quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.8% → 85.0%

Time: 13.9s

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: 51.8% 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: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+137}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.66 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+137)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.66e-97)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ 1.0 (fma 0.5 (/ a b_2) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+137) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.66e-97) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+137)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 1.66e-97)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+137], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.66e-97], 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], N[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.0000000000000001e137

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. unsub-neg 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in b_2 around -inf 94.3%

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

   if -2.0000000000000001e137 < b_2 < 1.6599999999999999e-97

   1. Initial program 82.7%

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

      1. +-commutative 82.7%

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

      2. unsub-neg 82.7%

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

   3. Simplified 82.7%

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

   if 1.6599999999999999e-97 < b_2

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

      2. unsub-neg 21.7%

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

   3. Simplified 21.7%

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

      1. clear-num 21.7%

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

      2. inv-pow 21.7%

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

      3. sub-neg 21.7%

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

      4. add-sqr-sqrt 17.7%

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

      5. hypot-define 24.4%

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

      6. *-commutative 24.4%

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

      7. distribute-rgt-neg-in 24.4%

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

   6. Applied egg-rr 24.4%

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

      1. unpow-1 24.4%

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

   8. Simplified 24.4%

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

   9. Taylor expanded in a around 0 0.0%

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

       1. fma-define 0.0%

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

       2. associate-*r/ 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 86.3%

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

       6. times-frac 86.3%

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

       7. metadata-eval 86.3%

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

   11. Simplified 86.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+137}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.66 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.15 \cdot 10^{-18}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.3 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.15e-18)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 3.3e-98)
     (/ (- (sqrt (* a (- c))) b_2) a)
     (/ 1.0 (fma 0.5 (/ a b_2) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.15e-18) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.3e-98) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.15e-18)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 3.3e-98)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.15e-18], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.3e-98], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.1500000000000002e-18

   1. Initial program 66.6%

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

      1. +-commutative 66.6%

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

      2. unsub-neg 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in b_2 around -inf 88.9%

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

   if -3.1500000000000002e-18 < b_2 < 3.3000000000000001e-98

   1. Initial program 78.7%

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

      1. +-commutative 78.7%

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

      2. unsub-neg 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in b_2 around 0 71.3%

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

      1. associate-*r* 71.3%

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

      2. neg-mul-1 71.3%

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

      3. *-commutative 71.3%

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

   7. Simplified 71.3%

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

   if 3.3000000000000001e-98 < b_2

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

      2. unsub-neg 21.7%

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

   3. Simplified 21.7%

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

      1. clear-num 21.7%

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

      2. inv-pow 21.7%

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

      3. sub-neg 21.7%

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

      4. add-sqr-sqrt 17.7%

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

      5. hypot-define 24.4%

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

      6. *-commutative 24.4%

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

      7. distribute-rgt-neg-in 24.4%

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

   6. Applied egg-rr 24.4%

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

      1. unpow-1 24.4%

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

   8. Simplified 24.4%

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

   9. Taylor expanded in a around 0 0.0%

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

       1. fma-define 0.0%

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

       2. associate-*r/ 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 86.3%

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

       6. times-frac 86.3%

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

       7. metadata-eval 86.3%

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

   11. Simplified 86.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.15 \cdot 10^{-18}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.3 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8 \cdot 10^{-18}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8e-18)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 5e-98) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e-18) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 5e-98) {
		tmp = (sqrt((a * -c)) - b_2) / 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 <= (-8d-18)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 5d-98) then
        tmp = (sqrt((a * -c)) - b_2) / 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 <= -8e-18) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 5e-98) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8e-18:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 5e-98:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8e-18)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 5e-98)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / 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 <= -8e-18)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 5e-98)
		tmp = (sqrt((a * -c)) - b_2) / 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, -8e-18], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 5e-98], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.0000000000000006e-18

   1. Initial program 66.6%

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

      1. +-commutative 66.6%

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

      2. unsub-neg 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in b_2 around -inf 88.9%

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

   if -8.0000000000000006e-18 < b_2 < 5.00000000000000018e-98

   1. Initial program 78.7%

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

      1. +-commutative 78.7%

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

      2. unsub-neg 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in b_2 around 0 71.3%

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

      1. associate-*r* 71.3%

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

      2. neg-mul-1 71.3%

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

      3. *-commutative 71.3%

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

   7. Simplified 71.3%

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

   if 5.00000000000000018e-98 < b_2

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

      2. unsub-neg 21.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in b_2 around inf 85.9%

