The quadratic formula (r2)

Percentage Accurate: 52.6% → 91.2%

Time: 13.1s

Alternatives: 11

Speedup: 2.5×

• could not determine a ground truth

  1. a = 1.3484360574028287e-255
  2. b = -2.129456041822372e+248
  3. c = 0.0009607559910349735

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: 52.6% 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: 91.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(c \cdot 0.5\right) \cdot -4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+153)
   (- (/ c b))
   (if (<= b -1.1e-302)
     (/ (* (* c 0.5) -4.0) (- b (sqrt (fma a (* c -4.0) (* b b)))))
     (if (<= b 6e+106)
       (fma (/ b a) -0.5 (/ (sqrt (fma b b (* c (* -4.0 a)))) (* a -2.0)))
       (/ b (- a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = -(c / b);
	} else if (b <= -1.1e-302) {
		tmp = ((c * 0.5) * -4.0) / (b - sqrt(fma(a, (c * -4.0), (b * b))));
	} else if (b <= 6e+106) {
		tmp = fma((b / a), -0.5, (sqrt(fma(b, b, (c * (-4.0 * a)))) / (a * -2.0)));
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(-Float64(c / b));
	elseif (b <= -1.1e-302)
		tmp = Float64(Float64(Float64(c * 0.5) * -4.0) / Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	elseif (b <= 6e+106)
		tmp = fma(Float64(b / a), -0.5, Float64(sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))) / Float64(a * -2.0)));
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+153], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, -1.1e-302], N[(N[(N[(c * 0.5), $MachinePrecision] * -4.0), $MachinePrecision] / N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e+106], N[(N[(b / a), $MachinePrecision] * -0.5 + N[(N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.00000000000000018e153

   1. Initial program 1.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 96.3

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

   5. Simplified 96.3%

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

   if -5.00000000000000018e153 < b < -1.10000000000000004e-302

   1. Initial program 51.3%

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

   4. Applied egg-rr 40.4%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 82.6

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

   7. Simplified 82.6%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. times-frac N/A

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

      6. *-inverses N/A

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

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

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. *-lowering-*.f64 88.9

          \[\leadsto \frac{\left(1 \cdot \left(c \cdot 0.5\right)\right) \cdot -4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, \color{blue}{b \cdot b}\right)}} \]

   9. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{\left(1 \cdot \left(c \cdot 0.5\right)\right) \cdot -4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}} \]

   if -1.10000000000000004e-302 < b < 6.0000000000000001e106

   1. Initial program 88.4%

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

   4. Applied egg-rr 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)} \]

   if 6.0000000000000001e106 < b

   1. Initial program 50.7%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

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

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

      3. /-lowering-/.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(c \cdot 0.5\right) \cdot -4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+148}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -6.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(c \cdot 0.5\right) \cdot -4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e+148)
   (- (/ c b))
   (if (<= b -6.8e-302)
     (/ (* (* c 0.5) -4.0) (- b (sqrt (fma a (* c -4.0) (* b b)))))
     (if (<= b 2e+104)
       (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
       (/ b (- a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e+148) {
		tmp = -(c / b);
	} else if (b <= -6.8e-302) {
		tmp = ((c * 0.5) * -4.0) / (b - sqrt(fma(a, (c * -4.0), (b * b))));
	} else if (b <= 2e+104) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e+148)
		tmp = Float64(-Float64(c / b));
	elseif (b <= -6.8e-302)
		tmp = Float64(Float64(Float64(c * 0.5) * -4.0) / Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	elseif (b <= 2e+104)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e+148], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, -6.8e-302], N[(N[(N[(c * 0.5), $MachinePrecision] * -4.0), $MachinePrecision] / N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+104], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.19999999999999998e148

   1. Initial program 1.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 96.3

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

   5. Simplified 96.3%

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

   if -4.19999999999999998e148 < b < -6.8e-302

   1. Initial program 51.3%

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

   4. Applied egg-rr 40.4%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 82.6

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

   7. Simplified 82.6%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. times-frac N/A

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

      6. *-inverses N/A

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

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

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. *-lowering-*.f64 88.9

          \[\leadsto \frac{\left(1 \cdot \left(c \cdot 0.5\right)\right) \cdot -4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, \color{blue}{b \cdot b}\right)}} \]

