quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.9% → 86.7%

Time: 10.8s

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: 52.9% 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: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{+114}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.7 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\mathsf{fma}\left(-1, b_2, t_0\right)}}{a}\\ \mathbf{elif}\;b_2 \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -3.8e+114)
     (/ (* -0.5 c) b_2)
     (if (<= b_2 -1.7e-130)
       (/ (/ (* c a) (fma -1.0 b_2 t_0)) a)
       (if (<= b_2 1.2e+77)
         (- (/ (- b_2) a) (/ t_0 a))
         (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -3.8e+114) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -1.7e-130) {
		tmp = ((c * a) / fma(-1.0, b_2, t_0)) / a;
	} else if (b_2 <= 1.2e+77) {
		tmp = (-b_2 / a) - (t_0 / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -3.8e+114)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -1.7e-130)
		tmp = Float64(Float64(Float64(c * a) / fma(-1.0, b_2, t_0)) / a);
	elseif (b_2 <= 1.2e+77)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64(t_0 / 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_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -3.8e+114], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -1.7e-130], N[(N[(N[(c * a), $MachinePrecision] / N[(-1.0 * b$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.2e+77], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(t$95$0 / a), $MachinePrecision]), $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 < -3.8000000000000001e114

   1. Initial program 4.0%

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

   2. Taylor expanded in b_2 around -inf 98.1%

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

      1. associate-*r/ 98.1%

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

   4. Simplified 98.1%

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

   if -3.8000000000000001e114 < b_2 < -1.70000000000000003e-130

   1. Initial program 40.3%

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

      1. sub-neg 40.3%

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

      2. +-commutative 40.3%

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

      3. *-commutative 40.3%

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

      4. distribute-lft-neg-in 40.3%

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

      5. fma-def 40.3%

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

   3. Applied egg-rr 40.3%

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

      1. flip-- 39.9%

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

      2. pow2 39.9%

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

      3. add-sqr-sqrt 40.3%

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

      4. neg-mul-1 40.3%

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

      5. fma-def 40.3%

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

   5. Applied egg-rr 40.3%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b_2\right)}^{2} - \mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}{\mathsf{fma}\left(-1, b_2, \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}\right)}}}{a} \]
   6. Step-by-step derivation

      1. unpow2 40.3%

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

      2. fma-udef 40.3%

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

      3. distribute-lft-neg-out 40.3%

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

      4. distribute-rgt-neg-out 40.3%

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

      5. +-commutative 40.3%

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

      6. distribute-rgt-neg-out 40.3%

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

      7. sub-neg 40.3%

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

      8. associate--r- 82.2%

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

      9. unpow2 82.2%

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

      10. sqr-neg 82.2%

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

      11. unpow2 82.2%

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

      12. +-inverses 82.2%

          \[\leadsto \frac{\frac{\color{blue}{0} + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}\right)}}{a} \]

      13. unpow2 82.2%

          \[\leadsto \frac{\frac{0 + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{\mathsf{fma}\left(-c, a, \color{blue}{{b_2}^{2}}\right)}\right)}}{a} \]

      14. fma-udef 82.2%

          \[\leadsto \frac{\frac{0 + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{\color{blue}{\left(-c\right) \cdot a + {b_2}^{2}}}\right)}}{a} \]

      15. distribute-lft-neg-out 82.2%

          \[\leadsto \frac{\frac{0 + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{\color{blue}{\left(-c \cdot a\right)} + {b_2}^{2}}\right)}}{a} \]

      16. distribute-rgt-neg-out 82.2%

          \[\leadsto \frac{\frac{0 + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)} + {b_2}^{2}}\right)}}{a} \]

      17. +-commutative 82.2%

          \[\leadsto \frac{\frac{0 + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{\color{blue}{{b_2}^{2} + c \cdot \left(-a\right)}}\right)}}{a} \]

      18. distribute-rgt-neg-out 82.2%

          \[\leadsto \frac{\frac{0 + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{{b_2}^{2} + \color{blue}{\left(-c \cdot a\right)}}\right)}}{a} \]

      19. sub-neg 82.2%

          \[\leadsto \frac{\frac{0 + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{\color{blue}{{b_2}^{2} - c \cdot a}}\right)}}{a} \]

