quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.5% → 84.9%

Time: 12.8s

Alternatives: 10

Speedup: 11.2×

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: 52.5% 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: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.36 \cdot 10^{+38}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 10^{-24}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right) - b\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.36e+38)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 6.5e-71)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (if (<= b_2 1e-24)
       (/ (* c -0.5) b_2)
       (if (<= b_2 1.5e-14)
         (* (/ 1.0 a) (- (hypot b_2 (sqrt (* a (- c)))) b_2))
         (/ c (fma 0.5 (* a (/ c b_2)) (* b_2 -2.0))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.36e+38) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 6.5e-71) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else if (b_2 <= 1e-24) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.5e-14) {
		tmp = (1.0 / a) * (hypot(b_2, sqrt((a * -c))) - b_2);
	} else {
		tmp = c / fma(0.5, (a * (c / b_2)), (b_2 * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.36e+38)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 6.5e-71)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	elseif (b_2 <= 1e-24)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.5e-14)
		tmp = Float64(Float64(1.0 / a) * Float64(hypot(b_2, sqrt(Float64(a * Float64(-c)))) - b_2));
	else
		tmp = Float64(c / fma(0.5, Float64(a * Float64(c / b_2)), Float64(b_2 * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.36e+38], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 6.5e-71], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1e-24], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.5e-14], N[(N[(1.0 / a), $MachinePrecision] * N[(N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.5 * N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -1.36000000000000002e38

   1. Initial program 61.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 61.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 61.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 95.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 95.8%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 95.8%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if -1.36000000000000002e38 < b_2 < 6.50000000000000005e-71

   1. Initial program 85.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 85.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 85.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 85.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   if 6.50000000000000005e-71 < b_2 < 9.99999999999999924e-25

   1. Initial program 15.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 15.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 15.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 15.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 76.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ 76.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

      2. *-commutative 76.0%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   7. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   if 9.99999999999999924e-25 < b_2 < 1.4999999999999999e-14

   1. Initial program 99.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right)} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{1}{a} \cdot \left(\sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(-a \cdot c\right)}} - b\_2\right) \]

      4. add-sqr-sqrt 100.0%

         \[\leadsto \frac{1}{a} \cdot \left(\sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b\_2\right) \]

      5. hypot-define 100.0%

         \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{-a \cdot c}\right)} - b\_2\right) \]

      6. *-commutative 100.0%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b\_2\right) \]

      7. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b\_2\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2\right)} \]

   if 1.4999999999999999e-14 < b_2

   1. Initial program 10.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 10.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 10.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 10.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 10.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}} \]

      2. inv-pow 10.8%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}\right)}^{-1}} \]

      3. sub-neg 10.8%

         \[\leadsto {\left(\frac{a}{\sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(-a \cdot c\right)}} - b\_2}\right)}^{-1} \]

      4. add-sqr-sqrt 7.8%

         \[\leadsto {\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b\_2}\right)}^{-1} \]

      5. hypot-define 31.0%

         \[\leadsto {\left(\frac{a}{\color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{-a \cdot c}\right)} - b\_2}\right)}^{-1} \]

      6. *-commutative 31.0%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b\_2}\right)}^{-1} \]

      7. distribute-rgt-neg-in 31.0%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b\_2}\right)}^{-1} \]

   6. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 31.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   8. Simplified 31.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   9. Taylor expanded in c around 0 0.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5 \cdot \frac{a \cdot c}{b\_2} + 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}}{c}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot c}{b\_2}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}}{c}} \]

       2. associate-/l* 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, \color{blue}{a \cdot \frac{c}{b\_2}}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}{c}} \]

       3. unpow2 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}\right)}{c}} \]

       4. rem-square-sqrt 91.2%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{-1}}\right)}{c}} \]

   11. Simplified 91.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}{c}}} \]
   12. Step-by-step derivation

       1. clear-num 92.1%

          \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}} \]

       2. div-inv 91.8%

          \[\leadsto \color{blue}{c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}} \]

       3. *-commutative 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \color{blue}{\frac{b\_2}{-1} \cdot 2}\right)} \]

       4. div-inv 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \color{blue}{\left(b\_2 \cdot \frac{1}{-1}\right)} \cdot 2\right)} \]

       5. metadata-eval 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \left(b\_2 \cdot \color{blue}{-1}\right) \cdot 2\right)} \]

       6. associate-*l* 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \color{blue}{b\_2 \cdot \left(-1 \cdot 2\right)}\right)} \]

       7. metadata-eval 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot \color{blue}{-2}\right)} \]