   9. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{\left(1 \cdot \left(c \cdot 0.5\right)\right) \cdot -4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}} \]

   if -6.8e-302 < b < 2e104

   1. Initial program 88.4%

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

   if 2e104 < b

   1. Initial program 50.7%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

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

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

      3. /-lowering-/.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+148}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -6.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(c \cdot 0.5\right) \cdot -4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-60}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-60)
   (- (/ c b))
   (if (<= b 5e+97)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (/ b (- a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-60) {
		tmp = -(c / b);
	} else if (b <= 5e+97) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = b / -a;
	}
	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 <= (-4.6d-60)) then
        tmp = -(c / b)
    else if (b <= 5d+97) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-60) {
		tmp = -(c / b);
	} else if (b <= 5e+97) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.6e-60:
		tmp = -(c / b)
	elif b <= 5e+97:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e-60)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 5e+97)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.6e-60)
		tmp = -(c / b);
	elseif (b <= 5e+97)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e-60], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 5e+97], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.6000000000000003e-60

   1. Initial program 15.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 92.4

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

   5. Simplified 92.4%

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

   if -4.6000000000000003e-60 < b < 4.99999999999999999e97

   1. Initial program 81.5%

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

   if 4.99999999999999999e97 < b

   1. Initial program 50.7%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

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

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

      3. /-lowering-/.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-60}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-50}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+108}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-50)
   (- (/ c b))
   (if (<= b 1.75e+108)
     (* (/ -0.5 a) (+ b (sqrt (fma b b (* c (* -4.0 a))))))
     (/ b (- a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-50) {
		tmp = -(c / b);
	} else if (b <= 1.75e+108) {
		tmp = (-0.5 / a) * (b + sqrt(fma(b, b, (c * (-4.0 * a)))));
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-50)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.75e+108)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(fma(b, b, Float64(c * Float64(-4.0 * a))))));
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-50], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.75e+108], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7999999999999999e-50

   1. Initial program 15.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 92.4

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

   5. Simplified 92.4%

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

   if -3.7999999999999999e-50 < b < 1.7500000000000001e108

   1. Initial program 81.5%

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

   4. Applied egg-rr 81.4%

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

   if 1.7500000000000001e108 < b

   1. Initial program 50.7%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

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

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

      3. /-lowering-/.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-50}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+108}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-57}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{-6}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-57)
   (- (/ c b))
   (if (<= b 1.52e-6)
     (/ (+ b (sqrt (* a (* c -4.0)))) (* a -2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-57) {
		tmp = -(c / b);
	} else if (b <= 1.52e-6) {
		tmp = (b + sqrt((a * (c * -4.0)))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-4.5d-57)) then
        tmp = -(c / b)
    else if (b <= 1.52d-6) then
        tmp = (b + sqrt((a * (c * (-4.0d0))))) / (a * (-2.0d0))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-57) {
		tmp = -(c / b);
	} else if (b <= 1.52e-6) {
		tmp = (b + Math.sqrt((a * (c * -4.0)))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-57:
		tmp = -(c / b)
	elif b <= 1.52e-6:
		tmp = (b + math.sqrt((a * (c * -4.0)))) / (a * -2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-57)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.52e-6)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -4.0)))) / Float64(a * -2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e-57)
		tmp = -(c / b);
	elseif (b <= 1.52e-6)
		tmp = (b + sqrt((a * (c * -4.0)))) / (a * -2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-57], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.52e-6], N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.49999999999999973e-57

   1. Initial program 15.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 92.4

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

   5. Simplified 92.4%

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

   if -4.49999999999999973e-57 < b < 1.52000000000000006e-6

   1. Initial program 76.1%

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

   3. Taylor expanded in b around 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. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 68.6

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

   5. Simplified 68.6%

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

      1. frac-2neg N/A

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

      2. *-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. distribute-neg-out N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 68.6