      20. unpow2 82.2%

          \[\leadsto \frac{\frac{0 + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{\color{blue}{b_2 \cdot b_2} - c \cdot a}\right)}}{a} \]

   7. Simplified 82.2%

      \[\leadsto \frac{\color{blue}{\frac{0 + c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}}}{a} \]

   if -1.70000000000000003e-130 < b_2 < 1.1999999999999999e77

   1. Initial program 83.9%

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

      1. add-sqr-sqrt 83.6%

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

      2. pow2 83.6%

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

      3. pow1/2 83.6%

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

      4. sqrt-pow1 83.6%

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

      5. metadata-eval 83.6%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 83.6%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
   4. Step-by-step derivation

      1. div-sub 83.6%

         \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{a}} \]

      2. pow-pow 83.9%

         \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}{a} \]

      3. metadata-eval 83.9%

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

      4. pow1/2 83.9%

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

      5. cancel-sign-sub-inv 83.9%

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

      6. *-commutative 83.9%

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

      7. add-sqr-sqrt 73.8%

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

      8. hypot-udef 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. hypot-udef 73.8%

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

      2. add-sqr-sqrt 83.9%

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

      3. *-commutative 83.9%

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

      4. cancel-sign-sub-inv 83.9%

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

      5. pow1/2 83.9%

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

      6. *-commutative 83.9%

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

   7. Applied egg-rr 83.9%

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

      1. sqr-pow 83.6%

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

      2. metadata-eval 83.6%

         \[\leadsto \frac{-b_2}{a} - \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.25}} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(\frac{0.5}{2}\right)}}{a} \]

      3. metadata-eval 83.6%

         \[\leadsto \frac{-b_2}{a} - \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.25}}}{a} \]

   9. Applied egg-rr 83.6%

      \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25}}}{a} \]
   10. Step-by-step derivation

       1. pow-sqr 83.9%

          \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(2 \cdot 0.25\right)}}}{a} \]

       2. metadata-eval 83.9%

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

       3. unpow1/2 83.9%

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

   11. Simplified 83.9%

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

   if 1.1999999999999999e77 < b_2

   1. Initial program 58.0%

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

   2. Taylor expanded in b_2 around inf 95.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{+114}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.7 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\mathsf{fma}\left(-1, b_2, \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}}{a}\\ \mathbf{elif}\;b_2 \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-51)
   (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))
   (if (<= b_2 1.3e+77)
     (- (/ (- b_2) a) (/ (sqrt (- (* b_2 b_2) (* c a))) 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 <= -1e-51) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 1.3e+77) {
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / 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 <= (-1d-51)) then
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    else if (b_2 <= 1.3d+77) then
        tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / 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 <= -1e-51) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 1.3e+77) {
		tmp = (-b_2 / a) - (Math.sqrt(((b_2 * b_2) - (c * a))) / 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 <= -1e-51:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	elif b_2 <= 1.3e+77:
		tmp = (-b_2 / a) - (math.sqrt(((b_2 * b_2) - (c * a))) / 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 <= -1e-51)
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	elseif (b_2 <= 1.3e+77)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) / 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 <= -1e-51)
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	elseif (b_2 <= 1.3e+77)
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / 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, -1e-51], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.3e+77], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $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 < -1e-51

   1. Initial program 19.9%

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

      1. sub-neg 19.9%

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

      2. +-commutative 19.9%

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

      3. *-commutative 19.9%

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

      4. distribute-lft-neg-in 19.9%

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

      5. fma-def 19.9%

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

   3. Applied egg-rr 19.9%

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

      1. clear-num 19.9%

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

      2. inv-pow 19.9%

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

   5. Applied egg-rr 19.9%

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

      1. unpow-1 19.9%

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

      2. unpow2 19.9%

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

      3. fma-udef 19.8%

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

      4. distribute-lft-neg-out 19.8%

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

      5. distribute-rgt-neg-out 19.8%

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

      6. +-commutative 19.8%

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

      7. distribute-rgt-neg-out 19.8%

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

      8. sub-neg 19.8%

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

      9. unpow2 19.8%

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

   7. Simplified 19.8%

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

   8. Taylor expanded in b_2 around -inf 88.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]

   if -1e-51 < b_2 < 1.3000000000000001e77

   1. Initial program 80.9%

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

      1. add-sqr-sqrt 80.6%

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

      2. pow2 80.6%

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

      3. pow1/2 80.6%

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

      4. sqrt-pow1 80.6%

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

      5. metadata-eval 80.6%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 80.6%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
   4. Step-by-step derivation