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

      1. associate-*r/ 85.9%

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

      2. *-commutative 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -8 \cdot 10^{-18}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-26}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 4.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-26)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 4.9e-98) (/ (sqrt (* a (- c))) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-26) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4.9e-98) {
		tmp = sqrt((a * -c)) / 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 <= (-3.2d-26)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 4.9d-98) then
        tmp = sqrt((a * -c)) / 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 <= -3.2e-26) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4.9e-98) {
		tmp = Math.sqrt((a * -c)) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.2e-26:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 4.9e-98:
		tmp = math.sqrt((a * -c)) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-26)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 4.9e-98)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / 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 <= -3.2e-26)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 4.9e-98)
		tmp = sqrt((a * -c)) / 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, -3.2e-26], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.9e-98], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.2000000000000001e-26

   1. Initial program 66.7%

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

      1. +-commutative 66.7%

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

      2. unsub-neg 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in b_2 around -inf 87.1%

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

   if -3.2000000000000001e-26 < b_2 < 4.90000000000000014e-98

   1. Initial program 79.2%

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

      1. +-commutative 79.2%

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

      2. unsub-neg 79.2%

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

   3. Simplified 79.2%

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

      1. prod-diff 78.9%

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

      2. *-commutative 78.9%

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

      3. fma-neg 78.9%

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

      4. prod-diff 78.9%

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

      5. *-commutative 78.9%

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

      6. fma-neg 78.9%

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

      7. associate-+l+ 78.8%

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

      8. pow2 78.8%

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

      9. *-commutative 78.8%

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

      10. fma-undefine 78.9%

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

      11. distribute-lft-neg-in 78.9%

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

      12. *-commutative 78.9%

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

      13. distribute-rgt-neg-in 78.9%

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

      14. fma-define 78.8%

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

      15. *-commutative 78.8%

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

      16. fma-undefine 78.9%

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

      17. distribute-lft-neg-in 78.9%

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

      18. *-commutative 78.9%

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

      19. distribute-rgt-neg-in 78.9%

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

   6. Applied egg-rr 78.8%

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

      1. associate-+l- 78.8%

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

      2. count-2 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in b_2 around 0 70.3%

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

       1. associate-*l/ 70.4%

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

       2. *-lft-identity 70.4%

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

       3. *-commutative 70.4%

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

       4. neg-mul-1 70.4%

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

       5. distribute-rgt-neg-in 70.4%

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

       6. fma-define 70.3%

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

       7. fma-define 70.4%

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

       8. distribute-rgt-neg-in 70.4%

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

       9. neg-mul-1 70.4%

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

       10. distribute-lft1-in 70.4%

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

       11. metadata-eval 70.4%

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

       12. mul0-lft 70.7%

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

       13. metadata-eval 70.7%

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

       14. neg-sub0 70.7%

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

       15. distribute-rgt-neg-in 70.7%

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

   11. Simplified 70.7%

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

   if 4.90000000000000014e-98 < b_2

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

      2. unsub-neg 21.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in b_2 around inf 85.9%

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

      1. associate-*r/ 85.9%

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

      2. *-commutative 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-26}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 4.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.1 \cdot 10^{-203}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 9.4 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.1e-203)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 9.4e-154) (sqrt (/ c (- a))) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.1e-203) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 9.4e-154) {
		tmp = sqrt((c / -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 <= (-2.1d-203)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 9.4d-154) then
        tmp = sqrt((c / -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 <= -2.1e-203) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 9.4e-154) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.1e-203:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 9.4e-154:
		tmp = math.sqrt((c / -a))
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.1e-203)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 9.4e-154)
		tmp = sqrt(Float64(c / Float64(-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 <= -2.1e-203)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 9.4e-154)
		tmp = sqrt((c / -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, -2.1e-203], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9.4e-154], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.10000000000000002e-203

   1. Initial program 71.0%

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

      1. +-commutative 71.0%

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

      2. unsub-neg 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in b_2 around -inf 72.6%