   13. Applied egg-rr 91.8%

       \[\leadsto \color{blue}{c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}} \]
   14. Step-by-step derivation

       1. associate-*r/ 92.1%

          \[\leadsto \color{blue}{\frac{c \cdot 1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}} \]

       2. *-rgt-identity 92.1%

          \[\leadsto \frac{\color{blue}{c}}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)} \]

   15. Simplified 92.1%

       \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.36 \cdot 10^{+38}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 10^{-24}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right) - b\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.36 \cdot 10^{+38}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.36e+38)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 3.9e-69)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (if (<= b_2 1.65e-21)
       (/ (* c -0.5) b_2)
       (if (<= b_2 2.8e-15)
         (/ (- (sqrt (* a (- c))) b_2) a)
         (/ c (fma 0.5 (* a (/ c b_2)) (* b_2 -2.0))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.36e+38) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 3.9e-69) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else if (b_2 <= 1.65e-21) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.8e-15) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = c / fma(0.5, (a * (c / b_2)), (b_2 * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.36e+38)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 3.9e-69)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	elseif (b_2 <= 1.65e-21)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 2.8e-15)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(c / fma(0.5, Float64(a * Float64(c / b_2)), Float64(b_2 * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.36e+38], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.9e-69], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.65e-21], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.8e-15], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c / N[(0.5 * N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -1.36000000000000002e38

   1. Initial program 61.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 61.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 61.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 95.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 95.8%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 95.8%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if -1.36000000000000002e38 < b_2 < 3.89999999999999981e-69

   1. Initial program 85.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 85.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 85.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 85.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   if 3.89999999999999981e-69 < b_2 < 1.65000000000000004e-21

   1. Initial program 15.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 15.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 15.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 15.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 76.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ 76.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

      2. *-commutative 76.0%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   7. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   if 1.65000000000000004e-21 < b_2 < 2.80000000000000014e-15

   1. Initial program 99.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0 99.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}} - b\_2}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 99.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}} - b\_2}{a} \]

      2. neg-mul-1 99.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right)} \cdot c} - b\_2}{a} \]

      3. *-commutative 99.7%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   if 2.80000000000000014e-15 < b_2

   1. Initial program 10.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 10.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 10.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 10.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 10.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}} \]

      2. inv-pow 10.8%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}\right)}^{-1}} \]

      3. sub-neg 10.8%

         \[\leadsto {\left(\frac{a}{\sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(-a \cdot c\right)}} - b\_2}\right)}^{-1} \]

      4. add-sqr-sqrt 7.8%

         \[\leadsto {\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b\_2}\right)}^{-1} \]

      5. hypot-define 31.0%

         \[\leadsto {\left(\frac{a}{\color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{-a \cdot c}\right)} - b\_2}\right)}^{-1} \]

      6. *-commutative 31.0%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b\_2}\right)}^{-1} \]

      7. distribute-rgt-neg-in 31.0%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b\_2}\right)}^{-1} \]

   6. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 31.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   8. Simplified 31.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   9. Taylor expanded in c around 0 0.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5 \cdot \frac{a \cdot c}{b\_2} + 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}}{c}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot c}{b\_2}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}}{c}} \]

       2. associate-/l* 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, \color{blue}{a \cdot \frac{c}{b\_2}}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}{c}} \]

       3. unpow2 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}\right)}{c}} \]

       4. rem-square-sqrt 91.2%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{-1}}\right)}{c}} \]

   11. Simplified 91.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}{c}}} \]
   12. Step-by-step derivation

       1. clear-num 92.1%

          \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}} \]

       2. div-inv 91.8%

          \[\leadsto \color{blue}{c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}} \]

       3. *-commutative 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \color{blue}{\frac{b\_2}{-1} \cdot 2}\right)} \]

       4. div-inv 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \color{blue}{\left(b\_2 \cdot \frac{1}{-1}\right)} \cdot 2\right)} \]

       5. metadata-eval 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \left(b\_2 \cdot \color{blue}{-1}\right) \cdot 2\right)} \]

       6. associate-*l* 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \color{blue}{b\_2 \cdot \left(-1 \cdot 2\right)}\right)} \]

       7. metadata-eval 91.8%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot \color{blue}{-2}\right)} \]

   13. Applied egg-rr 91.8%

       \[\leadsto \color{blue}{c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}} \]
   14. Step-by-step derivation

       1. associate-*r/ 92.1%

          \[\leadsto \color{blue}{\frac{c \cdot 1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}} \]

       2. *-rgt-identity 92.1%

          \[\leadsto \frac{\color{blue}{c}}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)} \]

