Quadratic roots, full range

Percentage Accurate: 52.6% → 90.5%

Time: 10.0s

Alternatives: 9

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{4 \cdot c}{b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+153)
   (- (/ b a))
   (if (<= b 2.4e-208)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (if (<= b 9.2e+107)
       (* (/ (* 4.0 c) (+ b (sqrt (fma -4.0 (* a c) (* b b))))) -0.5)
       (/ c (- b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+153) {
		tmp = -(b / a);
	} else if (b <= 2.4e-208) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if (b <= 9.2e+107) {
		tmp = ((4.0 * c) / (b + sqrt(fma(-4.0, (a * c), (b * b))))) * -0.5;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+153)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 2.4e-208)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	elseif (b <= 9.2e+107)
		tmp = Float64(Float64(Float64(4.0 * c) / Float64(b + sqrt(fma(-4.0, Float64(a * c), Float64(b * b))))) * -0.5);
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+153], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 2.4e-208], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2e+107], N[(N[(N[(4.0 * c), $MachinePrecision] / N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.50000000000000009e153

   1. Initial program 31.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.9

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

   5. Simplified 92.9%

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

   if -2.50000000000000009e153 < b < 2.3999999999999999e-208

   1. Initial program 88.9%

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

   if 2.3999999999999999e-208 < b < 9.2000000000000001e107

   1. Initial program 30.9%

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

   4. Applied egg-rr 30.9%

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

   6. Applied egg-rr 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift--.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

   8. Applied egg-rr 69.8%

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

   10. Applied egg-rr 84.9%

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

   if 9.2000000000000001e107 < b

   1. Initial program 4.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 94.6

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

   5. Simplified 94.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{4 \cdot c}{b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-a, \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+153)
   (- (/ b a))
   (if (<= b 9.8e-36)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (* c (fma (- a) (/ c (* b (* b b))) (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+153) {
		tmp = -(b / a);
	} else if (b <= 9.8e-36) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c * fma(-a, (c / (b * (b * b))), (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+153)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 9.8e-36)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * fma(Float64(-a), Float64(c / Float64(b * Float64(b * b))), Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+153], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 9.8e-36], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[((-a) * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.50000000000000009e153

   1. Initial program 31.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.9

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

   5. Simplified 92.9%

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

   if -2.50000000000000009e153 < b < 9.7999999999999994e-36

   1. Initial program 83.9%

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

   if 9.7999999999999994e-36 < b

   1. Initial program 12.9%

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

   4. Applied egg-rr 12.8%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)} \]

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(-1 \cdot a\right)} \cdot \frac{c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto c \cdot \left(\left(-1 \cdot a\right) \cdot \frac{c}{{b}^{3}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right) \]

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(-1 \cdot a, \frac{c}{{b}^{3}}, \frac{-1}{b}\right)} \]

      10. mul-1-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, \frac{c}{{b}^{3}}, \frac{-1}{b}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, \frac{c}{{b}^{3}}, \frac{-1}{b}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{\frac{c}{{b}^{3}}}, \frac{-1}{b}\right) \]

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 85.7

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

   7. Simplified 85.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-a, \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+139}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-a, \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e+139)
   (- (/ b a))
   (if (<= b 9.8e-36)
     (* (/ -0.5 a) (- b (sqrt (fma a (* c -4.0) (* b b)))))
     (* c (fma (- a) (/ c (* b (* b b))) (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e+139) {
		tmp = -(b / a);
	} else if (b <= 9.8e-36) {
		tmp = (-0.5 / a) * (b - sqrt(fma(a, (c * -4.0), (b * b))));
	} else {
		tmp = c * fma(-a, (c / (b * (b * b))), (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e+139)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 9.8e-36)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	else
		tmp = Float64(c * fma(Float64(-a), Float64(c / Float64(b * Float64(b * b))), Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e+139], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 9.8e-36], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[((-a) * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.5000000000000002e139

   1. Initial program 43.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.1

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

   5. Simplified 94.1%

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

   if -9.5000000000000002e139 < b < 9.7999999999999994e-36

   1. Initial program 83.2%

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

   4. Applied egg-rr 83.0%

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

   if 9.7999999999999994e-36 < b

   1. Initial program 12.9%

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

   4. Applied egg-rr 12.8%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)} \]

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(-1 \cdot a\right)} \cdot \frac{c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto c \cdot \left(\left(-1 \cdot a\right) \cdot \frac{c}{{b}^{3}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right) \]

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(-1 \cdot a, \frac{c}{{b}^{3}}, \frac{-1}{b}\right)} \]