      1. div-sub 80.6%

         \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{a}} \]

      2. pow-pow 81.0%

         \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}{a} \]

      3. metadata-eval 81.0%

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

      4. pow1/2 81.0%

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

      5. cancel-sign-sub-inv 81.0%

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

      6. *-commutative 81.0%

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

      7. add-sqr-sqrt 71.3%

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

      8. hypot-udef 72.8%

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

   5. Applied egg-rr 72.8%

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

      1. hypot-udef 71.3%

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

      2. add-sqr-sqrt 81.0%

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

      3. *-commutative 81.0%

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

      4. cancel-sign-sub-inv 81.0%

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

      5. pow1/2 81.0%

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

      6. *-commutative 81.0%

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

   7. Applied egg-rr 81.0%

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

      1. sqr-pow 80.6%

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

      2. metadata-eval 80.6%

         \[\leadsto \frac{-b_2}{a} - \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.25}} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(\frac{0.5}{2}\right)}}{a} \]

      3. metadata-eval 80.6%

         \[\leadsto \frac{-b_2}{a} - \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.25}}}{a} \]

   9. Applied egg-rr 80.6%

      \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25}}}{a} \]
   10. Step-by-step derivation

       1. pow-sqr 81.0%

          \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(2 \cdot 0.25\right)}}}{a} \]

       2. metadata-eval 81.0%

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

       3. unpow1/2 81.0%

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

   11. Simplified 81.0%

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

   if 1.3000000000000001e77 < b_2

   1. Initial program 58.0%

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

   2. Taylor expanded in b_2 around inf 95.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8e-53)
   (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))
   (if (<= b_2 1.45e+77)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) 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 <= -8e-53) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 1.45e+77) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / 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 <= (-8d-53)) then
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    else if (b_2 <= 1.45d+77) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / 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 <= -8e-53) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 1.45e+77) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / 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 <= -8e-53:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	elif b_2 <= 1.45e+77:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / 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 <= -8e-53)
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	elseif (b_2 <= 1.45e+77)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / 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 <= -8e-53)
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	elseif (b_2 <= 1.45e+77)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / 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, -8e-53], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.45e+77], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $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 < -8.00000000000000025e-53

