quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.6% → 66.9%

Time: 12.5s

Alternatives: 10

Speedup: 0.9×

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: 51.6% 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: 66.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.75 \cdot 10^{-135}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{a}{\mathsf{fma}\left(\frac{1}{b\_2}, b\_2 \cdot b\_2, \sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.75e-135)
   (/ (* a c) (* a (- (sqrt (- (* b_2 b_2) (* a c))) b_2)))
   (/
    -1.0
    (/ a (fma (/ 1.0 b_2) (* b_2 b_2) (sqrt (fma c (- a) (* b_2 b_2))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.75e-135) {
		tmp = (a * c) / (a * (sqrt(((b_2 * b_2) - (a * c))) - b_2));
	} else {
		tmp = -1.0 / (a / fma((1.0 / b_2), (b_2 * b_2), sqrt(fma(c, -a, (b_2 * b_2)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.75e-135)
		tmp = Float64(Float64(a * c) / Float64(a * Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2)));
	else
		tmp = Float64(-1.0 / Float64(a / fma(Float64(1.0 / b_2), Float64(b_2 * b_2), sqrt(fma(c, Float64(-a), Float64(b_2 * b_2))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.75e-135], N[(N[(a * c), $MachinePrecision] / N[(a * N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(a / N[(N[(1.0 / b$95$2), $MachinePrecision] * N[(b$95$2 * b$95$2), $MachinePrecision] + N[Sqrt[N[(c * (-a) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.75e-135

   1. Initial program 18.5%

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

   4. Applied egg-rr 10.9%

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

   5. Taylor expanded in b_2 around 0

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

      1. lower-*.f64 61.0

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

   7. Simplified 61.0%

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

   if -2.75e-135 < b_2

   1. Initial program 72.9%

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

   4. Applied egg-rr 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. inv-pow N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. *-inverses N/A

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

      12. associate-/r* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/r* N/A

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

      17. *-inverses N/A

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

      18. lower-/.f64 73.0

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

      19. lift--.f64 N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   6. Applied egg-rr 73.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.75 \cdot 10^{-135}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{a}{\mathsf{fma}\left(\frac{1}{b\_2}, b\_2 \cdot b\_2, \sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.75 \cdot 10^{-135}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{a}, b\_2, \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}{-a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.75e-135)
   (/ (* a c) (* a (- (sqrt (- (* b_2 b_2) (* a c))) b_2)))
   (fma (/ -1.0 a) b_2 (/ (sqrt (fma b_2 b_2 (- (* a c)))) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.75e-135) {
		tmp = (a * c) / (a * (sqrt(((b_2 * b_2) - (a * c))) - b_2));
	} else {
		tmp = fma((-1.0 / a), b_2, (sqrt(fma(b_2, b_2, -(a * c))) / -a));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.75e-135)
		tmp = Float64(Float64(a * c) / Float64(a * Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2)));
	else
		tmp = fma(Float64(-1.0 / a), b_2, Float64(sqrt(fma(b_2, b_2, Float64(-Float64(a * c)))) / Float64(-a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.75e-135], N[(N[(a * c), $MachinePrecision] / N[(a * N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / a), $MachinePrecision] * b$95$2 + N[(N[Sqrt[N[(b$95$2 * b$95$2 + (-N[(a * c), $MachinePrecision])), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.75e-135

