quadp (p42, positive)

Percentage Accurate: 52.0% → 87.6%

Time: 11.7s

Alternatives: 12

Speedup: 2.4×

• could not determine a ground truth

  1. a = 8.850494594133397e-289
  2. b = 20044445674380902000.0
  3. c = -8.842673131471462e-215

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.0% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+154)
   (/ (- (* a (/ c b)) b) a)
   (if (<= b 4.2e-159)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b 3.6e+26)
       (/
        (/ (* a (* c 4.0)) (+ b (sqrt (fma a (* c -4.0) (fma b b 0.0)))))
        (* a -2.0))
       (- 0.0 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+154) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 4.2e-159) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 3.6e+26) {
		tmp = ((a * (c * 4.0)) / (b + sqrt(fma(a, (c * -4.0), fma(b, b, 0.0))))) / (a * -2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+154)
		tmp = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
	elseif (b <= 4.2e-159)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= 3.6e+26)
		tmp = Float64(Float64(Float64(a * Float64(c * 4.0)) / Float64(b + sqrt(fma(a, Float64(c * -4.0), fma(b, b, 0.0))))) / Float64(a * -2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+154], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.2e-159], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e+26], N[(N[(N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.00000000000000015e154

   1. Initial program 38.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. associate-/l* N/A

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

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

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

      8. /-lowering-/.f64 100.0

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

   8. Simplified 100.0%

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

   if -4.00000000000000015e154 < b < 4.1999999999999998e-159

   1. Initial program 90.8%

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

   if 4.1999999999999998e-159 < b < 3.60000000000000024e26

   1. Initial program 56.8%

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

   4. Applied egg-rr 45.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right) \cdot \left(a \cdot -2\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 70.5

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

   7. Simplified 70.5%

      \[\leadsto \frac{\color{blue}{\left(a \cdot 4\right) \cdot c}}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right) \cdot \left(a \cdot -2\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

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

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

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

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

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. +-rgt-identity N/A

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

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

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

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

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

      13. +-rgt-identity N/A

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

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

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

      15. *-lowering-*.f64 82.2

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

   9. Applied egg-rr 82.2%

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

   if 3.60000000000000024e26 < b

   1. Initial program 11.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 91.3

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

   5. Simplified 91.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 91.3

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

   7. Applied egg-rr 91.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+51}:\\ \;\;\;\;c \cdot \frac{a \cdot 4}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right) \cdot \left(a \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+153)
   (/ (- (* a (/ c b)) b) a)
   (if (<= b 6e-127)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b 7e+51)
       (*
        c
        (/
         (* a 4.0)
         (* (+ b (sqrt (fma a (* c -4.0) (fma b b 0.0)))) (* a -2.0))))
       (- 0.0 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 6e-127) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 7e+51) {
		tmp = c * ((a * 4.0) / ((b + sqrt(fma(a, (c * -4.0), fma(b, b, 0.0)))) * (a * -2.0)));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
	elseif (b <= 6e-127)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= 7e+51)
		tmp = Float64(c * Float64(Float64(a * 4.0) / Float64(Float64(b + sqrt(fma(a, Float64(c * -4.0), fma(b, b, 0.0)))) * Float64(a * -2.0))));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+153], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6e-127], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e+51], N[(c * N[(N[(a * 4.0), $MachinePrecision] / N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.00000000000000018e153

   1. Initial program 38.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. associate-/l* N/A

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

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

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

      8. /-lowering-/.f64 100.0

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

   8. Simplified 100.0%

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

   if -5.00000000000000018e153 < b < 6.00000000000000017e-127

   1. Initial program 90.5%

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

   if 6.00000000000000017e-127 < b < 7e51

   1. Initial program 46.2%

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

   4. Applied egg-rr 36.8%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right) \cdot \left(a \cdot -2\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 67.4

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

   7. Simplified 67.4%

      \[\leadsto \frac{\color{blue}{\left(a \cdot 4\right) \cdot c}}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right) \cdot \left(a \cdot -2\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-rgt-identity N/A

