \cdot 0.5\right) \cdot \frac{1}{\frac{b_2}{a}}}\right)} \]

       15. associate-*l* 90.3%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{c \cdot \left(0.5 \cdot \frac{1}{\frac{b_2}{a}}\right)}\right)} \]

       16. associate-/r/ 90.3%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \color{blue}{\left(\frac{1}{b_2} \cdot a\right)}\right)\right)} \]

       17. *-commutative 90.3%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \color{blue}{\left(a \cdot \frac{1}{b_2}\right)}\right)\right)} \]

       18. associate-*r/ 90.3%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \color{blue}{\frac{a \cdot 1}{b_2}}\right)\right)} \]

       19. *-rgt-identity 90.3%

           \[\leadsto \frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \frac{\color{blue}{a}}{b_2}\right)\right)} \]

   16. Simplified 90.3%

       \[\leadsto \frac{a}{a} \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \frac{a}{b_2}\right)\right)}} \]

   if -9.8000000000000001e-79 < b_2 < 7.6999999999999996e127

   1. Initial program 83.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 7.6999999999999996e127 < b_2

   1. Initial program 45.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 97.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{a}{a} \cdot \frac{c}{\mathsf{fma}\left(b_2, -2, c \cdot \left(0.5 \cdot \frac{a}{b_2}\right)\right)}\\ \mathbf{elif}\;b_2 \leq 7.7 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.5e-83)
   (/ (* c -0.5) b_2)
   (if (<= b_2 7e+127)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-83) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 7e+127) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.5d-83)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 7d+127) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-83) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 7e+127) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.5e-83:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 7e+127:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.5e-83)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 7e+127)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.5e-83)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 7e+127)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.5e-83], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7e+127], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.49999999999999997e-83

   1. Initial program 16.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 90.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 90.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 90.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -4.49999999999999997e-83 < b_2 < 6.99999999999999956e127

   1. Initial program 83.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 6.99999999999999956e127 < b_2

   1. Initial program 45.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 97.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-79)
   (/ (* c -0.5) b_2)
   (if (<= b_2 7.5e-57)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-79) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 7.5e-57) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.2d-79)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 7.5d-57) then
        tmp = (-b_2 - sqrt((a * -c))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-79) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 7.5e-57) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.2e-79:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 7.5e-57:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.2e-79)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 7.5e-57)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.2e-79)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 7.5e-57)
		tmp = (-b_2 - sqrt((a * -c))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.2e-79], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7.5e-57], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.1999999999999999e-79

   1. Initial program 16.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 90.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 90.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 90.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -4.1999999999999999e-79 < b_2 < 7.49999999999999973e-57

   1. Initial program 76.4%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 74.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}}}{a} \]
   3. Step-by-step derivation

      1. mul-1-neg 74.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-c \cdot a}}}{a} \]

      2. distribute-rgt-neg-out 74.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   4. Simplified 74.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   if 7.49999999999999973e-57 < b_2

   1. Initial program 69.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 89.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-80}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-80)
   (/ (* c -0.5) b_2)
   (if (<= b_2 8e-142)
     (/ (- (sqrt (* a (- c)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-80) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 8e-142) {
		tmp = -sqrt((a * -c)) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-80)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 8d-142) then
        tmp = -sqrt((a * -c)) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-80) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 8e-142) {
		tmp = -Math.sqrt((a * -c)) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-80:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 8e-142:
		tmp = -math.sqrt((a * -c)) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-80)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 8e-142)
		tmp = Float64(Float64(-sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-80)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 8e-142)
		tmp = -sqrt((a * -c)) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-80], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 8e-142], N[((-N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.99999999999999985e-80

   1. Initial program 16.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 90.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 90.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 90.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -3.99999999999999985e-80 < b_2 < 8.0000000000000003e-142

   1. Initial program 76.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 75.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 75.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fma-neg 75.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 75.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 75.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fma-neg 75.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 75.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. *-commutative 75.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. fma-udef 75.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. distribute-lft-neg-in 75.7%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. *-commutative 75.7%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. distribute-rgt-neg-in 75.7%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. fma-def 75.5%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. *-commutative 75.5%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

      15. fma-udef 75.7%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}\right)}}{a} \]

      16. distribute-lft-neg-in 75.7%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)\right)}}{a} \]

      17. *-commutative 75.7%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)\right)}}{a} \]

      18. distribute-rgt-neg-in 75.7%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)\right)}}{a} \]

      19. fma-def 75.5%

          \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}\right)}}{a} \]

   3. Applied egg-rr 75.5%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 75.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      2. count-2 75.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

      3. *-commutative 75.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)}}{a} \]

   5. Simplified 75.5%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

   6. Taylor expanded in b_2 around 0 74.3%

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

      1. mul-1-neg 74.3%

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

      2. distribute-lft1-in 74.3%

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

      3. metadata-eval 74.3%

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

      4. mul0-lft 74.7%

         \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{0} - c \cdot a}}{a} \]

      5. metadata-eval 74.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{0} - c \cdot a}}{a} \]