   15. Simplified 92.1%

       \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.36 \cdot 10^{+38}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-5)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 8e-72)
     (/ (- (sqrt (* a (- c))) b_2) a)
     (/ c (fma 0.5 (* a (/ c b_2)) (* b_2 -2.0))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-5) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 8e-72) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = c / fma(0.5, (a * (c / b_2)), (b_2 * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-5)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 8e-72)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(c / fma(0.5, Float64(a * Float64(c / b_2)), Float64(b_2 * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-5], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 8e-72], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c / N[(0.5 * N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.9e-5

   1. Initial program 65.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 65.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 65.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 92.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 92.0%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 92.0%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if -2.9e-5 < b_2 < 7.9999999999999997e-72

   1. Initial program 85.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 85.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 85.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0 77.0%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}} - b\_2}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 77.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}} - b\_2}{a} \]

      2. neg-mul-1 77.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right)} \cdot c} - b\_2}{a} \]

      3. *-commutative 77.0%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   7. Simplified 77.0%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   if 7.9999999999999997e-72 < b_2

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 15.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 15.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 15.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 15.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}} \]

      2. inv-pow 15.5%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}\right)}^{-1}} \]

      3. sub-neg 15.5%

         \[\leadsto {\left(\frac{a}{\sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(-a \cdot c\right)}} - b\_2}\right)}^{-1} \]

      4. add-sqr-sqrt 12.9%

         \[\leadsto {\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b\_2}\right)}^{-1} \]

      5. hypot-define 33.2%

         \[\leadsto {\left(\frac{a}{\color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{-a \cdot c}\right)} - b\_2}\right)}^{-1} \]

      6. *-commutative 33.2%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b\_2}\right)}^{-1} \]

      7. distribute-rgt-neg-in 33.2%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b\_2}\right)}^{-1} \]

   6. Applied egg-rr 33.2%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 33.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   8. Simplified 33.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   9. Taylor expanded in c around 0 0.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5 \cdot \frac{a \cdot c}{b\_2} + 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}}{c}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot c}{b\_2}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}}{c}} \]

       2. associate-/l* 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, \color{blue}{a \cdot \frac{c}{b\_2}}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}{c}} \]

       3. unpow2 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}\right)}{c}} \]

       4. rem-square-sqrt 85.8%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{-1}}\right)}{c}} \]

   11. Simplified 85.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}{c}}} \]
   12. Step-by-step derivation

       1. clear-num 86.5%

          \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}} \]

       2. div-inv 86.3%

          \[\leadsto \color{blue}{c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}} \]

       3. *-commutative 86.3%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \color{blue}{\frac{b\_2}{-1} \cdot 2}\right)} \]

       4. div-inv 86.3%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \color{blue}{\left(b\_2 \cdot \frac{1}{-1}\right)} \cdot 2\right)} \]

       5. metadata-eval 86.3%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \left(b\_2 \cdot \color{blue}{-1}\right) \cdot 2\right)} \]

       6. associate-*l* 86.3%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, \color{blue}{b\_2 \cdot \left(-1 \cdot 2\right)}\right)} \]

       7. metadata-eval 86.3%

          \[\leadsto c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot \color{blue}{-2}\right)} \]

   13. Applied egg-rr 86.3%

       \[\leadsto \color{blue}{c \cdot \frac{1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}} \]
   14. Step-by-step derivation

       1. associate-*r/ 86.5%

          \[\leadsto \color{blue}{\frac{c \cdot 1}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}} \]

       2. *-rgt-identity 86.5%

          \[\leadsto \frac{\color{blue}{c}}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)} \]

   15. Simplified 86.5%

       \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, b\_2 \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e-5)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 3.9e-69)
     (/ (- (sqrt (* a (- c))) b_2) a)
     (/ 1.0 (+ (* -2.0 (/ b_2 c)) (* 0.5 (/ a b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-5) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 3.9e-69) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / 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 <= (-3.8d-5)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 3.9d-69) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = 1.0d0 / (((-2.0d0) * (b_2 / c)) + (0.5d0 * (a / 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 <= -3.8e-5) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 3.9e-69) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.8e-5:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 3.9e-69:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.8e-5)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 3.9e-69)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b_2 / c)) + Float64(0.5 * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.8e-5)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 3.9e-69)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.8e-5], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.9e-69], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.8000000000000002e-5

   1. Initial program 65.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 65.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 65.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 92.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 92.0%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 92.0%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if -3.8000000000000002e-5 < b_2 < 3.89999999999999981e-69

   1. Initial program 85.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 85.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 85.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0 77.0%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}} - b\_2}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 77.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}} - b\_2}{a} \]

