quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.5% → 84.1%

Time: 11.5s

Alternatives: 11

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: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\frac{b\_2 \cdot \left(-0.5 \cdot \left(a \cdot \frac{-c}{{b\_2}^{2}}\right) - 2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq -1.55 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{{b\_2}^{2} + \left(2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right) - a \cdot c\right)} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{a \cdot \left(-c\right)}, b\_2\right) - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+146)
   (/ (* b_2 (- (* -0.5 (* a (/ (- c) (pow b_2 2.0)))) 2.0)) a)
   (if (<= b_2 -1.55e-112)
     (/
      (-
       (sqrt (+ (pow b_2 2.0) (- (* 2.0 (fma a (- c) (* a c))) (* a c))))
       b_2)
      a)
     (if (<= b_2 1.3e-175)
       (/ (- (hypot (sqrt (* a (- c))) b_2) b_2) a)
       (/ (* -0.5 c) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+146) {
		tmp = (b_2 * ((-0.5 * (a * (-c / pow(b_2, 2.0)))) - 2.0)) / a;
	} else if (b_2 <= -1.55e-112) {
		tmp = (sqrt((pow(b_2, 2.0) + ((2.0 * fma(a, -c, (a * c))) - (a * c)))) - b_2) / a;
	} else if (b_2 <= 1.3e-175) {
		tmp = (hypot(sqrt((a * -c)), b_2) - 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 <= -1e+146)
		tmp = Float64(Float64(b_2 * Float64(Float64(-0.5 * Float64(a * Float64(Float64(-c) / (b_2 ^ 2.0)))) - 2.0)) / a);
	elseif (b_2 <= -1.55e-112)
		tmp = Float64(Float64(sqrt(Float64((b_2 ^ 2.0) + Float64(Float64(2.0 * fma(a, Float64(-c), Float64(a * c))) - Float64(a * c)))) - b_2) / a);
	elseif (b_2 <= 1.3e-175)
		tmp = Float64(Float64(hypot(sqrt(Float64(a * Float64(-c))), b_2) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+146], N[(N[(b$95$2 * N[(N[(-0.5 * N[(a * N[((-c) / N[Power[b$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -1.55e-112], N[(N[(N[Sqrt[N[(N[Power[b$95$2, 2.0], $MachinePrecision] + N[(N[(2.0 * N[(a * (-c) + N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.3e-175], N[(N[(N[Sqrt[N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] ^ 2 + b$95$2 ^ 2], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -9.99999999999999934e145

   1. Initial program 42.7%

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

      1. +-commutative 42.7%

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

      2. unsub-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in b_2 around -inf 70.4%