      6. neg-sub0 74.7%

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

      7. distribute-rgt-neg-out 74.7%

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

   8. Simplified 74.7%

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

   if 8.0000000000000003e-142 < b_2

   1. Initial program 69.7%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 87.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-80}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6: 67.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* c -0.5) b_2)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 31.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 69.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 69.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 71.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 68.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 7: 47.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{-302}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.5e-302) (* c (/ -0.5 b_2)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.5e-302) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.5d-302)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.5e-302) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.5e-302:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.5e-302)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.5e-302)
		tmp = c * (-0.5 / b_2);
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.5e-302], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.5000000000000001e-302

   1. Initial program 31.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. frac-2neg 31.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-a}} \]

      2. div-inv 31.0%

         \[\leadsto \color{blue}{\left(-\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a}} \]

   3. Applied egg-rr 40.3%

      \[\leadsto \color{blue}{\left(b_2 + \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right) \cdot \frac{1}{-a}} \]

   4. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 70.3%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 70.3%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 70.3%

         \[\leadsto \frac{\color{blue}{-0.5} \cdot c}{b_2} \]

      7. *-commutative 70.3%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

      8. *-rgt-identity 70.3%

         \[\leadsto \frac{\color{blue}{\left(c \cdot -0.5\right) \cdot 1}}{b_2} \]

      9. associate-*r/ 70.1%

         \[\leadsto \color{blue}{\left(c \cdot -0.5\right) \cdot \frac{1}{b_2}} \]

      10. associate-*l* 70.1%

          \[\leadsto \color{blue}{c \cdot \left(-0.5 \cdot \frac{1}{b_2}\right)} \]

      11. associate-*r/ 70.1%

          \[\leadsto c \cdot \color{blue}{\frac{-0.5 \cdot 1}{b_2}} \]

      12. metadata-eval 70.1%

          \[\leadsto c \cdot \frac{\color{blue}{-0.5}}{b_2} \]

   6. Simplified 70.1%

      \[\leadsto \color{blue}{c \cdot \frac{-0.5}{b_2}} \]

   if -3.5000000000000001e-302 < b_2

   1. Initial program 71.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 41.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}}}{a} \]
   3. Step-by-step derivation

      1. mul-1-neg 41.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-c \cdot a}}}{a} \]

      2. distribute-rgt-neg-out 41.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   4. Simplified 41.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   5. Taylor expanded in b_2 around inf 24.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg 24.0%

         \[\leadsto \color{blue}{-\frac{b_2}{a}} \]

      2. distribute-frac-neg 24.0%

         \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

   7. Simplified 24.0%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{-302}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 8: 47.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-303}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-303) (/ (* c -0.5) b_2) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-303) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7d-303)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-303) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7e-303:
		tmp = (c * -0.5) / b_2
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-303)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7e-303)
		tmp = (c * -0.5) / b_2;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-303], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7e-303

   1. Initial program 31.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 70.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.3%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 70.3%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -7e-303 < b_2

   1. Initial program 71.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 41.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}}}{a} \]
   3. Step-by-step derivation

      1. mul-1-neg 41.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-c \cdot a}}}{a} \]

      2. distribute-rgt-neg-out 41.1%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   4. Simplified 41.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   5. Taylor expanded in b_2 around inf 24.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg 24.0%

         \[\leadsto \color{blue}{-\frac{b_2}{a}} \]

      2. distribute-frac-neg 24.0%

         \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

   7. Simplified 24.0%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-303}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 9: 67.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-303}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-303) (/ (* c -0.5) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-303) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7d-303)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-303) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7e-303:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-303)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7e-303)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-303], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7e-303

   1. Initial program 31.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 70.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.3%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 70.3%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -7e-303 < b_2

   1. Initial program 71.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 66.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   3. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   4. Simplified 66.4%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-303}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 10: 15.4% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{-b_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (- b_2) a))

C

double code(double a, double b_2, double c) {
	return -b_2 / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = -b_2 / a
end function

Java

public static double code(double a, double b_2, double c) {
	return -b_2 / a;
}

Python

def code(a, b_2, c):
	return -b_2 / a

Julia

function code(a, b_2, c)
	return Float64(Float64(-b_2) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -b_2 / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[((-b$95$2) / a), $MachinePrecision]

Derivation

1. Initial program 50.5%

   \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

2. Taylor expanded in b_2 around 0 32.9%

   \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}}}{a} \]
3. Step-by-step derivation

   1. mul-1-neg 32.9%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-c \cdot a}}}{a} \]

   2. distribute-rgt-neg-out 32.9%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

4. Simplified 32.9%

   \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

5. Taylor expanded in b_2 around inf 12.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
6. Step-by-step derivation

   1. mul-1-neg 12.9%

      \[\leadsto \color{blue}{-\frac{b_2}{a}} \]

   2. distribute-frac-neg 12.9%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

7. Simplified 12.9%

   \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

8. Final simplification 12.9%

   \[\leadsto \frac{-b_2}{a} \]

Reproduce

herbie shell --seed 2023201 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))