      2. neg-mul-1 77.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right)} \cdot c} - b\_2}{a} \]

      3. *-commutative 77.0%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   7. Simplified 77.0%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   if 3.89999999999999981e-69 < b_2

   1. Initial program 15.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 15.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 15.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 15.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 15.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}} \]

      2. inv-pow 15.5%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}\right)}^{-1}} \]

      3. sub-neg 15.5%

         \[\leadsto {\left(\frac{a}{\sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(-a \cdot c\right)}} - b\_2}\right)}^{-1} \]

      4. add-sqr-sqrt 12.9%

         \[\leadsto {\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b\_2}\right)}^{-1} \]

      5. hypot-define 33.2%

         \[\leadsto {\left(\frac{a}{\color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{-a \cdot c}\right)} - b\_2}\right)}^{-1} \]

      6. *-commutative 33.2%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b\_2}\right)}^{-1} \]

      7. distribute-rgt-neg-in 33.2%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b\_2}\right)}^{-1} \]

   6. Applied egg-rr 33.2%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 33.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   8. Simplified 33.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   9. Taylor expanded in c around 0 0.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5 \cdot \frac{a \cdot c}{b\_2} + 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}}{c}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot c}{b\_2}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}}{c}} \]

       2. associate-/l* 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, \color{blue}{a \cdot \frac{c}{b\_2}}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}{c}} \]

       3. unpow2 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}\right)}{c}} \]

       4. rem-square-sqrt 85.8%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{-1}}\right)}{c}} \]

   11. Simplified 85.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}{c}}} \]

   12. Taylor expanded in a around 0 85.9%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.2 \cdot 10^{-307}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.2e-307)
   (/ (* b_2 -2.0) a)
   (/ 1.0 (+ (* -2.0 (/ b_2 c)) (* 0.5 (/ a b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.2e-307) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / 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 <= (-9.2d-307)) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = 1.0d0 / (((-2.0d0) * (b_2 / c)) + (0.5d0 * (a / 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 <= -9.2e-307) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9.2e-307:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9.2e-307)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b_2 / c)) + Float64(0.5 * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9.2e-307)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9.2e-307], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.1999999999999996e-307

   1. Initial program 72.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 72.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 72.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 72.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 72.0%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 72.0%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if -9.1999999999999996e-307 < b_2

   1. Initial program 33.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 33.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 33.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 33.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 33.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}} \]

      2. inv-pow 33.3%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}\right)}^{-1}} \]

      3. sub-neg 33.3%

         \[\leadsto {\left(\frac{a}{\sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(-a \cdot c\right)}} - b\_2}\right)}^{-1} \]

      4. add-sqr-sqrt 31.4%

         \[\leadsto {\left(\frac{a}{\sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b\_2}\right)}^{-1} \]

      5. hypot-define 46.3%

         \[\leadsto {\left(\frac{a}{\color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{-a \cdot c}\right)} - b\_2}\right)}^{-1} \]

      6. *-commutative 46.3%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b\_2}\right)}^{-1} \]

      7. distribute-rgt-neg-in 46.3%

         \[\leadsto {\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b\_2}\right)}^{-1} \]

   6. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 46.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   8. Simplified 46.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2}}} \]

   9. Taylor expanded in c around 0 0.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5 \cdot \frac{a \cdot c}{b\_2} + 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}}{c}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot c}{b\_2}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}}{c}} \]

       2. associate-/l* 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, \color{blue}{a \cdot \frac{c}{b\_2}}, 2 \cdot \frac{b\_2}{{\left(\sqrt{-1}\right)}^{2}}\right)}{c}} \]

       3. unpow2 0.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}\right)}{c}} \]

       4. rem-square-sqrt 66.8%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{\color{blue}{-1}}\right)}{c}} \]

   11. Simplified 66.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b\_2}, 2 \cdot \frac{b\_2}{-1}\right)}{c}}} \]

   12. Taylor expanded in a around 0 66.9%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.2 \cdot 10^{-307}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 23.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 7.2e+26) (/ b_2 (- a)) (* 0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e+26) {
		tmp = b_2 / -a;
	} else {
		tmp = 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 <= 7.2d+26) then
        tmp = b_2 / -a
    else
        tmp = 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 <= 7.2e+26) {
		tmp = b_2 / -a;
	} else {
		tmp = 0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 7.2e+26:
		tmp = b_2 / -a
	else:
		tmp = 0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 7.2e+26)
		tmp = Float64(b_2 / Float64(-a));
	else
		tmp = 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 <= 7.2e+26)
		tmp = b_2 / -a;
	else
		tmp = 0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 7.2e+26], N[(b$95$2 / (-a)), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 7.20000000000000048e26