   1. Initial program 19.9%

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

      1. sub-neg 19.9%

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

      2. +-commutative 19.9%

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

      3. *-commutative 19.9%

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

      4. distribute-lft-neg-in 19.9%

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

      5. fma-def 19.9%

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

   3. Applied egg-rr 19.9%

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

      1. clear-num 19.9%

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

      2. inv-pow 19.9%

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

   5. Applied egg-rr 19.9%

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

      1. unpow-1 19.9%

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

      2. unpow2 19.9%

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

      3. fma-udef 19.8%

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

      4. distribute-lft-neg-out 19.8%

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

      5. distribute-rgt-neg-out 19.8%

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

      6. +-commutative 19.8%

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

      7. distribute-rgt-neg-out 19.8%

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

      8. sub-neg 19.8%

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

      9. unpow2 19.8%

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

   7. Simplified 19.8%

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

   8. Taylor expanded in b_2 around -inf 88.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]

   if -8.00000000000000025e-53 < b_2 < 1.4500000000000001e77

   1. Initial program 80.9%

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

   if 1.4500000000000001e77 < b_2

   1. Initial program 58.0%

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

   2. Taylor expanded in b_2 around inf 95.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.35 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.8e-120)
   (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))
   (if (<= b_2 1.35e-31)
     (/ (- (- b_2) (sqrt (* c (- a)))) 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 <= -8.8e-120) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 1.35e-31) {
		tmp = (-b_2 - sqrt((c * -a))) / 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 <= (-8.8d-120)) then
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    else if (b_2 <= 1.35d-31) then
        tmp = (-b_2 - sqrt((c * -a))) / 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 <= -8.8e-120) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 1.35e-31) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / 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 <= -8.8e-120:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	elif b_2 <= 1.35e-31:
		tmp = (-b_2 - math.sqrt((c * -a))) / 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 <= -8.8e-120)
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	elseif (b_2 <= 1.35e-31)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / 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 <= -8.8e-120)
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	elseif (b_2 <= 1.35e-31)
		tmp = (-b_2 - sqrt((c * -a))) / 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, -8.8e-120], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.35e-31], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $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 < -8.8000000000000005e-120

   1. Initial program 22.7%

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

      1. sub-neg 22.7%

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

      2. +-commutative 22.7%

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

      3. *-commutative 22.7%

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

      4. distribute-lft-neg-in 22.7%

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

      5. fma-def 22.7%

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

   3. Applied egg-rr 22.7%

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

      1. clear-num 22.7%

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

      2. inv-pow 22.7%

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

   5. Applied egg-rr 22.7%

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

      1. unpow-1 22.7%

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

      2. unpow2 22.7%

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

      3. fma-udef 22.6%

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

      4. distribute-lft-neg-out 22.6%

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

      5. distribute-rgt-neg-out 22.6%

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

      6. +-commutative 22.6%

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

      7. distribute-rgt-neg-out 22.6%

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

      8. sub-neg 22.6%

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

      9. unpow2 22.6%

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

   7. Simplified 22.6%

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

   8. Taylor expanded in b_2 around -inf 84.5%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]

   if -8.8000000000000005e-120 < b_2 < 1.35000000000000007e-31

   1. Initial program 78.9%

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

   2. Taylor expanded in b_2 around 0 75.3%

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

      1. mul-1-neg 75.3%

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

      2. distribute-rgt-neg-out 75.3%

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

   4. Simplified 75.3%

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

   if 1.35000000000000007e-31 < b_2

   1. Initial program 68.8%

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

   2. Taylor expanded in b_2 around inf 90.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.35 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5: 66.8% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-312)
   (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))
   (+ (* -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-312) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} 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-312)) then
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    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-312) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} 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-312:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	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-312)
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	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-312)
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	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-312], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -5.0000000000022e-312

   1. Initial program 35.7%

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

      1. sub-neg 35.7%

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

      2. +-commutative 35.7%

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

      3. *-commutative 35.7%

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

      4. distribute-lft-neg-in 35.7%

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

      5. fma-def 35.7%

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

   3. Applied egg-rr 35.7%

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

      1. clear-num 35.7%

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

      2. inv-pow 35.7%

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

   5. Applied egg-rr 35.7%

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

      1. unpow-1 35.7%

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

      2. unpow2 35.7%

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

      3. fma-udef 35.7%

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

      4. distribute-lft-neg-out 35.7%

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

      5. distribute-rgt-neg-out 35.7%

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

      6. +-commutative 35.7%

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

      7. distribute-rgt-neg-out 35.7%

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

      8. sub-neg 35.7%

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

      9. unpow2 35.7%

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

   7. Simplified 35.7%

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

   8. Taylor expanded in b_2 around -inf 69.3%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]

   if -5.0000000000022e-312 < b_2

   1. Initial program 70.8%

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

   2. Taylor expanded in b_2 around inf 69.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6: 67.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-312)
   (/ (* -0.5 c) 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-312) {
		tmp = (-0.5 * c) / 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-312)) then
        tmp = ((-0.5d0) * c) / 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-312) {
		tmp = (-0.5 * c) / 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-312:
		tmp = (-0.5 * c) / 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-312)
		tmp = Float64(Float64(-0.5 * c) / 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-312)
		tmp = (-0.5 * c) / 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-312], N[(N[(-0.5 * c), $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 < -5.0000000000022e-312