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

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

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

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

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

      12. +-rgt-identity N/A

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

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

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

      14. *-lowering-*.f64 75.9

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

   9. Applied egg-rr 75.9%

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

   if 7e51 < b

   1. Initial program 10.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 95.3

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

   5. Simplified 95.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 95.3

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

   7. Applied egg-rr 95.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+51}:\\ \;\;\;\;c \cdot \frac{a \cdot 4}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right) \cdot \left(a \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+153)
   (/ (- (* a (/ c b)) b) a)
   (if (<= b 4.8e-9)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 4.8e-9) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - 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+153)) then
        tmp = ((a * (c / b)) - b) / a
    else if (b <= 4.8d-9) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - 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+153) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 4.8e-9) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+153:
		tmp = ((a * (c / b)) - b) / a
	elif b <= 4.8e-9:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
	elseif (b <= 4.8e-9)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - 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+153)
		tmp = ((a * (c / b)) - b) / a;
	elseif (b <= 4.8e-9)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+153], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.8e-9], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $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 < -5.00000000000000018e153

   1. Initial program 38.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. associate-/l* N/A

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

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

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

      8. /-lowering-/.f64 100.0

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

   8. Simplified 100.0%

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

   if -5.00000000000000018e153 < b < 4.8e-9

   1. Initial program 84.6%

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

   if 4.8e-9 < b

   1. Initial program 14.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.6

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

   5. Simplified 88.6%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 88.6

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

   7. Applied egg-rr 88.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+140)
   (/ (- (* a (/ c b)) b) a)
   (if (<= b 3.9e-10)
     (* (/ -0.5 a) (- b (sqrt (fma a (* c -4.0) (fma b b 0.0)))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+140) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 3.9e-10) {
		tmp = (-0.5 / a) * (b - sqrt(fma(a, (c * -4.0), fma(b, b, 0.0))));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+140)
		tmp = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
	elseif (b <= 3.9e-10)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(a, Float64(c * -4.0), fma(b, b, 0.0)))));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+140], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.9e-10], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999996e140

   1. Initial program 44.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. associate-/l* N/A

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

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

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

      8. /-lowering-/.f64 100.0

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

   8. Simplified 100.0%

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

   if -8.4999999999999996e140 < b < 3.9e-10

   1. Initial program 84.0%

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

   4. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)} \]

   if 3.9e-10 < b

   1. Initial program 14.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.6

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

   5. Simplified 88.6%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 88.6

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

   7. Applied egg-rr 88.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+140}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e+140)
   (/ (- (* a (/ c b)) b) a)
   (if (<= b 4.5e-5)
     (* (/ -0.5 a) (- b (sqrt (fma (* a c) -4.0 (* b b)))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e+140) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 4.5e-5) {
		tmp = (-0.5 / a) * (b - sqrt(fma((a * c), -4.0, (b * b))));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e+140)
		tmp = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
	elseif (b <= 4.5e-5)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e+140], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.5e-5], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8999999999999999e140

   1. Initial program 44.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. associate-/l* N/A

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

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

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

      8. /-lowering-/.f64 100.0

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

   8. Simplified 100.0%

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

   if -2.8999999999999999e140 < b < 4.50000000000000028e-5

   1. Initial program 84.0%

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

   4. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)} \]

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 83.8

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

   7. Simplified 83.8%

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

   if 4.50000000000000028e-5 < b

   1. Initial program 14.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 88.6

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

   5. Simplified 88.6%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 88.6

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

   7. Applied egg-rr 88.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+140}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.45 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.45e-105)
   (/ (- (* a (/ c b)) b) a)
   (if (<= b 5.2e-87)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.45e-105) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 5.2e-87) {
		tmp = (sqrt((a * (c * -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 <= (-3.45d-105)) then
        tmp = ((a * (c / b)) - b) / a
    else if (b <= 5.2d-87) then
        tmp = (sqrt((a * (c * (-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 <= -3.45e-105) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 5.2e-87) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.45e-105:
		tmp = ((a * (c / b)) - b) / a
	elif b <= 5.2e-87:
		tmp = (math.sqrt((a * (c * -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 <= -3.45e-105)
		tmp = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
	elseif (b <= 5.2e-87)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -3.45e-105)
		tmp = ((a * (c / b)) - b) / a;
	elseif (b <= 5.2e-87)
		tmp = (sqrt((a * (c * -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, -3.45e-105], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 5.2e-87], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $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 < -3.45000000000000014e-105