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

      1. associate-*r* 70.4%

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

      2. neg-mul-1 70.4%

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

      3. associate-/l* 92.5%

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

   7. Simplified 92.5%

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

   if -9.99999999999999934e145 < b_2 < -1.5499999999999999e-112

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. unsub-neg 96.1%

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

   3. Simplified 96.1%

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

      1. prod-diff 95.9%

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

      2. *-commutative 95.9%

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

      3. fmm-def 95.9%

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

      4. prod-diff 95.9%

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

      5. *-commutative 95.9%

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

      6. fmm-def 95.9%

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

      7. associate-+l+ 96.1%

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

      8. pow2 96.1%

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

      9. *-commutative 96.1%

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

      10. fma-undefine 95.9%

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

      11. distribute-lft-neg-in 95.9%

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

      12. *-commutative 95.9%

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

      13. distribute-rgt-neg-in 95.9%

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

      14. fma-define 96.1%

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

      15. *-commutative 96.1%

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

      16. fma-undefine 95.9%

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

      17. distribute-lft-neg-in 95.9%

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

      18. *-commutative 95.9%

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

      19. distribute-rgt-neg-in 95.9%

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

   6. Applied egg-rr 96.1%

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

      1. associate-+l- 96.1%

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

      2. count-2 96.1%

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

   8. Simplified 96.1%

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

   if -1.5499999999999999e-112 < b_2 < 1.3e-175

   1. Initial program 62.3%

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

      1. +-commutative 62.3%

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

      2. unsub-neg 62.3%

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

   3. Simplified 62.3%

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

      1. sub-neg 62.3%

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

      2. +-commutative 62.3%

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

      3. add-sqr-sqrt 62.3%

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

      4. hypot-define 69.4%

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

      5. *-commutative 69.4%

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

      6. distribute-rgt-neg-in 69.4%

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

   6. Applied egg-rr 69.4%

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

   if 1.3e-175 < b_2

   1. Initial program 21.2%

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

      1. +-commutative 21.2%

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

      2. unsub-neg 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in b_2 around inf 81.4%

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

      1. associate-*r/ 81.5%

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

      2. *-commutative 81.5%

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

   7. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\frac{b\_2 \cdot \left(-0.5 \cdot \left(a \cdot \frac{-c}{{b\_2}^{2}}\right) - 2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq -1.55 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{{b\_2}^{2} + \left(2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right) - a \cdot c\right)} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{a \cdot \left(-c\right)}, b\_2\right) - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\frac{b\_2 \cdot \left(-0.5 \cdot \left(a \cdot \frac{-c}{{b\_2}^{2}}\right) - 2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq -4 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{a \cdot \left(-c\right)}, b\_2\right) - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+146)
   (/ (* b_2 (- (* -0.5 (* a (/ (- c) (pow b_2 2.0)))) 2.0)) a)
   (if (<= b_2 -4e-99)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (if (<= b_2 1.3e-175)
       (/ (- (hypot (sqrt (* a (- c))) b_2) b_2) a)
       (/ (* -0.5 c) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+146) {
		tmp = (b_2 * ((-0.5 * (a * (-c / pow(b_2, 2.0)))) - 2.0)) / a;
	} else if (b_2 <= -4e-99) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else if (b_2 <= 1.3e-175) {
		tmp = (hypot(sqrt((a * -c)), b_2) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+146) {
		tmp = (b_2 * ((-0.5 * (a * (-c / Math.pow(b_2, 2.0)))) - 2.0)) / a;
	} else if (b_2 <= -4e-99) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else if (b_2 <= 1.3e-175) {
		tmp = (Math.hypot(Math.sqrt((a * -c)), b_2) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e+146:
		tmp = (b_2 * ((-0.5 * (a * (-c / math.pow(b_2, 2.0)))) - 2.0)) / a
	elif b_2 <= -4e-99:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	elif b_2 <= 1.3e-175:
		tmp = (math.hypot(math.sqrt((a * -c)), b_2) - 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 <= -1e+146)
		tmp = Float64(Float64(b_2 * Float64(Float64(-0.5 * Float64(a * Float64(Float64(-c) / (b_2 ^ 2.0)))) - 2.0)) / a);
	elseif (b_2 <= -4e-99)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	elseif (b_2 <= 1.3e-175)
		tmp = Float64(Float64(hypot(sqrt(Float64(a * Float64(-c))), b_2) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e+146)
		tmp = (b_2 * ((-0.5 * (a * (-c / (b_2 ^ 2.0)))) - 2.0)) / a;
	elseif (b_2 <= -4e-99)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	elseif (b_2 <= 1.3e-175)
		tmp = (hypot(sqrt((a * -c)), b_2) - 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, -1e+146], N[(N[(b$95$2 * N[(N[(-0.5 * N[(a * N[((-c) / N[Power[b$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -4e-99], 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, 1.3e-175], N[(N[(N[Sqrt[N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] ^ 2 + b$95$2 ^ 2], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -9.99999999999999934e145

   1. Initial program 42.7%

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

      1. +-commutative 42.7%

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

      2. unsub-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in b_2 around -inf 70.4%

