quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.3% → 86.9%

Time: 10.6s

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: 52.3% 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: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left|b\_2 \cdot \sqrt{\mathsf{fma}\left(a, \frac{\frac{c}{b\_2}}{-b\_2}, 1\right)}\right| - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+28)
   (/ (- (fabs (* b_2 (sqrt (fma a (/ (/ c b_2) (- b_2)) 1.0)))) b_2) a)
   (if (<= b_2 4.2e-83)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+28) {
		tmp = (fabs((b_2 * sqrt(fma(a, ((c / b_2) / -b_2), 1.0)))) - b_2) / a;
	} else if (b_2 <= 4.2e-83) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+28)
		tmp = Float64(Float64(abs(Float64(b_2 * sqrt(fma(a, Float64(Float64(c / b_2) / Float64(-b_2)), 1.0)))) - b_2) / a);
	elseif (b_2 <= 4.2e-83)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - 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, -5e+28], N[(N[(N[Abs[N[(b$95$2 * N[Sqrt[N[(a * N[(N[(c / b$95$2), $MachinePrecision] / (-b$95$2)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 4.2e-83], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.99999999999999957e28

   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

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. unpow2 N/A

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

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

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

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

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

      16. neg-lowering-neg.f64 68.1

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

   5. Simplified 68.1%

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

      1. rem-square-sqrt N/A

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

      2. rem-sqrt-square N/A

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

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

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

      4. sqrt-prod N/A

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

      5. sqrt-prod N/A

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

      6. rem-square-sqrt N/A

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

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

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

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

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

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

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

      10. distribute-rgt-neg-out N/A

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

      11. distribute-frac-neg2 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 98.9

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

   7. Applied egg-rr 98.9%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   if -4.99999999999999957e28 < b_2 < 4.1999999999999998e-83

   1. Initial program 83.4%

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

   if 4.1999999999999998e-83 < b_2

   1. Initial program 21.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 88.5

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

   5. Simplified 88.5%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left|b\_2 \cdot \sqrt{\mathsf{fma}\left(a, \frac{\frac{c}{b\_2}}{-b\_2}, 1\right)}\right| - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+149}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 4.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+149)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 4.4e-83)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+149) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 4.4e-83) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d+149)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 4.4d-83) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+149) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 4.4e-83) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e+149:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 4.4e-83:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+149)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 4.4e-83)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e+149)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 4.4e-83)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+149], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 4.4e-83], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.00000000000000005e149

   1. Initial program 49.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 98.3

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

   5. Simplified 98.3%

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

   if -1.00000000000000005e149 < b_2 < 4.40000000000000015e-83

   1. Initial program 87.3%

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

   if 4.40000000000000015e-83 < b_2

   1. Initial program 21.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 88.5

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

   5. Simplified 88.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+149}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 4.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.25 \cdot 10^{-110}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 6.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{-a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.25e-110)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 6.4e-83) (/ (- (sqrt (- (* a c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.25e-110) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 6.4e-83) {
		tmp = (sqrt(-(a * c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.25d-110)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 6.4d-83) then
        tmp = (sqrt(-(a * c)) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.25e-110) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 6.4e-83) {
		tmp = (Math.sqrt(-(a * c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.25e-110:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 6.4e-83:
		tmp = (math.sqrt(-(a * c)) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.25e-110)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 6.4e-83)
		tmp = Float64(Float64(sqrt(Float64(-Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.25e-110)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 6.4e-83)
		tmp = (sqrt(-(a * c)) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.25e-110], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 6.4e-83], N[(N[(N[Sqrt[(-N[(a * c), $MachinePrecision])], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.25e-110

   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 84.5

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

   5. Simplified 84.5%

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

   if -1.25e-110 < b_2 < 6.4000000000000002e-83

   1. Initial program 78.6%

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

   3. Taylor expanded in b_2 around 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. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 73.2

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

   5. Simplified 73.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. distribute-rgt-neg-out N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. *-lowering-*.f64 73.2

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

   7. Applied egg-rr 73.2%

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

   if 6.4000000000000002e-83 < b_2

   1. Initial program 21.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 88.5

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

   5. Simplified 88.5%

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

Alternative 4: 67.5% 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 75.4%

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

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 39.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 66.7

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

   5. Simplified 66.7%

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

Alternative 5: 67.4% 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 75.4%

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

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 39.1%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. unpow2 N/A

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

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

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

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

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

      16. neg-lowering-neg.f64 19.9

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

   5. Simplified 19.9%

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

   6. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 66.5

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

   8. Simplified 66.5%

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

Alternative 6: 67.3% 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 75.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 68.1

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

   7. Applied egg-rr 68.1%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 39.1%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. unpow2 N/A

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

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

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

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

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

      16. neg-lowering-neg.f64 19.9

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

   5. Simplified 19.9%

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

   6. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 67.4%

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

Alternative 7: 47.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (- (/ b_2 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 / 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 / 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 / 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 / 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 / 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 / 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 / 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 75.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 46.7

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

   5. Simplified 46.7%

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

   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-lowering-neg.f64 32.9

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

   8. Simplified 32.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 39.1%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. unpow2 N/A

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

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

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

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

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

      16. neg-lowering-neg.f64 19.9

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

   5. Simplified 19.9%

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

   6. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 48.4%

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

Alternative 8: 23.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.3 \cdot 10^{-89}:\\ \;\;\;\;-\frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 2.3e-89) (- (/ b_2 a)) 0.0))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.3e-89) {
		tmp = -(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 <= 2.3d-89) then
        tmp = -(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 <= 2.3e-89) {
		tmp = -(b_2 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.3e-89:
		tmp = -(b_2 / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.3e-89)
		tmp = Float64(-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 <= 2.3e-89)
		tmp = -(b_2 / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.3e-89], (-N[(b$95$2 / a), $MachinePrecision]), 0.0]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.3e-89

   1. Initial program 76.0%

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

   3. Taylor expanded in b_2 around 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. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 51.7

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

   5. Simplified 51.7%

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

   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-lowering-neg.f64 27.2

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

   8. Simplified 27.2%

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

   if 2.3e-89 < b_2

   1. Initial program 23.7%

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

   3. Taylor expanded in b_2 around inf

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

   5. Simplified 28.2%

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

   6. Taylor expanded in b_2 around 0

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

   8. Simplified 28.2%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.3 \cdot 10^{-89}:\\ \;\;\;\;-\frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
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 58.7%

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

3. Taylor expanded in b_2 around inf

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

5. Simplified 11.3%

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

6. Taylor expanded in b_2 around 0

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

8. Simplified 11.3%

   \[\leadsto \color{blue}{0} \]
9. 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 2024199 
(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))