   1. Initial program 35.7%

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

   2. Taylor expanded in b_2 around -inf 68.4%

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

      1. associate-*r/ 68.4%

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

   4. Simplified 68.4%

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

   if -5.0000000000022e-312 < b_2

   1. Initial program 70.8%

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

   2. Taylor expanded in b_2 around inf 69.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 7: 47.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e-308) (/ (* -0.5 c) b_2) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e-308) {
		tmp = (-0.5 * c) / 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 <= (-1.8d-308)) then
        tmp = ((-0.5d0) * c) / 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 <= -1.8e-308) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.8e-308:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.8e-308)
		tmp = Float64(Float64(-0.5 * c) / 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 <= -1.8e-308)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.8e-308], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.7999999999999999e-308

   1. Initial program 35.7%

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

   2. Taylor expanded in b_2 around -inf 68.4%

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

      1. associate-*r/ 68.4%

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

   4. Simplified 68.4%

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

   if -1.7999999999999999e-308 < b_2

   1. Initial program 70.8%

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

      1. add-sqr-sqrt 70.6%

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

      2. pow2 70.6%

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

      3. pow1/2 70.6%

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

      4. sqrt-pow1 70.6%

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

      5. metadata-eval 70.6%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 70.6%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around inf 27.9%

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

      1. mul-1-neg 27.9%

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

      2. distribute-frac-neg 27.9%

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

   6. Simplified 27.9%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 8: 66.8% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-312) (/ (* -0.5 c) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-312) {
		tmp = (-0.5 * c) / 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 <= (-5d-312)) then
        tmp = ((-0.5d0) * c) / 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 <= -5e-312) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-312:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-312)
		tmp = Float64(Float64(-0.5 * c) / 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 <= -5e-312)
		tmp = (-0.5 * c) / 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, -5e-312], N[(N[(-0.5 * c), $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 < -5.0000000000022e-312

   1. Initial program 35.7%

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

   2. Taylor expanded in b_2 around -inf 68.4%

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

      1. associate-*r/ 68.4%

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

   4. Simplified 68.4%

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

   if -5.0000000000022e-312 < b_2

   1. Initial program 70.8%

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

   2. Taylor expanded in b_2 around inf 69.4%

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

      1. *-commutative 69.4%

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

   4. Simplified 69.4%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 9: 23.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-8}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-8) (/ 0.0 a) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-8) {
		tmp = 0.0 / a;
	} 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-8)) then
        tmp = 0.0d0 / a
    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-8) {
		tmp = 0.0 / a;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7e-8:
		tmp = 0.0 / a
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-8)
		tmp = Float64(0.0 / a);
	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-8)
		tmp = 0.0 / a;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-8], N[(0.0 / a), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7.00000000000000048e-8

   1. Initial program 16.0%

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

      1. add-sqr-sqrt 12.6%

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

      2. pow2 12.6%

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

      3. pow1/2 12.6%

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

      4. sqrt-pow1 12.7%

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

      5. metadata-eval 12.7%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 12.7%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around -inf 45.0%

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

      1. distribute-lft1-in 45.0%

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

      2. metadata-eval 45.0%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 45.0%

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

   6. Simplified 45.0%

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

   if -7.00000000000000048e-8 < b_2

   1. Initial program 68.7%

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

      1. add-sqr-sqrt 68.5%

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

      2. pow2 68.5%

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

      3. pow1/2 68.5%

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

      4. sqrt-pow1 68.5%

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

      5. metadata-eval 68.5%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 68.5%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around inf 20.8%

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

      1. mul-1-neg 20.8%

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

      2. distribute-frac-neg 20.8%

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

   6. Simplified 20.8%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-8}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 10: 11.1% accurate, 37.3× speedup

Math

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

FPCore

(FPCore (a b_2 c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b_2, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0 / a;
}

Python

def code(a, b_2, c):
	return 0.0 / a

Julia

function code(a, b_2, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 53.1%

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

   1. add-sqr-sqrt 51.9%

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

   2. pow2 51.9%

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

   3. pow1/2 51.9%

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

   4. sqrt-pow1 51.9%

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

   5. metadata-eval 51.9%

      \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

3. Applied egg-rr 51.9%

   \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

4. Taylor expanded in b_2 around -inf 15.4%

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

   1. distribute-lft1-in 15.4%

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

   2. metadata-eval 15.4%

      \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

   3. mul0-lft 15.4%

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

6. Simplified 15.4%

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

7. Final simplification 15.4%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023238 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))