      10. mul-1-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, \frac{c}{{b}^{3}}, \frac{-1}{b}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, \frac{c}{{b}^{3}}, \frac{-1}{b}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{\frac{c}{{b}^{3}}}, \frac{-1}{b}\right) \]

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 85.7

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

   7. Simplified 85.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+139}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-a, \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.62e-74)
   (- (/ b a))
   (if (<= b 8.5e-67)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.62e-74) {
		tmp = -(b / a);
	} else if (b <= 8.5e-67) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} 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 <= (-1.62d-74)) then
        tmp = -(b / a)
    else if (b <= 8.5d-67) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    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 <= -1.62e-74) {
		tmp = -(b / a);
	} else if (b <= 8.5e-67) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.62e-74:
		tmp = -(b / a)
	elif b <= 8.5e-67:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.62e-74)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 8.5e-67)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.62e-74)
		tmp = -(b / a);
	elseif (b <= 8.5e-67)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.62e-74], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 8.5e-67], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.62000000000000007e-74

   1. Initial program 77.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.8

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

   5. Simplified 84.8%

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

   if -1.62000000000000007e-74 < b < 8.49999999999999993e-67

   1. Initial program 74.6%

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

   3. Taylor expanded in b around 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. lower-*.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. lower-*.f64 74.6

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

   5. Simplified 74.6%

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

   if 8.49999999999999993e-67 < b

   1. Initial program 13.7%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 85.1

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

   5. Simplified 85.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.62e-74)
   (- (/ b a))
   (if (<= b 8.5e-67)
     (* (/ -0.5 a) (- b (sqrt (* -4.0 (* a c)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.62e-74) {
		tmp = -(b / a);
	} else if (b <= 8.5e-67) {
		tmp = (-0.5 / a) * (b - sqrt((-4.0 * (a * c))));
	} 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 <= (-1.62d-74)) then
        tmp = -(b / a)
    else if (b <= 8.5d-67) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((-4.0d0) * (a * c))))
    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 <= -1.62e-74) {
		tmp = -(b / a);
	} else if (b <= 8.5e-67) {
		tmp = (-0.5 / a) * (b - Math.sqrt((-4.0 * (a * c))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.62e-74:
		tmp = -(b / a)
	elif b <= 8.5e-67:
		tmp = (-0.5 / a) * (b - math.sqrt((-4.0 * (a * c))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.62e-74)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 8.5e-67)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(-4.0 * Float64(a * c)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.62e-74)
		tmp = -(b / a);
	elseif (b <= 8.5e-67)
		tmp = (-0.5 / a) * (b - sqrt((-4.0 * (a * c))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.62e-74], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 8.5e-67], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.62000000000000007e-74

   1. Initial program 77.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.8

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

   5. Simplified 84.8%

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

   if -1.62000000000000007e-74 < b < 8.49999999999999993e-67

   1. Initial program 74.6%

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

   4. Applied egg-rr 74.4%

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

   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. lower-*.f64 N/A

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

      3. lower-*.f64 74.3

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

   7. Simplified 74.3%

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

   if 8.49999999999999993e-67 < b

   1. Initial program 13.7%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 85.1

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

   5. Simplified 85.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (- (/ b a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = -(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 <= 5.1d-272) then
        tmp = -(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 <= 5.1e-272) {
		tmp = -(b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = -(b / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(-Float64(b / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = -(b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], (-N[(b / a), $MachinePrecision]), N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 77.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 55.3

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

   5. Simplified 55.3%

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

   if 5.0999999999999998e-272 < b

   1. Initial program 24.8%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.8

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

   5. Simplified 71.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1150000:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1150000.0) (- (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1150000.0) {
		tmp = -(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 <= 1150000.0d0) then
        tmp = -(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 <= 1150000.0) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1150000.0:
		tmp = -(b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1150000.0)
		tmp = Float64(-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 <= 1150000.0)
		tmp = -(b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1150000.0], (-N[(b / a), $MachinePrecision]), N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15e6

   1. Initial program 70.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 41.7

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

   5. Simplified 41.7%

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

   if 1.15e6 < b

   1. Initial program 11.5%

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

   4. Applied egg-rr 5.4%

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

   5. Taylor expanded in b around -inf

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

      1. lower-/.f64 23.3

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

   7. Simplified 23.3%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1150000:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.3% 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 51.6%

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

4. Applied egg-rr 32.6%

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

5. Taylor expanded in b around -inf

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

   1. lower-/.f64 9.7

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

7. Simplified 9.7%

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

Alternative 9: 2.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

4. Applied egg-rr 32.6%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 2.7

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

7. Simplified 2.7%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))