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

   if -2.10000000000000002e-203 < b_2 < 9.4000000000000003e-154

   1. Initial program 78.0%

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

      1. +-commutative 78.0%

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

      2. unsub-neg 78.0%

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

   3. Simplified 78.0%

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

      1. prod-diff 77.7%

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

      2. *-commutative 77.7%

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

      3. fma-neg 77.7%

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

      4. prod-diff 77.7%

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

      5. *-commutative 77.7%

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

      6. fma-neg 77.7%

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

      7. associate-+l+ 77.6%

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

      8. pow2 77.6%

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

      9. *-commutative 77.6%

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

      10. fma-undefine 77.7%

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

      11. distribute-lft-neg-in 77.7%

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

      12. *-commutative 77.7%

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

      13. distribute-rgt-neg-in 77.7%

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

      14. fma-define 77.6%

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

      15. *-commutative 77.6%

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

      16. fma-undefine 77.7%

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

      17. distribute-lft-neg-in 77.7%

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

      18. *-commutative 77.7%

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

      19. distribute-rgt-neg-in 77.7%

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

   6. Applied egg-rr 77.6%

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

      1. associate-+l- 77.6%

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

      2. count-2 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in a around inf 38.5%

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

       1. *-commutative 38.5%

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

       2. distribute-rgt1-in 38.5%

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

       3. metadata-eval 38.5%

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

       4. mul0-lft 38.5%

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

       5. metadata-eval 38.5%

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

       6. neg-sub0 38.5%

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

   11. Simplified 38.5%

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

   if 9.4000000000000003e-154 < b_2

   1. Initial program 26.7%

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

      1. +-commutative 26.7%

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

      2. unsub-neg 26.7%

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

   3. Simplified 26.7%

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

   5. Taylor expanded in b_2 around inf 80.6%

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

      1. associate-*r/ 80.6%

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

      2. *-commutative 80.6%

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

   7. Simplified 80.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.1 \cdot 10^{-203}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 9.4 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 3.3 \cdot 10^{+29}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 3.3e+29) (* -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 <= 3.3e+29) {
		tmp = -2.0 * (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 <= 3.3d+29) then
        tmp = (-2.0d0) * (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 <= 3.3e+29) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = 0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 3.3e+29:
		tmp = -2.0 * (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 <= 3.3e+29)
		tmp = Float64(-2.0 * Float64(b_2 / 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 <= 3.3e+29)
		tmp = -2.0 * (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, 3.3e+29], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 3.29999999999999984e29

   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 48.6%

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

   if 3.29999999999999984e29 < b_2

   1. Initial program 15.6%

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

      1. +-commutative 15.6%

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

      2. unsub-neg 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in c around 0 86.7%

      \[\leadsto \color{blue}{c \cdot \left(-0.125 \cdot \frac{a \cdot c}{{b\_2}^{3}} - 0.5 \cdot \frac{1}{b\_2}\right)} \]

   6. Taylor expanded in a around 0 93.0%

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

      1. associate-*r/ 93.3%

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

      2. frac-2neg 93.3%

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

      3. distribute-lft-neg-in 93.3%

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

      4. add-sqr-sqrt 43.5%

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

      5. sqrt-unprod 48.7%

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

      6. sqr-neg 48.7%

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

      7. sqrt-unprod 14.6%

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

      8. add-sqr-sqrt 31.4%

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

   8. Applied egg-rr 31.4%

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

      1. *-commutative 31.4%

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

      2. neg-mul-1 31.4%

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

      3. times-frac 31.4%

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

      4. metadata-eval 31.4%

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

   10. Simplified 31.4%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 3.3 \cdot 10^{+29}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.4 \cdot 10^{-307}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.4e-307) (* -2.0 (/ b_2 a)) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.4e-307) {
		tmp = -2.0 * (b_2 / 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 <= 2.4d-307) then
        tmp = (-2.0d0) * (b_2 / 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 <= 2.4e-307) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.4e-307:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.4e-307)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.4e-307)
		tmp = -2.0 * (b_2 / 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, 2.4e-307], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.40000000000000018e-307

   1. Initial program 71.4%

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

      1. +-commutative 71.4%

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

      2. unsub-neg 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in b_2 around -inf 65.7%

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

   if 2.40000000000000018e-307 < b_2

   1. Initial program 35.6%

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

      1. +-commutative 35.6%

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

      2. unsub-neg 35.6%

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

   3. Simplified 35.6%

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

   5. Taylor expanded in c around 0 60.9%

      \[\leadsto \color{blue}{c \cdot \left(-0.125 \cdot \frac{a \cdot c}{{b\_2}^{3}} - 0.5 \cdot \frac{1}{b\_2}\right)} \]

