quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.1% → 85.6%

Time: 9.8s

Alternatives: 9

Speedup: 1.7×

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: 52.1% 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: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.5 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.5e+122)
   (fma c (/ 0.5 b_2) (/ (* b_2 -2.0) a))
   (if (<= b_2 1.1e-126)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.5e+122) {
		tmp = fma(c, (0.5 / b_2), ((b_2 * -2.0) / a));
	} else if (b_2 <= 1.1e-126) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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 <= -9.5e+122)
		tmp = fma(c, Float64(0.5 / b_2), Float64(Float64(b_2 * -2.0) / a));
	elseif (b_2 <= 1.1e-126)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9.5e+122], N[(c * N[(0.5 / b$95$2), $MachinePrecision] + N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.1e-126], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $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 < -9.49999999999999986e122

   1. Initial program 53.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto 0 - b\_2 \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b\_2}^{2}}} + 2 \cdot \frac{1}{a}\right) \]

      6. *-commutative N/A

         \[\leadsto 0 - b\_2 \cdot \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right) \]

      7. associate-/l* N/A

         \[\leadsto 0 - b\_2 \cdot \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b\_2}^{2}}} + 2 \cdot \frac{1}{a}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto 0 - b\_2 \cdot \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b\_2}^{2}}, 2 \cdot \frac{1}{a}\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b\_2}^{2}}}, 2 \cdot \frac{1}{a}\right) \]

      10. unpow2 N/A

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b\_2 \cdot b\_2}}, 2 \cdot \frac{1}{a}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b\_2 \cdot b\_2}}, 2 \cdot \frac{1}{a}\right) \]

      12. associate-*r/ N/A

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b\_2 \cdot b\_2}, \color{blue}{\frac{2 \cdot 1}{a}}\right) \]

      13. metadata-eval N/A

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b\_2 \cdot b\_2}, \frac{\color{blue}{2}}{a}\right) \]

      14. /-lowering-/.f64 95.4

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{-0.5}{b\_2 \cdot b\_2}, \color{blue}{\frac{2}{a}}\right) \]

   5. Simplified 95.4%

      \[\leadsto \color{blue}{0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{-0.5}{b\_2 \cdot b\_2}, \frac{2}{a}\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2} - 2 \cdot \frac{b\_2}{a}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 96.2

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

   8. Simplified 96.2%

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

   if -9.49999999999999986e122 < b_2 < 1.10000000000000007e-126

   1. Initial program 87.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 1.10000000000000007e-126 < b_2

   1. Initial program 14.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 85.3

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

   5. Simplified 85.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.5 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, 0 - a, 0\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-98)
   (fma c (/ 0.5 b_2) (/ (* b_2 -2.0) a))
   (if (<= b_2 2e-126)
     (/ (- (sqrt (fma c (- 0.0 a) 0.0)) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.1e-98) {
		tmp = fma(c, (0.5 / b_2), ((b_2 * -2.0) / a));
	} else if (b_2 <= 2e-126) {
		tmp = (sqrt(fma(c, (0.0 - a), 0.0)) - 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 <= -3.1e-98)
		tmp = fma(c, Float64(0.5 / b_2), Float64(Float64(b_2 * -2.0) / a));
	elseif (b_2 <= 2e-126)
		tmp = Float64(Float64(sqrt(fma(c, Float64(0.0 - a), 0.0)) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.1e-98], N[(c * N[(0.5 / b$95$2), $MachinePrecision] + N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2e-126], N[(N[(N[Sqrt[N[(c * N[(0.0 - a), $MachinePrecision] + 0.0), $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 < -3.1e-98

   1. Initial program 71.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto 0 - b\_2 \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b\_2}^{2}}} + 2 \cdot \frac{1}{a}\right) \]

      6. *-commutative N/A

         \[\leadsto 0 - b\_2 \cdot \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right) \]

      7. associate-/l* N/A

         \[\leadsto 0 - b\_2 \cdot \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b\_2}^{2}}} + 2 \cdot \frac{1}{a}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto 0 - b\_2 \cdot \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b\_2}^{2}}, 2 \cdot \frac{1}{a}\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b\_2}^{2}}}, 2 \cdot \frac{1}{a}\right) \]

      10. unpow2 N/A

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b\_2 \cdot b\_2}}, 2 \cdot \frac{1}{a}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b\_2 \cdot b\_2}}, 2 \cdot \frac{1}{a}\right) \]

      12. associate-*r/ N/A

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b\_2 \cdot b\_2}, \color{blue}{\frac{2 \cdot 1}{a}}\right) \]

      13. metadata-eval N/A

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b\_2 \cdot b\_2}, \frac{\color{blue}{2}}{a}\right) \]

      14. /-lowering-/.f64 86.3

          \[\leadsto 0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{-0.5}{b\_2 \cdot b\_2}, \color{blue}{\frac{2}{a}}\right) \]

   5. Simplified 86.3%

      \[\leadsto \color{blue}{0 - b\_2 \cdot \mathsf{fma}\left(c, \frac{-0.5}{b\_2 \cdot b\_2}, \frac{2}{a}\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2} - 2 \cdot \frac{b\_2}{a}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 87.0