   1. Initial program 70.5%

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

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

   6. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. associate-/l* N/A

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

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

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

      8. /-lowering-/.f64 92.7

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

   8. Simplified 92.7%

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

   if -3.45000000000000014e-105 < b < 5.20000000000000005e-87

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 80.7

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

   5. Simplified 80.7%

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

   if 5.20000000000000005e-87 < b

   1. Initial program 20.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 83.2

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

   5. Simplified 83.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 83.2

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

   7. Applied egg-rr 83.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.45 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(a \cdot c\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-105)
   (/ (- (* a (/ c b)) b) a)
   (if (<= b 6.3e-87)
     (* (/ -0.5 a) (- b (sqrt (* (* a c) -4.0))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-105) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 6.3e-87) {
		tmp = (-0.5 / a) * (b - sqrt(((a * c) * -4.0)));
	} 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 <= (-3.7d-105)) then
        tmp = ((a * (c / b)) - b) / a
    else if (b <= 6.3d-87) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((a * c) * (-4.0d0))))
    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 <= -3.7e-105) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 6.3e-87) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((a * c) * -4.0)));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-105:
		tmp = ((a * (c / b)) - b) / a
	elif b <= 6.3e-87:
		tmp = (-0.5 / a) * (b - math.sqrt(((a * c) * -4.0)))
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-105)
		tmp = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
	elseif (b <= 6.3e-87)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(a * c) * -4.0))));
	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 <= -3.7e-105)
		tmp = ((a * (c / b)) - b) / a;
	elseif (b <= 6.3e-87)
		tmp = (-0.5 / a) * (b - sqrt(((a * c) * -4.0)));
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-105], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6.3e-87], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.70000000000000008e-105

   1. Initial program 70.5%

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

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

   6. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. associate-/l* N/A

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

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

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

      8. /-lowering-/.f64 92.7

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

   8. Simplified 92.7%

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

   if -3.70000000000000008e-105 < b < 6.29999999999999976e-87

   1. Initial program 80.9%

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

   4. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \]
   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 80.4

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

   7. Simplified 80.4%

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

   if 6.29999999999999976e-87 < b

   1. Initial program 20.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 83.2

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

   5. Simplified 83.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 83.2

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

   7. Applied egg-rr 83.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(a \cdot c\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- (/ c b) (/ b a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (c / b) - (b / 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 <= (-4d-310)) then
        tmp = (c / b) - (b / 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 <= -4e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(c / b) - Float64(b / 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 <= -4e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 73.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 70.6

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

   5. Simplified 70.6%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 70.9

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

   8. Simplified 70.9%

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

   if -3.999999999999988e-310 < b

   1. Initial program 39.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 62.2

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

   5. Simplified 62.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 62.2

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

   7. Applied egg-rr 62.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- 0.0 (/ b a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = 0.0 - (b / 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 <= (-4d-310)) then
        tmp = 0.0d0 - (b / 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 <= -4e-310) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = 0.0 - (b / a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(0.0 - Float64(b / 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 <= -4e-310)
		tmp = 0.0 - (b / a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 73.2%

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

   4. Applied egg-rr 73.2%

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

   5. Taylor expanded in b around -inf

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

      1. /-lowering-/.f64 69.9

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

   7. Simplified 69.9%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. mul-1-neg N/A

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

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

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

      5. /-lowering-/.f64 70.1

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

   9. Applied egg-rr 70.1%

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

   if -3.999999999999988e-310 < b

   1. Initial program 39.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 62.2

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

   5. Simplified 62.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   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 62.2

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

   7. Applied egg-rr 62.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 0.3) (- 0.0 (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.3) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = 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 <= 0.3d0) then
        tmp = 0.0d0 - (b / a)
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.3) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.3:
		tmp = 0.0 - (b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.3)
		tmp = Float64(0.0 - Float64(b / a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.3)
		tmp = 0.0 - (b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.3], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.299999999999999989

   1. Initial program 73.2%

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

   4. Applied egg-rr 73.1%

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

   5. Taylor expanded in b around -inf

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

      1. /-lowering-/.f64 50.1

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

   7. Simplified 50.1%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. mul-1-neg N/A

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

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

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

      5. /-lowering-/.f64 50.3

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

   9. Applied egg-rr 50.3%

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

   if 0.299999999999999989 < b

   1. Initial program 14.3%

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

   4. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}} \]

   5. Taylor expanded in b around -inf

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

      1. /-lowering-/.f64 21.4

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

   7. Simplified 21.4%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.7% 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 56.4%

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

4. Applied egg-rr 36.2%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}} \]

5. Taylor expanded in b around -inf

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

   1. /-lowering-/.f64 8.1

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

7. Simplified 8.1%

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

Alternative 12: 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 56.4%

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

4. Applied egg-rr 36.2%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}} \]

5. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 2.7

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

7. Simplified 2.7%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024197 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))