   1. Initial program 18.5%

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

   4. Applied egg-rr 10.9%

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

   5. Taylor expanded in b_2 around 0

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

      1. lower-*.f64 61.0

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

   7. Simplified 61.0%

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

   if -2.75e-135 < b_2

   1. Initial program 72.9%

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

   4. Applied egg-rr 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lower-*.f64 72.9

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

   6. Applied egg-rr 72.9%

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

   8. Applied egg-rr 69.3%

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

   10. Applied egg-rr 72.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.75 \cdot 10^{-135}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{a}, b\_2, \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}{-a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.75 \cdot 10^{-135}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{a}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.75e-135)
   (/ (* a c) (* a (- (sqrt (- (* b_2 b_2) (* a c))) b_2)))
   (/ -1.0 (/ a (+ b_2 (sqrt (fma b_2 b_2 (- (* a c)))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.75e-135) {
		tmp = (a * c) / (a * (sqrt(((b_2 * b_2) - (a * c))) - b_2));
	} else {
		tmp = -1.0 / (a / (b_2 + sqrt(fma(b_2, b_2, -(a * c)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.75e-135)
		tmp = Float64(Float64(a * c) / Float64(a * Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2)));
	else
		tmp = Float64(-1.0 / Float64(a / Float64(b_2 + sqrt(fma(b_2, b_2, Float64(-Float64(a * c)))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.75e-135], N[(N[(a * c), $MachinePrecision] / N[(a * N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(a / N[(b$95$2 + N[Sqrt[N[(b$95$2 * b$95$2 + (-N[(a * c), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.75e-135

   1. Initial program 18.5%

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

   4. Applied egg-rr 10.9%

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

   5. Taylor expanded in b_2 around 0

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

      1. lower-*.f64 61.0

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

   7. Simplified 61.0%

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

   if -2.75e-135 < b_2

   1. Initial program 72.9%

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

   4. Applied egg-rr 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lower-*.f64 72.9

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

   6. Applied egg-rr 72.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.75 \cdot 10^{-135}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{a}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{a \cdot \left(\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{a}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+155)
   (/ 2.0 (* a (* (* b_2 b_2) (* b_2 b_2))))
   (/ -1.0 (/ a (+ b_2 (sqrt (fma b_2 b_2 (- (* a c)))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+155) {
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)));
	} else {
		tmp = -1.0 / (a / (b_2 + sqrt(fma(b_2, b_2, -(a * c)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+155)
		tmp = Float64(2.0 / Float64(a * Float64(Float64(b_2 * b_2) * Float64(b_2 * b_2))));
	else
		tmp = Float64(-1.0 / Float64(a / Float64(b_2 + sqrt(fma(b_2, b_2, Float64(-Float64(a * c)))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+155], N[(2.0 / N[(a * N[(N[(b$95$2 * b$95$2), $MachinePrecision] * N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(a / N[(b$95$2 + N[Sqrt[N[(b$95$2 * b$95$2 + (-N[(a * c), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.00000000000000001e155

   1. Initial program 1.7%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 0.0

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

   7. Simplified 0.0%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{a \cdot {b\_2}^{4}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{a \cdot {b\_2}^{4}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2}{a \cdot {b\_2}^{\color{blue}{\left(3 + 1\right)}}} \]

      4. pow-plus N/A

         \[\leadsto \frac{2}{a \cdot \color{blue}{\left({b\_2}^{3} \cdot b\_2\right)}} \]

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 41.1

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

   10. Simplified 41.1%

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

   if -2.00000000000000001e155 < b_2

   1. Initial program 61.1%

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

   4. Applied egg-rr 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lower-*.f64 61.2

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

   6. Applied egg-rr 61.2%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{a \cdot \left(\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{a}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{a \cdot \left(\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{elif}\;b\_2 \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{b\_2 + \sqrt{-a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.25e+40)
   (/ 2.0 (* a (* (* b_2 b_2) (* b_2 b_2))))
   (if (<= b_2 3.3e-19) (/ (+ b_2 (sqrt (- (* a c)))) (- a)) (/ b_2 (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e+40) {
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)));
	} else if (b_2 <= 3.3e-19) {
		tmp = (b_2 + sqrt(-(a * c))) / -a;
	} else {
		tmp = b_2 / -a;
	}
	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 <= (-2.25d+40)) then
        tmp = 2.0d0 / (a * ((b_2 * b_2) * (b_2 * b_2)))
    else if (b_2 <= 3.3d-19) then
        tmp = (b_2 + sqrt(-(a * c))) / -a
    else
        tmp = b_2 / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e+40) {
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)));
	} else if (b_2 <= 3.3e-19) {
		tmp = (b_2 + Math.sqrt(-(a * c))) / -a;
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.25e+40:
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)))
	elif b_2 <= 3.3e-19:
		tmp = (b_2 + math.sqrt(-(a * c))) / -a
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.25e+40)
		tmp = Float64(2.0 / Float64(a * Float64(Float64(b_2 * b_2) * Float64(b_2 * b_2))));
	elseif (b_2 <= 3.3e-19)
		tmp = Float64(Float64(b_2 + sqrt(Float64(-Float64(a * c)))) / Float64(-a));
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.25e+40)
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)));
	elseif (b_2 <= 3.3e-19)
		tmp = (b_2 + sqrt(-(a * c))) / -a;
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.25e+40], N[(2.0 / N[(a * N[(N[(b$95$2 * b$95$2), $MachinePrecision] * N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.3e-19], N[(N[(b$95$2 + N[Sqrt[(-N[(a * c), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.25000000000000016e40