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

      1. associate-*r* 70.4%

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

      2. neg-mul-1 70.4%

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

      3. associate-/l* 92.5%

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

   7. Simplified 92.5%

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

   if -9.99999999999999934e145 < b_2 < -4.0000000000000001e-99

   1. Initial program 95.8%

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

      1. +-commutative 95.8%

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

      2. unsub-neg 95.8%

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

   3. Simplified 95.8%

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

   if -4.0000000000000001e-99 < b_2 < 1.3e-175

   1. Initial program 64.1%

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

      1. +-commutative 64.1%

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

      2. unsub-neg 64.1%

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

   3. Simplified 64.1%

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

      1. sub-neg 64.1%

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

      2. +-commutative 64.1%

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

      3. add-sqr-sqrt 64.1%

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

      4. hypot-define 70.8%

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

      5. *-commutative 70.8%

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

      6. distribute-rgt-neg-in 70.8%

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

   6. Applied egg-rr 70.8%

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

   if 1.3e-175 < b_2

   1. Initial program 21.2%

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

      1. +-commutative 21.2%

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

      2. unsub-neg 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in b_2 around inf 81.4%

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

      1. associate-*r/ 81.5%

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

      2. *-commutative 81.5%

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

   7. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\frac{b\_2 \cdot \left(-0.5 \cdot \left(a \cdot \frac{-c}{{b\_2}^{2}}\right) - 2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq -4 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{a \cdot \left(-c\right)}, b\_2\right) - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.02 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+146)
   (- (* -0.5 (/ c (- b_2))) (* 2.0 (/ b_2 a)))
   (if (<= b_2 1.02e-156)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+146) {
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 1.02e-156) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= (-1d+146)) then
        tmp = ((-0.5d0) * (c / -b_2)) - (2.0d0 * (b_2 / a))
    else if (b_2 <= 1.02d-156) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= -1e+146) {
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 1.02e-156) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e+146:
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a))
	elif b_2 <= 1.02e-156:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - 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 <= -1e+146)
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(-b_2))) - Float64(2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 1.02e-156)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e+146)
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	elseif (b_2 <= 1.02e-156)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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, -1e+146], N[(N[(-0.5 * N[(c / (-b$95$2)), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.02e-156], 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[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.99999999999999934e145

   1. Initial program 42.7%

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

      1. +-commutative 42.7%

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

      2. unsub-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in b_2 around -inf 92.3%

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

   6. Taylor expanded in c around 0 92.5%

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

   if -9.99999999999999934e145 < b_2 < 1.02e-156

   1. Initial program 77.8%

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

      1. +-commutative 77.8%

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

      2. unsub-neg 77.8%

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

   3. Simplified 77.8%

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

   if 1.02e-156 < b_2

   1. Initial program 20.6%

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

      1. +-commutative 20.6%

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

      2. unsub-neg 20.6%

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

   3. Simplified 20.6%

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

   5. Taylor expanded in b_2 around inf 81.9%

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

      1. associate-*r/ 82.0%

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

      2. *-commutative 82.0%

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

   7. Simplified 82.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.02 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{+146}:\\ \;\;\;\;\frac{b\_2 \cdot \left(-0.5 \cdot \left(a \cdot \frac{-c}{{b\_2}^{2}}\right) - 2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.06e+146)
   (/ (* b_2 (- (* -0.5 (* a (/ (- c) (pow b_2 2.0)))) 2.0)) a)
   (if (<= b_2 1.1e-156)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.06e+146) {
		tmp = (b_2 * ((-0.5 * (a * (-c / pow(b_2, 2.0)))) - 2.0)) / a;
	} else if (b_2 <= 1.1e-156) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= (-1.06d+146)) then
        tmp = (b_2 * (((-0.5d0) * (a * (-c / (b_2 ** 2.0d0)))) - 2.0d0)) / a
    else if (b_2 <= 1.1d-156) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= -1.06e+146) {
		tmp = (b_2 * ((-0.5 * (a * (-c / Math.pow(b_2, 2.0)))) - 2.0)) / a;
	} else if (b_2 <= 1.1e-156) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.06e+146:
		tmp = (b_2 * ((-0.5 * (a * (-c / math.pow(b_2, 2.0)))) - 2.0)) / a
	elif b_2 <= 1.1e-156:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - 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 <= -1.06e+146)
		tmp = Float64(Float64(b_2 * Float64(Float64(-0.5 * Float64(a * Float64(Float64(-c) / (b_2 ^ 2.0)))) - 2.0)) / a);
	elseif (b_2 <= 1.1e-156)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.06e+146)
		tmp = (b_2 * ((-0.5 * (a * (-c / (b_2 ^ 2.0)))) - 2.0)) / a;
	elseif (b_2 <= 1.1e-156)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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, -1.06e+146], N[(N[(b$95$2 * N[(N[(-0.5 * N[(a * N[((-c) / N[Power[b$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.1e-156], 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[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.06000000000000005e146