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

   7. Applied egg-rr 68.6%

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

   if 1.52000000000000006e-6 < b

   1. Initial program 66.7%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   4. 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 94.0

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

   5. Simplified 94.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-57}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{-6}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.65 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e-54)
   (- (/ c b))
   (if (<= b 3.65e-7)
     (* (/ -0.5 a) (+ b (sqrt (* -4.0 (* c a)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-54) {
		tmp = -(c / b);
	} else if (b <= 3.65e-7) {
		tmp = (-0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-2.2d-54)) then
        tmp = -(c / b)
    else if (b <= 3.65d-7) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((-4.0d0) * (c * a))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-54) {
		tmp = -(c / b);
	} else if (b <= 3.65e-7) {
		tmp = (-0.5 / a) * (b + Math.sqrt((-4.0 * (c * a))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e-54:
		tmp = -(c / b)
	elif b <= 3.65e-7:
		tmp = (-0.5 / a) * (b + math.sqrt((-4.0 * (c * a))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e-54)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 3.65e-7)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e-54)
		tmp = -(c / b);
	elseif (b <= 3.65e-7)
		tmp = (-0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e-54], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 3.65e-7], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2e-54

   1. Initial program 15.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 92.4

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

   5. Simplified 92.4%

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

   if -2.2e-54 < b < 3.65e-7

   1. Initial program 76.1%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 68.6

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

   7. Simplified 68.6%

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

   if 3.65e-7 < b

   1. Initial program 66.7%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   4. 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 94.0

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

   5. Simplified 94.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.65 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -(c / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	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-310)) then
        tmp = -(c / b)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -(c / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -(c / b);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], (-N[(c / b), $MachinePrecision]), N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 33.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 67.7

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

   5. Simplified 67.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 71.6%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   4. 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 70.2

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

   5. Simplified 70.2%

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

4. Final simplification 68.8%

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

Alternative 8: 68.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-309) (- (/ c b)) (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-309) {
		tmp = -(c / b);
	} else {
		tmp = b / -a;
	}
	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-309)) then
        tmp = -(c / b)
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-309) {
		tmp = -(c / b);
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-309:
		tmp = -(c / b)
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-309)
		tmp = Float64(-Float64(c / b));
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-309)
		tmp = -(c / b);
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-309], (-N[(c / b), $MachinePrecision]), N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.9999999999999995e-309

   1. Initial program 33.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 67.7

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

   5. Simplified 67.7%

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

   if -4.9999999999999995e-309 < b

   1. Initial program 71.6%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

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

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

      3. /-lowering-/.f64 70.0

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

   5. Simplified 70.0%

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

4. Final simplification 68.8%

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

Alternative 9: 43.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -2.2e+73) (/ c b) (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+73) {
		tmp = c / b;
	} else {
		tmp = b / -a;
	}
	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 <= (-2.2d+73)) then
        tmp = c / b
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+73) {
		tmp = c / b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e+73:
		tmp = c / b
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+73)
		tmp = Float64(c / b);
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e+73)
		tmp = c / b;
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+73], N[(c / b), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.2e73

   1. Initial program 13.8%

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

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. /-lowering-/.f64 2.7

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

   5. Simplified 2.7%

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

   6. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 36.6

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

   8. Simplified 36.6%

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

   if -2.2e73 < b

   1. Initial program 66.5%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

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

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

      3. /-lowering-/.f64 46.6

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

   5. Simplified 46.6%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.3% 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 50.9%

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

3. Taylor expanded in b around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-neg-frac N/A

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

   4. metadata-eval N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. associate-*r/ N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

   12. mul-1-neg N/A

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

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

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

   14. /-lowering-/.f64 32.8

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

5. Simplified 32.8%

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

6. Taylor expanded in b around 0

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

   1. /-lowering-/.f64 12.8

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

8. Simplified 12.8%

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

Alternative 11: 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 50.9%

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

4. Applied egg-rr 32.6%

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

5. Taylor expanded in b around -inf

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

   1. /-lowering-/.f64 2.5

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

7. Simplified 2.5%

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

Developer Target 1: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp = 0.0;
	if (b < 0.0)
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-b - t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024198 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

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