Quadratic roots, full range

Percentage Accurate: 52.1% → 86.6%

Time: 10.5s

Alternatives: 10

Speedup: 2.4×

• could not determine a ground truth

  1. a = 1.101589474340118e-251
  2. b = 4.742562691606788e+282
  3. c = 5279029330136948.0

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(\frac{c}{b}, \frac{a}{b \cdot -0.25}, 1\right)}, 0\right)\right| - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 0.0022:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+85)
   (/
    (- (fabs (fma b (sqrt (fma (/ c b) (/ a (* b -0.25)) 1.0)) 0.0)) b)
    (* a 2.0))
   (if (<= b 0.0022)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+85) {
		tmp = (fabs(fma(b, sqrt(fma((c / b), (a / (b * -0.25)), 1.0)), 0.0)) - b) / (a * 2.0);
	} else if (b <= 0.0022) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+85)
		tmp = Float64(Float64(abs(fma(b, sqrt(fma(Float64(c / b), Float64(a / Float64(b * -0.25)), 1.0)), 0.0)) - b) / Float64(a * 2.0));
	elseif (b <= 0.0022)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+85], N[(N[(N[Abs[N[(b * N[Sqrt[N[(N[(c / b), $MachinePrecision] * N[(a / N[(b * -0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + 0.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0022], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000001e85

   1. Initial program 50.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(a \cdot c, \frac{-4}{{b}^{2}}, 1\right)}}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{a \cdot c}, \frac{-4}{{b}^{2}}, 1\right)}}{2 \cdot a} \]

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

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(b \cdot b\right) \cdot \mathsf{fma}\left(a \cdot c, \color{blue}{\frac{-4}{{b}^{2}}}, 1\right)}}{2 \cdot a} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(b \cdot b\right) \cdot \mathsf{fma}\left(a \cdot c, \frac{-4}{\color{blue}{b \cdot b}}, 1\right)}}{2 \cdot a} \]

      12. *-lowering-*.f64 50.1

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b \cdot b\right) \cdot \mathsf{fma}\left(a \cdot c, \frac{-4}{\color{blue}{b \cdot b}}, 1\right)}}{2 \cdot a} \]

   5. Simplified 50.1%

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

      1. rem-square-sqrt N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)} \cdot \sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)}}}}{2 \cdot a} \]

      2. rem-sqrt-square N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left|\sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)}\right|}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left|\sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)}\right|}}{2 \cdot a} \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{\sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)} + 0}\right|}{2 \cdot a} \]

      5. sqrt-prod N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{\sqrt{b \cdot b} \cdot \sqrt{\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1}} + 0\right|}{2 \cdot a} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{\left(\sqrt{b} \cdot \sqrt{b}\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1} + 0\right|}{2 \cdot a} \]

      7. rem-square-sqrt N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{b} \cdot \sqrt{\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1} + 0\right|}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{\mathsf{fma}\left(b, \sqrt{\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1}, 0\right)}\right|}{2 \cdot a} \]

   7. Applied egg-rr 97.4%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left|\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a, \frac{c}{\left(b \cdot b\right) \cdot -0.25}, 1\right)}, 0\right)\right|}}{2 \cdot a} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\mathsf{fma}\left(b, \sqrt{\color{blue}{\frac{a \cdot c}{\left(b \cdot b\right) \cdot \frac{-1}{4}}} + 1}, 0\right)\right|}{2 \cdot a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\mathsf{fma}\left(b, \sqrt{\frac{\color{blue}{c \cdot a}}{\left(b \cdot b\right) \cdot \frac{-1}{4}} + 1}, 0\right)\right|}{2 \cdot a} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\mathsf{fma}\left(b, \sqrt{\frac{c \cdot a}{\color{blue}{b \cdot \left(b \cdot \frac{-1}{4}\right)}} + 1}, 0\right)\right|}{2 \cdot a} \]

      4. times-frac N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\mathsf{fma}\left(b, \sqrt{\color{blue}{\frac{c}{b} \cdot \frac{a}{b \cdot \frac{-1}{4}}} + 1}, 0\right)\right|}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\mathsf{fma}\left(b, \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c}{b}, \frac{a}{b \cdot \frac{-1}{4}}, 1\right)}}, 0\right)\right|}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{b}}, \frac{a}{b \cdot \frac{-1}{4}}, 1\right)}, 0\right)\right|}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(\frac{c}{b}, \color{blue}{\frac{a}{b \cdot \frac{-1}{4}}}, 1\right)}, 0\right)\right|}{2 \cdot a} \]