   1. Initial program 42.7%

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

      1. +-commutative 42.7%

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

      2. unsub-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in b_2 around -inf 70.4%

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

      1. associate-*r* 70.4%

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

      2. neg-mul-1 70.4%

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

      3. associate-/l* 92.5%

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

   7. Simplified 92.5%

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

   if -1.06000000000000005e146 < b_2 < 1.1e-156

   1. Initial program 77.8%

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

      1. +-commutative 77.8%

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

      2. unsub-neg 77.8%

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

   3. Simplified 77.8%

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

   if 1.1e-156 < b_2

   1. Initial program 20.6%

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

      1. +-commutative 20.6%

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

      2. unsub-neg 20.6%

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

   3. Simplified 20.6%

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

   5. Taylor expanded in b_2 around inf 81.9%

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

      1. associate-*r/ 82.0%

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

      2. *-commutative 82.0%

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

   7. Simplified 82.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{+146}:\\ \;\;\;\;\frac{b\_2 \cdot \left(-0.5 \cdot \left(a \cdot \frac{-c}{{b\_2}^{2}}\right) - 2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.4 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.4e-59)
   (- (* -0.5 (/ c (- b_2))) (* 2.0 (/ b_2 a)))
   (if (<= b_2 1.1e-156) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.4e-59) {
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 1.1e-156) {
		tmp = (sqrt((a * -c)) - 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 <= (-3.4d-59)) then
        tmp = ((-0.5d0) * (c / -b_2)) - (2.0d0 * (b_2 / a))
    else if (b_2 <= 1.1d-156) then
        tmp = (sqrt((a * -c)) - 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 <= -3.4e-59) {
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 1.1e-156) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.4e-59:
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a))
	elif b_2 <= 1.1e-156:
		tmp = (math.sqrt((a * -c)) - 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 <= -3.4e-59)
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(-b_2))) - Float64(2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 1.1e-156)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.4e-59)
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	elseif (b_2 <= 1.1e-156)
		tmp = (sqrt((a * -c)) - 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, -3.4e-59], N[(N[(-0.5 * N[(c / (-b$95$2)), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.1e-156], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.40000000000000018e-59

   1. Initial program 68.7%

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

      1. +-commutative 68.7%

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

      2. unsub-neg 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in b_2 around -inf 84.0%

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

   6. Taylor expanded in c around 0 84.3%

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

   if -3.40000000000000018e-59 < b_2 < 1.1e-156

   1. Initial program 69.5%

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

      1. +-commutative 69.5%

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

      2. unsub-neg 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in b_2 around 0 59.6%

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

      1. associate-*r* 59.6%

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

      2. neg-mul-1 59.6%

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

      3. *-commutative 59.6%

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

   7. Simplified 59.6%

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

   if 1.1e-156 < b_2

   1. Initial program 20.6%

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

      1. +-commutative 20.6%

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

      2. unsub-neg 20.6%

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

   3. Simplified 20.6%

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

   5. Taylor expanded in b_2 around inf 81.9%

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

      1. associate-*r/ 82.0%

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

      2. *-commutative 82.0%

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

   7. Simplified 82.0%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.4 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-311)
   (- (* -0.5 (/ c (- b_2))) (* 2.0 (/ b_2 a)))
   (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-311) {
		tmp = (-0.5 * (c / -b_2)) - (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 <= (-4d-311)) then
        tmp = ((-0.5d0) * (c / -b_2)) - (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 <= -4e-311) {
		tmp = (-0.5 * (c / -b_2)) - (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 <= -4e-311:
		tmp = (-0.5 * (c / -b_2)) - (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 <= -4e-311)
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(-b_2))) - Float64(2.0 * Float64(b_2 / a)));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-311)
		tmp = (-0.5 * (c / -b_2)) - (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, -4e-311], N[(N[(-0.5 * N[(c / (-b$95$2)), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.99999999999979e-311