   6. Taylor expanded in a around 0 67.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.4 \cdot 10^{-307}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310) (* -2.0 (/ b_2 a)) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = -2.0 * (b_2 / 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 <= (-2d-310)) then
        tmp = (-2.0d0) * (b_2 / 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 <= -2e-310) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(-2.0 * Float64(b_2 / 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 <= -2e-310)
		tmp = -2.0 * (b_2 / 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, -2e-310], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 71.4%

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

      1. +-commutative 71.4%

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

      2. unsub-neg 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in b_2 around -inf 65.7%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 35.6%

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

      1. +-commutative 35.6%

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

      2. unsub-neg 35.6%

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

   3. Simplified 35.6%

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

   5. Taylor expanded in b_2 around inf 67.9%

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

      1. associate-*r/ 67.9%

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

      2. *-commutative 67.9%

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

   7. Simplified 67.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.3% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{b\_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -2.0 (/ b_2 a)))

C

double code(double a, double b_2, double c) {
	return -2.0 * (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 = (-2.0d0) * (b_2 / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Python

def code(a, b_2, c):
	return -2.0 * (b_2 / a)

Julia

function code(a, b_2, c)
	return Float64(-2.0 * Float64(b_2 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -2.0 * (b_2 / a);
end

Wolfram

code[a_, b$95$2_, c_] := N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. +-commutative 54.6%

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

   2. unsub-neg 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in b_2 around -inf 36.3%

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

6. Final simplification 36.3%

   \[\leadsto -2 \cdot \frac{b\_2}{a} \]
7. Add Preprocessing

Alternative 10: 15.0% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{-b\_2}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. +-commutative 54.6%

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

   2. unsub-neg 54.6%

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

3. Simplified 54.6%

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

   1. prod-diff 54.3%

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

   2. *-commutative 54.3%

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

   3. fma-neg 54.3%

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

   4. prod-diff 54.3%

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

   5. *-commutative 54.3%

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

   6. fma-neg 54.3%

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

   7. associate-+l+ 54.2%

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

   8. pow2 54.2%

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

   9. *-commutative 54.2%

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

   10. fma-undefine 54.3%

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

   11. distribute-lft-neg-in 54.3%

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

   12. *-commutative 54.3%

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

   13. distribute-rgt-neg-in 54.3%

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

   14. fma-define 54.2%

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

   15. *-commutative 54.2%

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

   16. fma-undefine 54.3%

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

   17. distribute-lft-neg-in 54.3%

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

   18. *-commutative 54.3%

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

   19. distribute-rgt-neg-in 54.3%

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

6. Applied egg-rr 54.2%

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

   1. associate-+l- 54.2%

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

   2. count-2 54.2%

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

8. Simplified 54.2%

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

9. Taylor expanded in c around inf 21.9%

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

    1. +-commutative 21.9%

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

    2. mul-1-neg 21.9%

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

    3. unsub-neg 21.9%

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

    4. associate-*l/ 22.0%

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

    5. *-lft-identity 22.0%

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

    6. *-commutative 22.0%

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

    7. distribute-rgt1-in 22.0%

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

    8. metadata-eval 22.0%

       \[\leadsto c \cdot \left(\frac{\sqrt{\frac{\left(\color{blue}{0} \cdot a\right) \cdot 2 - a}{c}}}{a} - \frac{b\_2}{a \cdot c}\right) \]

    9. mul0-lft 22.0%

       \[\leadsto c \cdot \left(\frac{\sqrt{\frac{\color{blue}{0} \cdot 2 - a}{c}}}{a} - \frac{b\_2}{a \cdot c}\right) \]

    10. metadata-eval 22.0%

        \[\leadsto c \cdot \left(\frac{\sqrt{\frac{\color{blue}{0} - a}{c}}}{a} - \frac{b\_2}{a \cdot c}\right) \]

    11. neg-sub0 22.0%

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

11. Simplified 22.0%

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

12. Taylor expanded in b_2 around inf 17.6%

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

    1. associate-*r/ 17.6%

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

    2. neg-mul-1 17.6%

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

14. Simplified 17.6%

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

15. Final simplification 17.6%

    \[\leadsto \frac{-b\_2}{a} \]
16. 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 2024085 
(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))