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

   8. Simplified 87.0%

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

   if -3.1e-98 < b_2 < 1.9999999999999999e-126

   1. Initial program 84.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 83.4

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

   5. Simplified 83.4%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{0 - a \cdot c}}}{a} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

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

      6. +-commutative N/A

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

      7. sub0-neg N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 83.4

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

   7. Applied egg-rr 83.4%

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

   if 1.9999999999999999e-126 < b_2

   1. Initial program 14.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 85.3

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

   5. Simplified 85.3%

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

Alternative 3: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, 0 - a, 0\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.5e-99)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.25e-126)
     (/ (- (sqrt (fma c (- 0.0 a) 0.0)) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.5e-99) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.25e-126) {
		tmp = (sqrt(fma(c, (0.0 - a), 0.0)) - 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 <= -6.5e-99)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.25e-126)
		tmp = Float64(Float64(sqrt(fma(c, Float64(0.0 - a), 0.0)) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.5e-99], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.25e-126], N[(N[(N[Sqrt[N[(c * N[(0.0 - a), $MachinePrecision] + 0.0), $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 < -6.50000000000000033e-99

   1. Initial program 71.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.6

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

   5. Simplified 86.6%

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

   if -6.50000000000000033e-99 < b_2 < 1.25000000000000001e-126

   1. Initial program 84.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 83.4

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

   5. Simplified 83.4%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{0 - a \cdot c}}}{a} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

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

      6. +-commutative N/A

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

      7. sub0-neg N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 83.4

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

   7. Applied egg-rr 83.4%

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

   if 1.25000000000000001e-126 < b_2

   1. Initial program 14.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 85.3

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

   5. Simplified 85.3%

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

Alternative 4: 67.7% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* 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 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		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, -5e-310], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 74.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 30.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 68.8

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

   5. Simplified 68.8%

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

Alternative 5: 67.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* 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 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 74.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 30.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 68.8

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

   5. Simplified 68.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 68.6

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

   7. Applied egg-rr 68.6%

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

4. Final simplification 67.9%

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

Alternative 6: 67.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* 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 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 74.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 67.0

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

   7. Applied egg-rr 67.0%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 30.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 68.8

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

   5. Simplified 68.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 68.6

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

   7. Applied egg-rr 68.6%

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

4. Final simplification 67.8%

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

Alternative 7: 43.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.45 \cdot 10^{-290}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.45e-290) (* b_2 (/ -2.0 a)) 0.0))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.45e-290) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = 0.0;
	}
	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 <= (-2.45d-290)) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.45e-290) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.45e-290:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.45e-290)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.45e-290)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.45e-290], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.45e-290

   1. Initial program 74.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 68.2

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

   5. Simplified 68.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 68.0

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

   7. Applied egg-rr 68.0%

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

   if -2.45e-290 < b_2

   1. Initial program 31.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{{b\_2}^{2}}}}{a} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 4.3

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

   5. Simplified 4.3%

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

   6. Taylor expanded in b_2 around 0

      \[\leadsto \color{blue}{0} \]
   7. Step-by-step derivation

   8. Simplified 18.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.45 \cdot 10^{-290}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 23.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.45 \cdot 10^{-290}:\\ \;\;\;\;0 - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.45e-290) (- 0.0 (/ b_2 a)) 0.0))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.45e-290) {
		tmp = 0.0 - (b_2 / a);
	} else {
		tmp = 0.0;
	}
	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 <= (-2.45d-290)) then
        tmp = 0.0d0 - (b_2 / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.45e-290) {
		tmp = 0.0 - (b_2 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.45e-290:
		tmp = 0.0 - (b_2 / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.45e-290)
		tmp = Float64(0.0 - Float64(b_2 / a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.45e-290)
		tmp = 0.0 - (b_2 / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.45e-290], N[(0.0 - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.45e-290

   1. Initial program 74.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 41.7

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

   5. Simplified 41.7%

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

   6. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\_2\right)}}{a} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - b\_2}}{a} \]

      3. --lowering--.f64 28.8

         \[\leadsto \frac{\color{blue}{0 - b\_2}}{a} \]

   8. Simplified 28.8%

      \[\leadsto \frac{\color{blue}{0 - b\_2}}{a} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\_2\right)}}{a} \]

      2. neg-lowering-neg.f64 28.8

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

   10. Applied egg-rr 28.8%

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

   if -2.45e-290 < b_2

   1. Initial program 31.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{{b\_2}^{2}}}}{a} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 4.3

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

   5. Simplified 4.3%

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

   6. Taylor expanded in b_2 around 0

      \[\leadsto \color{blue}{0} \]
   7. Step-by-step derivation

   8. Simplified 18.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.45 \cdot 10^{-290}:\\ \;\;\;\;0 - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 11.2% accurate, 40.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 52.0%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in b_2 around inf

   \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{{b\_2}^{2}}}}{a} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. *-lowering-*.f64 26.9

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

5. Simplified 26.9%

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

6. Taylor expanded in b_2 around 0

   \[\leadsto \color{blue}{0} \]
7. Step-by-step derivation

8. Simplified 11.1%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.2× 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 2024197 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

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