      8. *-lowering-*.f64 100.0

         \[\leadsto \frac{\left(-b\right) + \left|\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(\frac{c}{b}, \frac{a}{\color{blue}{b \cdot -0.25}}, 1\right)}, 0\right)\right|}{2 \cdot a} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{\left(-b\right) + \left|\mathsf{fma}\left(b, \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c}{b}, \frac{a}{b \cdot -0.25}, 1\right)}}, 0\right)\right|}{2 \cdot a} \]

   if -5.0000000000000001e85 < b < 0.00220000000000000013

   1. Initial program 79.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 0.00220000000000000013 < b

   1. Initial program 12.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 95.2

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

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 95.2

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

   7. Applied egg-rr 95.2%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(\frac{c}{b}, \frac{a}{b \cdot -0.25}, 1\right)}, 0\right)\right| - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 0.0022:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a, \frac{c}{-0.25 \cdot \left(b \cdot b\right)}, 1\right)}, 0\right)\right| - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 0.0215:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+87)
   (/
    (- (fabs (fma b (sqrt (fma a (/ c (* -0.25 (* b b))) 1.0)) 0.0)) b)
    (* a 2.0))
   (if (<= b 0.0215)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+87) {
		tmp = (fabs(fma(b, sqrt(fma(a, (c / (-0.25 * (b * b))), 1.0)), 0.0)) - b) / (a * 2.0);
	} else if (b <= 0.0215) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+87)
		tmp = Float64(Float64(abs(fma(b, sqrt(fma(a, Float64(c / Float64(-0.25 * Float64(b * b))), 1.0)), 0.0)) - b) / Float64(a * 2.0));
	elseif (b <= 0.0215)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+87], N[(N[(N[Abs[N[(b * N[Sqrt[N[(a * N[(c / N[(-0.25 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + 0.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0215], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999998e87

   1. Initial program 50.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(a \cdot c, \frac{-4}{{b}^{2}}, 1\right)}}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{a \cdot c}, \frac{-4}{{b}^{2}}, 1\right)}}{2 \cdot a} \]

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

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(b \cdot b\right) \cdot \mathsf{fma}\left(a \cdot c, \color{blue}{\frac{-4}{{b}^{2}}}, 1\right)}}{2 \cdot a} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(b \cdot b\right) \cdot \mathsf{fma}\left(a \cdot c, \frac{-4}{\color{blue}{b \cdot b}}, 1\right)}}{2 \cdot a} \]

      12. *-lowering-*.f64 50.1

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(b \cdot b\right) \cdot \mathsf{fma}\left(a \cdot c, \frac{-4}{\color{blue}{b \cdot b}}, 1\right)}}{2 \cdot a} \]

   5. Simplified 50.1%

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

      1. rem-square-sqrt N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)} \cdot \sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)}}}}{2 \cdot a} \]

      2. rem-sqrt-square N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left|\sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)}\right|}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left|\sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)}\right|}}{2 \cdot a} \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{\sqrt{\left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1\right)} + 0}\right|}{2 \cdot a} \]

      5. sqrt-prod N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{\sqrt{b \cdot b} \cdot \sqrt{\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1}} + 0\right|}{2 \cdot a} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{\left(\sqrt{b} \cdot \sqrt{b}\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1} + 0\right|}{2 \cdot a} \]

      7. rem-square-sqrt N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{b} \cdot \sqrt{\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1} + 0\right|}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \left|\color{blue}{\mathsf{fma}\left(b, \sqrt{\left(a \cdot c\right) \cdot \frac{-4}{b \cdot b} + 1}, 0\right)}\right|}{2 \cdot a} \]