   1. Initial program 69.0%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 69.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 69.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0 46.2%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}} - b\_2}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 46.2%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}} - b\_2}{a} \]

      2. neg-mul-1 46.2%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right)} \cdot c} - b\_2}{a} \]

      3. *-commutative 46.2%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   7. Simplified 46.2%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   8. Taylor expanded in b_2 around inf 21.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 21.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot b\_2}{a}} \]

      2. neg-mul-1 21.3%

         \[\leadsto \frac{\color{blue}{-b\_2}}{a} \]

   10. Simplified 21.3%

       \[\leadsto \color{blue}{\frac{-b\_2}{a}} \]

   if 7.20000000000000048e26 < b_2

   1. Initial program 10.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 10.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 10.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 10.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 39.4%

      \[\leadsto \frac{\color{blue}{\left(b\_2 + -0.5 \cdot \frac{a \cdot c}{b\_2}\right)} - b\_2}{a} \]

   6. Taylor expanded in b_2 around 0 79.5%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b\_2}}}{a} \]
   7. Step-by-step derivation

      1. associate-*r/ 84.7%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b\_2}\right)}}{a} \]

      2. associate-*r* 84.7%

         \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot a\right) \cdot \frac{c}{b\_2}}}{a} \]

      3. *-commutative 84.7%

         \[\leadsto \frac{\color{blue}{\left(a \cdot -0.5\right)} \cdot \frac{c}{b\_2}}{a} \]

   8. Simplified 84.7%

      \[\leadsto \frac{\color{blue}{\left(a \cdot -0.5\right) \cdot \frac{c}{b\_2}}}{a} \]
   9. Step-by-step derivation

      1. div-inv 84.6%

         \[\leadsto \color{blue}{\left(\left(a \cdot -0.5\right) \cdot \frac{c}{b\_2}\right) \cdot \frac{1}{a}} \]

      2. associate-*l* 84.6%

         \[\leadsto \color{blue}{\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right)} \cdot \frac{1}{a} \]

      3. associate-*l* 77.8%

         \[\leadsto \color{blue}{a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{1}{a}\right)} \]

      4. frac-2neg 77.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \color{blue}{\frac{-1}{-a}}\right) \]

      5. metadata-eval 77.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{\color{blue}{-1}}{-a}\right) \]

      6. add-sqr-sqrt 44.5%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}\right) \]

      7. sqrt-unprod 41.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}\right) \]

      8. sqr-neg 41.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\sqrt{\color{blue}{a \cdot a}}}\right) \]

      9. sqrt-unprod 14.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\right) \]

      10. add-sqr-sqrt 36.2%

          \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{a}}\right) \]

   10. Applied egg-rr 36.2%

       \[\leadsto \color{blue}{a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{a}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 36.2%

          \[\leadsto \color{blue}{\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{-1}{a}} \]

       2. *-commutative 36.2%

          \[\leadsto \color{blue}{\frac{-1}{a} \cdot \left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right)} \]

       3. associate-*r/ 36.2%

          \[\leadsto \frac{-1}{a} \cdot \left(a \cdot \color{blue}{\frac{-0.5 \cdot c}{b\_2}}\right) \]

       4. associate-/l* 36.1%

          \[\leadsto \frac{-1}{a} \cdot \color{blue}{\frac{a \cdot \left(-0.5 \cdot c\right)}{b\_2}} \]

       5. times-frac 36.1%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(a \cdot \left(-0.5 \cdot c\right)\right)}{a \cdot b\_2}} \]

       6. neg-mul-1 36.1%

          \[\leadsto \frac{\color{blue}{-a \cdot \left(-0.5 \cdot c\right)}}{a \cdot b\_2} \]

       7. distribute-rgt-neg-in 36.1%

          \[\leadsto \frac{\color{blue}{a \cdot \left(--0.5 \cdot c\right)}}{a \cdot b\_2} \]

       8. times-frac 36.0%

          \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{--0.5 \cdot c}{b\_2}} \]

       9. *-inverses 36.0%

          \[\leadsto \color{blue}{1} \cdot \frac{--0.5 \cdot c}{b\_2} \]

       10. distribute-neg-frac 36.0%

           \[\leadsto 1 \cdot \color{blue}{\left(-\frac{-0.5 \cdot c}{b\_2}\right)} \]

       11. associate-*r/ 36.0%

           \[\leadsto 1 \cdot \left(-\color{blue}{-0.5 \cdot \frac{c}{b\_2}}\right) \]

       12. distribute-rgt-neg-in 36.0%

           \[\leadsto \color{blue}{-1 \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)} \]