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

      2. unsub-neg 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in b_2 around -inf 57.5%

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

   6. Taylor expanded in c around 0 61.9%

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

   if -3.99999999999979e-311 < b_2

   1. Initial program 29.6%

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

      1. +-commutative 29.6%

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

      2. unsub-neg 29.6%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in b_2 around inf 68.3%

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

      1. associate-*r/ 68.3%

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

      2. *-commutative 68.3%

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

   7. Simplified 68.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 6.2 \cdot 10^{-299}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 6.2e-299) (* 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 <= 6.2e-299) {
		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 <= 6.2d-299) 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 <= 6.2e-299) {
		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 <= 6.2e-299:
		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 <= 6.2e-299)
		tmp = Float64(b_2 * Float64(-2.0 / 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 <= 6.2e-299)
		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, 6.2e-299], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 6.1999999999999999e-299

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. unsub-neg 70.1%

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

   3. Simplified 70.1%

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

      1. clear-num 70.0%

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

      2. associate-/r/ 69.9%

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

      3. sub-neg 69.9%

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

      4. add-sqr-sqrt 55.5%

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

      5. hypot-define 64.0%

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

      6. *-commutative 64.0%

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

      7. distribute-rgt-neg-in 64.0%

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

   6. Applied egg-rr 64.0%

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

   7. Taylor expanded in b_2 around -inf 59.5%

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

      1. metadata-eval 59.5%

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

      2. distribute-lft-neg-in 59.5%

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

      3. associate-*r/ 59.5%

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

      4. *-commutative 59.5%

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

      5. associate-/l* 59.4%

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

      6. metadata-eval 59.4%

         \[\leadsto -b\_2 \cdot \frac{\color{blue}{2 \cdot 1}}{a} \]

      7. associate-*r/ 59.4%

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

      8. distribute-rgt-neg-in 59.4%

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

      9. associate-*r/ 59.4%

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

      10. metadata-eval 59.4%

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

      11. distribute-neg-frac 59.4%

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

      12. metadata-eval 59.4%

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

   9. Simplified 59.4%

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

   if 6.1999999999999999e-299 < b_2

   1. Initial program 28.2%

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

      1. +-commutative 28.2%

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

      2. unsub-neg 28.2%

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

   3. Simplified 28.2%

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

      1. clear-num 28.2%

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

      2. associate-/r/ 28.2%

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

      3. sub-neg 28.2%

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

      4. add-sqr-sqrt 26.4%

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

      5. hypot-define 39.5%

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

      6. *-commutative 39.5%

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

      7. distribute-rgt-neg-in 39.5%

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

   6. Applied egg-rr 39.5%

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

   7. Taylor expanded in a around 0 0.0%

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

      1. metadata-eval 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 70.3%

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

      5. neg-mul-1 70.3%

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

      6. times-frac 70.3%

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

      7. distribute-rgt-neg-in 70.3%

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

      8. neg-mul-1 70.3%

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

      9. times-frac 70.3%

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

      10. metadata-eval 70.3%

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

      11. *-lft-identity 70.3%

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

      12. *-commutative 70.3%

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

      13. associate-/l* 70.1%

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

   9. Simplified 70.1%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 6.2 \cdot 10^{-299}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 3 \cdot 10^{-299}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 3e-299) (/ (* 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 <= 3e-299) {
		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 <= 3d-299) 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 <= 3e-299) {
		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 <= 3e-299:
		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 <= 3e-299)
		tmp = Float64(Float64(b_2 * -2.0) / 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 <= 3e-299)
		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, 3e-299], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.99999999999999984e-299

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. unsub-neg 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in b_2 around -inf 59.5%