   7. Applied egg-rr 97.4%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left|\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a, \frac{c}{\left(b \cdot b\right) \cdot -0.25}, 1\right)}, 0\right)\right|}}{2 \cdot a} \]

   if -4.9999999999999998e87 < b < 0.021499999999999998

   1. Initial program 79.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 0.021499999999999998 < b

   1. Initial program 12.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 95.2

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

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 95.2

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

   7. Applied egg-rr 95.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a, \frac{c}{-0.25 \cdot \left(b \cdot b\right)}, 1\right)}, 0\right)\right| - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 0.0215:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 0.0022:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+157)
   (/ b (- 0.0 a))
   (if (<= b 0.0022)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+157) {
		tmp = b / (0.0 - a);
	} else if (b <= 0.0022) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d+157)) then
        tmp = b / (0.0d0 - a)
    else if (b <= 0.0022d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+157) {
		tmp = b / (0.0 - a);
	} else if (b <= 0.0022) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+157:
		tmp = b / (0.0 - a)
	elif b <= 0.0022:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+157)
		tmp = Float64(b / Float64(0.0 - a));
	elseif (b <= 0.0022)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+157)
		tmp = b / (0.0 - a);
	elseif (b <= 0.0022)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+157], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0022], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.99999999999999976e157

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 96.3

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

   5. Simplified 96.3%

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

   if -4.99999999999999976e157 < b < 0.00220000000000000013

   1. Initial program 82.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 0.00220000000000000013 < b

   1. Initial program 12.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 95.2

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

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 95.2

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

   7. Applied egg-rr 95.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 0.0022:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+138}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 0.01:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.05e+138)
   (/ b (- 0.0 a))
   (if (<= b 0.01)
     (* (/ -0.5 a) (- b (sqrt (fma b b (* c (* a -4.0))))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e+138) {
		tmp = b / (0.0 - a);
	} else if (b <= 0.01) {
		tmp = (-0.5 / a) * (b - sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.05e+138)
		tmp = Float64(b / Float64(0.0 - a));
	elseif (b <= 0.01)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.05e+138], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.01], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05000000000000003e138

   1. Initial program 38.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 96.7

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

   5. Simplified 96.7%

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

   if -1.05000000000000003e138 < b < 0.0100000000000000002

   1. Initial program 81.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   if 0.0100000000000000002 < b

   1. Initial program 12.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 95.2

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

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 95.2

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

   7. Applied egg-rr 95.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+138}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 0.01:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)\\ \mathbf{elif}\;b \leq 0.0023:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-21)
   (fma b (/ c (* b b)) (/ b (- 0.0 a)))
   (if (<= b 0.0023)
     (/ (- b (sqrt (* c (* a -4.0)))) (* a -2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-21) {
		tmp = fma(b, (c / (b * b)), (b / (0.0 - a)));
	} else if (b <= 0.0023) {
		tmp = (b - sqrt((c * (a * -4.0)))) / (a * -2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-21)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(b / Float64(0.0 - a)));
	elseif (b <= 0.0023)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-21], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0023], N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.49999999999999968e-21

   1. Initial program 61.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. remove-double-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. /-lowering-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-sub0 N/A

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

      19. --lowering--.f64 92.5

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

   5. Simplified 92.5%

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

   if -4.49999999999999968e-21 < b < 0.0023

   1. Initial program 74.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied egg-rr 74.7%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

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

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

      3. *-lowering-*.f64 69.3

         \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}\right) \]

   7. Simplified 69.3%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\left(a \cdot c\right) \cdot -4}\right) \cdot \frac{\frac{-1}{2}}{a}} \]

      2. clear-num N/A

         \[\leadsto \left(b - \sqrt{\left(a \cdot c\right) \cdot -4}\right) \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-1}{2}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{b - \sqrt{\left(a \cdot c\right) \cdot -4}}{\frac{a}{\frac{-1}{2}}}} \]

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

         \[\leadsto \color{blue}{\frac{b - \sqrt{\left(a \cdot c\right) \cdot -4}}{\frac{a}{\frac{-1}{2}}}} \]

      5. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{b - \sqrt{\left(a \cdot c\right) \cdot -4}}}{\frac{a}{\frac{-1}{2}}} \]

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

         \[\leadsto \frac{b - \color{blue}{\sqrt{\left(a \cdot c\right) \cdot -4}}}{\frac{a}{\frac{-1}{2}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}{\frac{a}{\frac{-1}{2}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

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

         \[\leadsto \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

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

          \[\leadsto \frac{b - \sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      11. div-inv N/A

          \[\leadsto \frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{\color{blue}{a \cdot \frac{1}{\frac{-1}{2}}}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot \color{blue}{-2}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}} \]