   1. Initial program 7.4%

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

   4. Applied egg-rr 1.1%

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

   5. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 3.6

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

   7. Simplified 3.6%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{a \cdot {b\_2}^{4}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{a \cdot {b\_2}^{4}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2}{a \cdot {b\_2}^{\color{blue}{\left(3 + 1\right)}}} \]

      4. pow-plus N/A

         \[\leadsto \frac{2}{a \cdot \color{blue}{\left({b\_2}^{3} \cdot b\_2\right)}} \]

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 29.2

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

   10. Simplified 29.2%

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

   if -2.25000000000000016e40 < b_2 < 3.2999999999999998e-19

   1. Initial program 67.3%

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

   4. Applied egg-rr 67.4%

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

   5. Taylor expanded in b_2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 58.3

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

   7. Simplified 58.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l/ N/A

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

      7. neg-mul-1 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 58.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.2

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

   9. Applied egg-rr 58.2%

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

   if 3.2999999999999998e-19 < b_2

   1. Initial program 69.9%

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

   3. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 41.3

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

   5. Simplified 41.3%

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

   6. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.9

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

   8. Simplified 48.9%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{a \cdot \left(\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{elif}\;b\_2 \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{b\_2 + \sqrt{-a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{a \cdot \left(\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+155)
   (/ 2.0 (* a (* (* b_2 b_2) (* b_2 b_2))))
   (/ (- (- b_2) (sqrt (fma b_2 b_2 (- (* a c))))) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+155) {
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)));
	} else {
		tmp = (-b_2 - sqrt(fma(b_2, b_2, -(a * c)))) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+155)
		tmp = Float64(2.0 / Float64(a * Float64(Float64(b_2 * b_2) * Float64(b_2 * b_2))));
	else
		tmp = Float64(Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(-Float64(a * c))))) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+155], N[(2.0 / N[(a * N[(N[(b$95$2 * b$95$2), $MachinePrecision] * N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + (-N[(a * c), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.00000000000000001e155

   1. Initial program 1.7%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 0.0

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

   7. Simplified 0.0%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{a \cdot {b\_2}^{4}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{a \cdot {b\_2}^{4}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2}{a \cdot {b\_2}^{\color{blue}{\left(3 + 1\right)}}} \]

      4. pow-plus N/A

         \[\leadsto \frac{2}{a \cdot \color{blue}{\left({b\_2}^{3} \cdot b\_2\right)}} \]

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 41.1

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

   10. Simplified 41.1%

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

   if -2.00000000000000001e155 < b_2

   1. Initial program 61.1%

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

   4. Applied egg-rr 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied egg-rr 61.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. metadata-eval N/A

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

      11. neg-mul-1 N/A

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

      12. lower-/.f64 N/A

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

   8. Applied egg-rr 61.2%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{a \cdot \left(\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 23.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -660:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(b\_2 \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -660.0) (/ (* a c) (* a (* b_2 (* b_2 b_2)))) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -660.0) {
		tmp = (a * c) / (a * (b_2 * (b_2 * b_2)));
	} else {
		tmp = b_2 / -a;
	}
	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 <= (-660.0d0)) then
        tmp = (a * c) / (a * (b_2 * (b_2 * b_2)))
    else
        tmp = b_2 / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -660.0) {
		tmp = (a * c) / (a * (b_2 * (b_2 * b_2)));
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -660.0:
		tmp = (a * c) / (a * (b_2 * (b_2 * b_2)))
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -660.0)
		tmp = Float64(Float64(a * c) / Float64(a * Float64(b_2 * Float64(b_2 * b_2))));
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -660.0)
		tmp = (a * c) / (a * (b_2 * (b_2 * b_2)));
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -660.0], N[(N[(a * c), $MachinePrecision] / N[(a * N[(b$95$2 * N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -660