       13. distribute-lft-neg-in 36.0%

           \[\leadsto \color{blue}{\left(-1\right) \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)} \]

       14. metadata-eval 36.0%

           \[\leadsto \color{blue}{-1} \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right) \]

       15. neg-mul-1 36.0%

           \[\leadsto \color{blue}{--0.5 \cdot \frac{c}{b\_2}} \]

       16. distribute-lft-neg-in 36.0%

           \[\leadsto \color{blue}{\left(--0.5\right) \cdot \frac{c}{b\_2}} \]

       17. metadata-eval 36.0%

           \[\leadsto \color{blue}{0.5} \cdot \frac{c}{b\_2} \]

   12. Simplified 36.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 7 \cdot 10^{+26}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 7e+26) (* b_2 (/ -2.0 a)) (* 0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7e+26) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = 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 <= 7d+26) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = 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 <= 7e+26) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = 0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 7e+26:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = 0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 7e+26)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = 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 <= 7e+26)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = 0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 7e+26], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 6.9999999999999998e26

   1. Initial program 69.0%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 69.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 69.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 68.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}} \]

      2. associate-/r/ 69.0%

         \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right)} \]

      3. sub-neg 69.0%

         \[\leadsto \frac{1}{a} \cdot \left(\sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(-a \cdot c\right)}} - b\_2\right) \]

      4. add-sqr-sqrt 53.3%

         \[\leadsto \frac{1}{a} \cdot \left(\sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b\_2\right) \]

      5. hypot-define 60.0%

         \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{-a \cdot c}\right)} - b\_2\right) \]

      6. *-commutative 60.0%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b\_2\right) \]

      7. distribute-rgt-neg-in 60.0%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b\_2\right) \]

   6. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2\right)} \]

   7. Taylor expanded in b_2 around -inf 48.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 48.3%

         \[\leadsto \color{blue}{\frac{-2 \cdot b\_2}{a}} \]

      2. *-commutative 48.3%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

      3. associate-/l* 48.2%

         \[\leadsto \color{blue}{b\_2 \cdot \frac{-2}{a}} \]

   9. Simplified 48.2%

      \[\leadsto \color{blue}{b\_2 \cdot \frac{-2}{a}} \]

   if 6.9999999999999998e26 < b_2

   1. Initial program 10.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 10.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 10.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 10.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 39.4%

      \[\leadsto \frac{\color{blue}{\left(b\_2 + -0.5 \cdot \frac{a \cdot c}{b\_2}\right)} - b\_2}{a} \]

   6. Taylor expanded in b_2 around 0 79.5%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b\_2}}}{a} \]
   7. Step-by-step derivation

      1. associate-*r/ 84.7%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b\_2}\right)}}{a} \]

      2. associate-*r* 84.7%

         \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot a\right) \cdot \frac{c}{b\_2}}}{a} \]

      3. *-commutative 84.7%

         \[\leadsto \frac{\color{blue}{\left(a \cdot -0.5\right)} \cdot \frac{c}{b\_2}}{a} \]

   8. Simplified 84.7%

      \[\leadsto \frac{\color{blue}{\left(a \cdot -0.5\right) \cdot \frac{c}{b\_2}}}{a} \]
   9. Step-by-step derivation

      1. div-inv 84.6%

         \[\leadsto \color{blue}{\left(\left(a \cdot -0.5\right) \cdot \frac{c}{b\_2}\right) \cdot \frac{1}{a}} \]

      2. associate-*l* 84.6%

         \[\leadsto \color{blue}{\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right)} \cdot \frac{1}{a} \]

      3. associate-*l* 77.8%

         \[\leadsto \color{blue}{a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{1}{a}\right)} \]

      4. frac-2neg 77.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \color{blue}{\frac{-1}{-a}}\right) \]

      5. metadata-eval 77.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{\color{blue}{-1}}{-a}\right) \]

      6. add-sqr-sqrt 44.5%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}\right) \]

      7. sqrt-unprod 41.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}\right) \]

      8. sqr-neg 41.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\sqrt{\color{blue}{a \cdot a}}}\right) \]

      9. sqrt-unprod 14.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\right) \]

      10. add-sqr-sqrt 36.2%

          \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{a}}\right) \]

   10. Applied egg-rr 36.2%

       \[\leadsto \color{blue}{a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{a}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 36.2%

          \[\leadsto \color{blue}{\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{-1}{a}} \]

       2. *-commutative 36.2%

          \[\leadsto \color{blue}{\frac{-1}{a} \cdot \left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right)} \]

