quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.9% → 85.9%

Time: 7.0s

Alternatives: 7

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: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b\_2} \cdot a, \frac{0.5}{b\_2}, -2\right) \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.26 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+143)
   (/ (* (fma (* (/ c b_2) a) (/ 0.5 b_2) -2.0) b_2) a)
   (if (<= b_2 1.26e-64)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+143) {
		tmp = (fma(((c / b_2) * a), (0.5 / b_2), -2.0) * b_2) / a;
	} else if (b_2 <= 1.26e-64) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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 <= -5e+143)
		tmp = Float64(Float64(fma(Float64(Float64(c / b_2) * a), Float64(0.5 / b_2), -2.0) * b_2) / a);
	elseif (b_2 <= 1.26e-64)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+143], N[(N[(N[(N[(N[(c / b$95$2), $MachinePrecision] * a), $MachinePrecision] * N[(0.5 / b$95$2), $MachinePrecision] + -2.0), $MachinePrecision] * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.26e-64], 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[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.00000000000000012e143

   1. Initial program 43.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 96.6

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

   8. Applied rewrites 96.6%

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

   10. Applied rewrites 96.6%

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

   if -5.00000000000000012e143 < b_2 < 1.2599999999999999e-64

   1. Initial program 85.0%

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

   if 1.2599999999999999e-64 < b_2

   1. Initial program 14.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b\_2} \cdot a, \frac{0.5}{b\_2}, -2\right) \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.26 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b\_2} \cdot 0.5, c, -2 \cdot b\_2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.26 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+143)
   (/ (fma (* (/ a b_2) 0.5) c (* -2.0 b_2)) a)
   (if (<= b_2 1.26e-64)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+143) {
		tmp = fma(((a / b_2) * 0.5), c, (-2.0 * b_2)) / a;
	} else if (b_2 <= 1.26e-64) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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 <= -5e+143)
		tmp = Float64(fma(Float64(Float64(a / b_2) * 0.5), c, Float64(-2.0 * b_2)) / a);
	elseif (b_2 <= 1.26e-64)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+143], N[(N[(N[(N[(a / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision] * c + N[(-2.0 * b$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.26e-64], 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[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.00000000000000012e143

   1. Initial program 43.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 96.6

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 96.6%

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

   if -5.00000000000000012e143 < b_2 < 1.2599999999999999e-64

   1. Initial program 85.0%

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

   if 1.2599999999999999e-64 < b_2

   1. Initial program 14.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b\_2} \cdot 0.5, c, -2 \cdot b\_2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.26 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b\_2} \cdot 0.5, c, -2 \cdot b\_2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.26 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.2e-91)
   (/ (fma (* (/ a b_2) 0.5) c (* -2.0 b_2)) a)
   (if (<= b_2 1.26e-64) (/ (- (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 <= -7.2e-91) {
		tmp = fma(((a / b_2) * 0.5), c, (-2.0 * b_2)) / a;
	} else if (b_2 <= 1.26e-64) {
		tmp = (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 <= -7.2e-91)
		tmp = Float64(fma(Float64(Float64(a / b_2) * 0.5), c, Float64(-2.0 * b_2)) / a);
	elseif (b_2 <= 1.26e-64)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.2e-91], N[(N[(N[(N[(a / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision] * c + N[(-2.0 * b$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.26e-64], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.2000000000000001e-91

   1. Initial program 68.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 2.5

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

   5. Applied rewrites 2.5%

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

   6. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 83.5

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 83.5%

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

   if -7.2000000000000001e-91 < b_2 < 1.2599999999999999e-64

   1. Initial program 79.1%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if 1.2599999999999999e-64 < b_2

   1. Initial program 14.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b\_2} \cdot 0.5, c, -2 \cdot b\_2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.26 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 1.26 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.2e-91)
   (* (/ b_2 a) -2.0)
   (if (<= b_2 1.26e-64) (/ (- (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 <= -7.2e-91) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 1.26e-64) {
		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 <= (-7.2d-91)) then
        tmp = (b_2 / a) * (-2.0d0)
    else if (b_2 <= 1.26d-64) 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 <= -7.2e-91) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 1.26e-64) {
		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 <= -7.2e-91:
		tmp = (b_2 / a) * -2.0
	elif b_2 <= 1.26e-64:
		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 <= -7.2e-91)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	elseif (b_2 <= 1.26e-64)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.2e-91)
		tmp = (b_2 / a) * -2.0;
	elseif (b_2 <= 1.26e-64)
		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, -7.2e-91], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[b$95$2, 1.26e-64], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.2000000000000001e-91

   1. Initial program 68.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if -7.2000000000000001e-91 < b_2 < 1.2599999999999999e-64

   1. Initial program 79.1%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if 1.2599999999999999e-64 < b_2

   1. Initial program 14.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 1.26 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* (/ b_2 a) -2.0) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (b_2 / a) * -2.0;
	} 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 <= (-5d-310)) then
        tmp = (b_2 / a) * (-2.0d0)
    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 <= -5e-310) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 72.1%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 30.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

4. Final simplification 69.5%

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

Alternative 6: 68.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b\_2} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2e-309) (* (/ b_2 a) -2.0) (* (/ -0.5 b_2) c)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2e-309) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = (-0.5 / b_2) * c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 2d-309) then
        tmp = (b_2 / a) * (-2.0d0)
    else
        tmp = ((-0.5d0) / b_2) * c
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2e-309) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = (-0.5 / b_2) * c;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2e-309:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = (-0.5 / b_2) * c
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2e-309)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	else
		tmp = Float64(Float64(-0.5 / b_2) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2e-309)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = (-0.5 / b_2) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2e-309], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 / b$95$2), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.9999999999999988e-309

   1. Initial program 72.1%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if 1.9999999999999988e-309 < b_2

   1. Initial program 30.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 73.2%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b\_2} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

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 = (b_2 / a) * (-2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.7%

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

3. Taylor expanded in b_2 around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 33.3

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

5. Applied rewrites 33.3%

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

6. Final simplification 33.3%

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

Developer Target 1: 99.7% 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 2024254 
(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))