   1. Initial program 8.9%

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

   4. Applied egg-rr 2.5%

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

   5. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 3.7

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

   7. Simplified 3.7%

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

   8. Taylor expanded in b_2 around 0

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

      1. lower-*.f64 28.2

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

   10. Simplified 28.2%

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

   if -660 < b_2

   1. Initial program 69.3%

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

   3. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 22.8

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

   5. Simplified 22.8%

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

   6. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.2

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

   8. Simplified 27.2%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -660:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(b\_2 \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 23.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15:\\ \;\;\;\;\frac{2}{a \cdot \left(\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15) (/ 2.0 (* a (* (* b_2 b_2) (* b_2 b_2)))) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15) {
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)));
	} else {
		tmp = b_2 / -a;
	}
	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 <= (-1.15d0)) then
        tmp = 2.0d0 / (a * ((b_2 * b_2) * (b_2 * b_2)))
    else
        tmp = b_2 / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15) {
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)));
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15:
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)))
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15)
		tmp = Float64(2.0 / Float64(a * Float64(Float64(b_2 * b_2) * Float64(b_2 * b_2))));
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.15)
		tmp = 2.0 / (a * ((b_2 * b_2) * (b_2 * b_2)));
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15], N[(2.0 / N[(a * N[(N[(b$95$2 * b$95$2), $MachinePrecision] * N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.1499999999999999

   1. Initial program 10.1%

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

   4. Applied egg-rr 2.5%

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

   5. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{\color{blue}{a \cdot {b\_2}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 3.7

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

   7. Simplified 3.7%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{a \cdot {b\_2}^{4}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{a \cdot {b\_2}^{4}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2}{a \cdot {b\_2}^{\color{blue}{\left(3 + 1\right)}}} \]

      4. pow-plus N/A

         \[\leadsto \frac{2}{a \cdot \color{blue}{\left({b\_2}^{3} \cdot b\_2\right)}} \]

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 27.5

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

   10. Simplified 27.5%

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

   if -1.1499999999999999 < b_2

   1. Initial program 69.2%

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

   3. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 22.9

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

   5. Simplified 22.9%

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

   6. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.3

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

   8. Simplified 27.3%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15:\\ \;\;\;\;\frac{2}{a \cdot \left(\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 14.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 16.9

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

5. Simplified 16.9%

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

6. Taylor expanded in b_2 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 20.0

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

8. Simplified 20.0%

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

9. Final simplification 20.0%

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

Alternative 10: 5.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ -1.0 a))

C

double code(double a, double b_2, double c) {
	return -1.0 / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-1.0d0) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return -1.0 / a;
}

Python

def code(a, b_2, c):
	return -1.0 / a

Julia

function code(a, b_2, c)
	return Float64(-1.0 / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -1.0 / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(-1.0 / a), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 16.9

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

5. Simplified 16.9%

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

6. Taylor expanded in b_2 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 20.0

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

8. Simplified 20.0%

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

   1. lift-neg.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. neg-mul-1 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-neg.f64 N/A

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

   8. metadata-eval N/A

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

   9. frac-2neg N/A

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

   10. /-rgt-identity N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 20.0

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

10. Applied egg-rr 20.0%

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

11. Taylor expanded in a around 0

    \[\leadsto \color{blue}{\frac{-1}{a}} \]
12. Step-by-step derivation

    1. lower-/.f64 5.4

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

13. Simplified 5.4%

    \[\leadsto \color{blue}{\frac{-1}{a}} \]
14. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{c}{t\_1 - b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + t\_1}{-a}\\ \end{array} \end{array} \]

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) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

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 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	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 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	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 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	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(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	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 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	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[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024214 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :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) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

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