       3. associate-*r/ 36.2%

          \[\leadsto \frac{-1}{a} \cdot \left(a \cdot \color{blue}{\frac{-0.5 \cdot c}{b\_2}}\right) \]

       4. associate-/l* 36.1%

          \[\leadsto \frac{-1}{a} \cdot \color{blue}{\frac{a \cdot \left(-0.5 \cdot c\right)}{b\_2}} \]

       5. times-frac 36.1%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(a \cdot \left(-0.5 \cdot c\right)\right)}{a \cdot b\_2}} \]

       6. neg-mul-1 36.1%

          \[\leadsto \frac{\color{blue}{-a \cdot \left(-0.5 \cdot c\right)}}{a \cdot b\_2} \]

       7. distribute-rgt-neg-in 36.1%

          \[\leadsto \frac{\color{blue}{a \cdot \left(--0.5 \cdot c\right)}}{a \cdot b\_2} \]

       8. times-frac 36.0%

          \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{--0.5 \cdot c}{b\_2}} \]

       9. *-inverses 36.0%

          \[\leadsto \color{blue}{1} \cdot \frac{--0.5 \cdot c}{b\_2} \]

       10. distribute-neg-frac 36.0%

           \[\leadsto 1 \cdot \color{blue}{\left(-\frac{-0.5 \cdot c}{b\_2}\right)} \]

       11. associate-*r/ 36.0%

           \[\leadsto 1 \cdot \left(-\color{blue}{-0.5 \cdot \frac{c}{b\_2}}\right) \]

       12. distribute-rgt-neg-in 36.0%

           \[\leadsto \color{blue}{-1 \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)} \]

       13. distribute-lft-neg-in 36.0%

           \[\leadsto \color{blue}{\left(-1\right) \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)} \]

       14. metadata-eval 36.0%

           \[\leadsto \color{blue}{-1} \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right) \]

       15. neg-mul-1 36.0%

           \[\leadsto \color{blue}{--0.5 \cdot \frac{c}{b\_2}} \]

       16. distribute-lft-neg-in 36.0%

           \[\leadsto \color{blue}{\left(--0.5\right) \cdot \frac{c}{b\_2}} \]

       17. metadata-eval 36.0%

           \[\leadsto \color{blue}{0.5} \cdot \frac{c}{b\_2} \]

   12. Simplified 36.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 7 \cdot 10^{+26}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5.3e+29) (/ (* b_2 -2.0) a) (* 0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5.3e+29) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = 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 <= 5.3d+29) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = 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 <= 5.3e+29) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = 0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 5.3e+29:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = 0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 5.3e+29)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = 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 <= 5.3e+29)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = 0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 5.3e+29], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 5.3e29

   1. Initial program 69.0%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 69.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 69.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 48.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 48.3%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 48.3%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if 5.3e29 < b_2

   1. Initial program 10.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 10.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 10.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 10.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 39.4%

      \[\leadsto \frac{\color{blue}{\left(b\_2 + -0.5 \cdot \frac{a \cdot c}{b\_2}\right)} - b\_2}{a} \]

   6. Taylor expanded in b_2 around 0 79.5%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b\_2}}}{a} \]
   7. Step-by-step derivation

      1. associate-*r/ 84.7%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b\_2}\right)}}{a} \]

      2. associate-*r* 84.7%

         \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot a\right) \cdot \frac{c}{b\_2}}}{a} \]

      3. *-commutative 84.7%

         \[\leadsto \frac{\color{blue}{\left(a \cdot -0.5\right)} \cdot \frac{c}{b\_2}}{a} \]

   8. Simplified 84.7%

      \[\leadsto \frac{\color{blue}{\left(a \cdot -0.5\right) \cdot \frac{c}{b\_2}}}{a} \]
   9. Step-by-step derivation

      1. div-inv 84.6%

         \[\leadsto \color{blue}{\left(\left(a \cdot -0.5\right) \cdot \frac{c}{b\_2}\right) \cdot \frac{1}{a}} \]

      2. associate-*l* 84.6%

         \[\leadsto \color{blue}{\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right)} \cdot \frac{1}{a} \]

      3. associate-*l* 77.8%

         \[\leadsto \color{blue}{a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{1}{a}\right)} \]

      4. frac-2neg 77.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \color{blue}{\frac{-1}{-a}}\right) \]

      5. metadata-eval 77.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{\color{blue}{-1}}{-a}\right) \]

      6. add-sqr-sqrt 44.5%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}\right) \]

      7. sqrt-unprod 41.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}\right) \]

      8. sqr-neg 41.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\sqrt{\color{blue}{a \cdot a}}}\right) \]