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

      1. *-commutative 59.5%

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

   7. Simplified 59.5%

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

   if 2.99999999999999984e-299 < b_2

   1. Initial program 28.2%

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

      1. +-commutative 28.2%

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

      2. unsub-neg 28.2%

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

   3. Simplified 28.2%

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

      1. clear-num 28.2%

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

      2. associate-/r/ 28.2%

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

      3. sub-neg 28.2%

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

      4. add-sqr-sqrt 26.4%

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

      5. hypot-define 39.5%

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

      6. *-commutative 39.5%

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

      7. distribute-rgt-neg-in 39.5%

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

   6. Applied egg-rr 39.5%

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

   7. Taylor expanded in a around 0 0.0%

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

      1. metadata-eval 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 70.3%

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

      5. neg-mul-1 70.3%

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

      6. times-frac 70.3%

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

      7. distribute-rgt-neg-in 70.3%

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

      8. neg-mul-1 70.3%

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

      9. times-frac 70.3%

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

      10. metadata-eval 70.3%

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

      11. *-lft-identity 70.3%

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

      12. *-commutative 70.3%

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

      13. associate-/l* 70.1%

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

   9. Simplified 70.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 3 \cdot 10^{-299}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 3 \cdot 10^{-299}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 3e-299) (/ (* 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 <= 3e-299) {
		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 <= 3d-299) 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 <= 3e-299) {
		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 <= 3e-299:
		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 <= 3e-299)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 3e-299)
		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, 3e-299], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.99999999999999984e-299

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. unsub-neg 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in b_2 around -inf 59.5%

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

      1. *-commutative 59.5%

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

   7. Simplified 59.5%

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

   if 2.99999999999999984e-299 < b_2

   1. Initial program 28.2%

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

      1. +-commutative 28.2%

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

      2. unsub-neg 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in b_2 around inf 70.3%

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

      1. associate-*r/ 70.3%

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

      2. *-commutative 70.3%

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

   7. Simplified 70.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 3 \cdot 10^{-299}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.5% accurate, 22.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = b_2 * ((-2.0d0) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

   1. +-commutative 48.8%

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

   2. unsub-neg 48.8%

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

3. Simplified 48.8%

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

   1. clear-num 48.7%

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

   2. associate-/r/ 48.7%

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

   3. sub-neg 48.7%

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

   4. add-sqr-sqrt 40.7%

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

   5. hypot-define 51.6%

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

   6. *-commutative 51.6%

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

   7. distribute-rgt-neg-in 51.6%

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

6. Applied egg-rr 51.6%

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

7. Taylor expanded in b_2 around -inf 30.5%

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

   1. metadata-eval 30.5%

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

   2. distribute-lft-neg-in 30.5%

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

   3. associate-*r/ 30.5%

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

   4. *-commutative 30.5%

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

   5. associate-/l* 30.4%

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

   6. metadata-eval 30.4%

      \[\leadsto -b\_2 \cdot \frac{\color{blue}{2 \cdot 1}}{a} \]

   7. associate-*r/ 30.4%

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

   8. distribute-rgt-neg-in 30.4%

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

   9. associate-*r/ 30.4%

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

   10. metadata-eval 30.4%

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

   11. distribute-neg-frac 30.4%

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

   12. metadata-eval 30.4%

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

9. Simplified 30.4%

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

10. Final simplification 30.4%

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

Alternative 11: 15.4% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

   1. +-commutative 48.8%

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

   2. unsub-neg 48.8%

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

3. Simplified 48.8%

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

5. Taylor expanded in b_2 around 0 32.1%

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

   1. associate-*r* 32.1%

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

   2. neg-mul-1 32.1%

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

   3. *-commutative 32.1%

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

7. Simplified 32.1%

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

8. Taylor expanded in b_2 around inf 13.4%

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

   1. associate-*r/ 13.4%

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

   2. neg-mul-1 13.4%

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

10. Simplified 13.4%

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

11. Final simplification 13.4%

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

Developer target: 99.7% 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 2024130 
(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))