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

          \[\leadsto \frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{\color{blue}{a \cdot \left(\mathsf{neg}\left(2\right)\right)}} \]

      15. metadata-eval 69.4

          \[\leadsto \frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot \color{blue}{-2}} \]

   9. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}} \]

   if 0.0023 < b

   1. Initial program 12.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 95.2

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

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 95.2

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

   7. Applied egg-rr 95.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)\\ \mathbf{elif}\;b \leq 0.0023:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)\\ \mathbf{elif}\;b \leq 0.0022:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-16)
   (fma b (/ c (* b b)) (/ b (- 0.0 a)))
   (if (<= b 0.0022)
     (* (/ -0.5 a) (- b (sqrt (* -4.0 (* c a)))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-16) {
		tmp = fma(b, (c / (b * b)), (b / (0.0 - a)));
	} else if (b <= 0.0022) {
		tmp = (-0.5 / a) * (b - sqrt((-4.0 * (c * a))));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-16)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(b / Float64(0.0 - a)));
	elseif (b <= 0.0022)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(-4.0 * Float64(c * a)))));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-16], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0022], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5000000000000002e-16

   1. Initial program 61.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. remove-double-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. /-lowering-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-sub0 N/A

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

      19. --lowering--.f64 92.5

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

   5. Simplified 92.5%

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

   if -4.5000000000000002e-16 < b < 0.00220000000000000013

   1. Initial program 74.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied egg-rr 74.7%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

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

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

      3. *-lowering-*.f64 69.3

         \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}\right) \]

   7. Simplified 69.3%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

   if 0.00220000000000000013 < b

   1. Initial program 12.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 95.2

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

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 95.2

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

   7. Applied egg-rr 95.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, \frac{b}{0 - a}\right)\\ \mathbf{elif}\;b \leq 0.0022:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.5e-285) (/ b (- 0.0 a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.5e-285) {
		tmp = b / (0.0 - a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7.5d-285) then
        tmp = b / (0.0d0 - a)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.5e-285) {
		tmp = b / (0.0 - a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.5e-285:
		tmp = b / (0.0 - a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.5e-285)
		tmp = Float64(b / Float64(0.0 - a));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.5e-285)
		tmp = b / (0.0 - a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.5e-285], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4999999999999999e-285

   1. Initial program 68.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 66.8

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

   5. Simplified 66.8%

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

   if 7.4999999999999999e-285 < b

   1. Initial program 30.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 72.0

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

   5. Simplified 72.0%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 72.0

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

   7. Applied egg-rr 72.0%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (- 0.0 (/ c b)))

C

double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Fortran

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

Java

public static double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Python

def code(a, b, c):
	return 0.0 - (c / b)

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

   4. /-lowering-/.f64 33.7

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

5. Simplified 33.7%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

   3. /-lowering-/.f64 33.7

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

7. Applied egg-rr 33.7%

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

8. Final simplification 33.7%

   \[\leadsto 0 - \frac{c}{b} \]
9. Add Preprocessing

Alternative 9: 10.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

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

Fortran

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

Java

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

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

   4. /-lowering-/.f64 33.7

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

5. Simplified 33.7%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {\left(\frac{c}{b}\right)}^{3}}{0 \cdot 0 + \left(\frac{c}{b} \cdot \frac{c}{b} + 0 \cdot \frac{c}{b}\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {\left(\frac{c}{b}\right)}^{3}}{0 \cdot 0 + \left(\frac{c}{b} \cdot \frac{c}{b} + 0 \cdot \frac{c}{b}\right)} \]

   3. div-sub N/A

      \[\leadsto \color{blue}{\frac{0}{0 \cdot 0 + \left(\frac{c}{b} \cdot \frac{c}{b} + 0 \cdot \frac{c}{b}\right)} - \frac{{\left(\frac{c}{b}\right)}^{3}}{0 \cdot 0 + \left(\frac{c}{b} \cdot \frac{c}{b} + 0 \cdot \frac{c}{b}\right)}} \]

7. Applied egg-rr 8.1%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
8. Add Preprocessing

Alternative 10: 2.5% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (/ b a))

C

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

Fortran

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

Java

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

Python

def code(a, b, c):
	return b / a

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied egg-rr 32.6%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}} \]

5. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 2.3

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

7. Simplified 2.3%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))