      9. sqrt-unprod 14.8%

         \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\right) \]

      10. add-sqr-sqrt 36.2%

          \[\leadsto a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{\color{blue}{a}}\right) \]

   10. Applied egg-rr 36.2%

       \[\leadsto \color{blue}{a \cdot \left(\left(-0.5 \cdot \frac{c}{b\_2}\right) \cdot \frac{-1}{a}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 36.2%

          \[\leadsto \color{blue}{\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{-1}{a}} \]

       2. *-commutative 36.2%

          \[\leadsto \color{blue}{\frac{-1}{a} \cdot \left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right)} \]

       3. associate-*r/ 36.2%

          \[\leadsto \frac{-1}{a} \cdot \left(a \cdot \color{blue}{\frac{-0.5 \cdot c}{b\_2}}\right) \]

       4. associate-/l* 36.1%

          \[\leadsto \frac{-1}{a} \cdot \color{blue}{\frac{a \cdot \left(-0.5 \cdot c\right)}{b\_2}} \]

       5. times-frac 36.1%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(a \cdot \left(-0.5 \cdot c\right)\right)}{a \cdot b\_2}} \]

       6. neg-mul-1 36.1%

          \[\leadsto \frac{\color{blue}{-a \cdot \left(-0.5 \cdot c\right)}}{a \cdot b\_2} \]

       7. distribute-rgt-neg-in 36.1%

          \[\leadsto \frac{\color{blue}{a \cdot \left(--0.5 \cdot c\right)}}{a \cdot b\_2} \]

       8. times-frac 36.0%

          \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{--0.5 \cdot c}{b\_2}} \]

       9. *-inverses 36.0%

          \[\leadsto \color{blue}{1} \cdot \frac{--0.5 \cdot c}{b\_2} \]

       10. distribute-neg-frac 36.0%

           \[\leadsto 1 \cdot \color{blue}{\left(-\frac{-0.5 \cdot c}{b\_2}\right)} \]

       11. associate-*r/ 36.0%

           \[\leadsto 1 \cdot \left(-\color{blue}{-0.5 \cdot \frac{c}{b\_2}}\right) \]

       12. distribute-rgt-neg-in 36.0%

           \[\leadsto \color{blue}{-1 \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)} \]

       13. distribute-lft-neg-in 36.0%

           \[\leadsto \color{blue}{\left(-1\right) \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)} \]

       14. metadata-eval 36.0%

           \[\leadsto \color{blue}{-1} \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right) \]

       15. neg-mul-1 36.0%

           \[\leadsto \color{blue}{--0.5 \cdot \frac{c}{b\_2}} \]

       16. distribute-lft-neg-in 36.0%

           \[\leadsto \color{blue}{\left(--0.5\right) \cdot \frac{c}{b\_2}} \]

       17. metadata-eval 36.0%

           \[\leadsto \color{blue}{0.5} \cdot \frac{c}{b\_2} \]

   12. Simplified 36.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.7 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.7e-308) (/ (* b_2 -2.0) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.7e-308) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / 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 <= 2.7d-308) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c * (-0.5d0)) / 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 <= 2.7e-308) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.7e-308:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.7e-308)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.7e-308)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.7e-308], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.70000000000000015e-308

   1. Initial program 72.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 72.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 72.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 71.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 71.4%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if 2.70000000000000015e-308 < b_2

   1. Initial program 32.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 32.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 32.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 32.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 66.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

      2. *-commutative 66.8%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   7. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.7 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 15.2% 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(b_2 / Float64(-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 51.2%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 51.2%

      \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

   2. unsub-neg 51.2%

      \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

3. Simplified 51.2%

   \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
4. Add Preprocessing

5. Taylor expanded in b_2 around 0 32.9%

   \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}} - b\_2}{a} \]
6. Step-by-step derivation

   1. associate-*r* 32.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}} - b\_2}{a} \]

   2. neg-mul-1 32.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right)} \cdot c} - b\_2}{a} \]

   3. *-commutative 32.9%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

7. Simplified 32.9%

   \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

8. Taylor expanded in b_2 around inf 15.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 15.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot b\_2}{a}} \]

   2. neg-mul-1 15.6%

      \[\leadsto \frac{\color{blue}{-b\_2}}{a} \]

10. Simplified 15.6%

    \[\leadsto \color{blue}{\frac{-b\_2}{a}} \]

11. Final simplification 15.6%

    \[\leadsto \frac{b\_2}{-a} \]
12. Add Preprocessing

Developer target: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	end
	tmp_3 = tmp_2;
end

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2) a) (/ (- c) (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))))))

  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))