Cubic critical

Percentage Accurate: 50.7% → 85.1%

Time: 10.3s

Alternatives: 16

Speedup: 2.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.3333333333333333, \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{3 \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e+118)
   (/ (/ b -1.5) a)
   (if (<= b 1.32e-107)
     (fma
      (/ b a)
      -0.3333333333333333
      (/ (sqrt (fma b b (* a (* c -3.0)))) (* 3.0 a)))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e+118) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.32e-107) {
		tmp = fma((b / a), -0.3333333333333333, (sqrt(fma(b, b, (a * (c * -3.0)))) / (3.0 * a)));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e+118)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 1.32e-107)
		tmp = fma(Float64(b / a), -0.3333333333333333, Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) / Float64(3.0 * a)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e+118], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.32e-107], N[(N[(b / a), $MachinePrecision] * -0.3333333333333333 + N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.69999999999999987e118

   1. Initial program 56.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 99.7

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\frac{b}{-1.5}}{\color{blue}{a}} \]

   if -3.69999999999999987e118 < b < 1.3200000000000001e-107

   1. Initial program 85.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 85.9

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

   4. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. sub-div N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. sub-neg N/A

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

   6. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -0.3333333333333333, \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{a \cdot 3}\right)} \]

   if 1.3200000000000001e-107 < b

   1. Initial program 21.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 86.1

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.3333333333333333, \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{3 \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+144)
   (/ b (* a -1.5))
   (if (<= b 6e-128)
     (/ (/ (- b (sqrt (fma a (* c -3.0) (* b b)))) a) -3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+144) {
		tmp = b / (a * -1.5);
	} else if (b <= 6e-128) {
		tmp = ((b - sqrt(fma(a, (c * -3.0), (b * b)))) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+144)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 6e-128)
		tmp = Float64(Float64(Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+144], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-128], N[(N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999998e144

   1. Initial program 49.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 99.7

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.8%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]

   if -8.4999999999999998e144 < b < 5.99999999999999956e-128

   1. Initial program 88.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 88.1%

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

   if 5.99999999999999956e-128 < b

   1. Initial program 23.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e+118)
   (/ (/ b -1.5) a)
   (if (<= b 6e-128)
     (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e+118) {
		tmp = (b / -1.5) / a;
	} else if (b <= 6e-128) {
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.7d+118)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 6d-128) then
        tmp = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e+118) {
		tmp = (b / -1.5) / a;
	} else if (b <= 6e-128) {
		tmp = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e+118:
		tmp = (b / -1.5) / a
	elif b <= 6e-128:
		tmp = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e+118)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 6e-128)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.7e+118)
		tmp = (b / -1.5) / a;
	elseif (b <= 6e-128)
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e+118], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6e-128], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.69999999999999987e118

   1. Initial program 56.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 99.7

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\frac{b}{-1.5}}{\color{blue}{a}} \]

   if -3.69999999999999987e118 < b < 5.99999999999999956e-128

   1. Initial program 87.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   if 5.99999999999999956e-128 < b

   1. Initial program 23.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e+118)
   (/ (/ b -1.5) a)
   (if (<= b 6e-128)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e+118) {
		tmp = (b / -1.5) / a;
	} else if (b <= 6e-128) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e+118)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 6e-128)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e+118], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6e-128], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.69999999999999987e118

   1. Initial program 56.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 99.7

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\frac{b}{-1.5}}{\color{blue}{a}} \]

   if -3.69999999999999987e118 < b < 5.99999999999999956e-128

   1. Initial program 87.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 87.3

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

   4. Applied rewrites 87.3%

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

   if 5.99999999999999956e-128 < b

   1. Initial program 23.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e+118)
   (/ (/ b -1.5) a)
   (if (<= b 6e-128)
     (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e+118) {
		tmp = (b / -1.5) / a;
	} else if (b <= 6e-128) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e+118)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 6e-128)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e+118], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6e-128], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.69999999999999987e118

   1. Initial program 56.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 99.7

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\frac{b}{-1.5}}{\color{blue}{a}} \]

   if -3.69999999999999987e118 < b < 5.99999999999999956e-128

   1. Initial program 87.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 87.3

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

   4. Applied rewrites 87.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 87.2

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 87.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 87.2

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

   6. Applied rewrites 87.2%

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

   if 5.99999999999999956e-128 < b

   1. Initial program 23.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e+118)
   (/ (/ b -1.5) a)
   (if (<= b 6e-128)
     (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e+118) {
		tmp = (b / -1.5) / a;
	} else if (b <= 6e-128) {
		tmp = (sqrt(fma(a, (c * -3.0), (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e+118)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 6e-128)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -3.0), Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e+118], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6e-128], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.69999999999999987e118

   1. Initial program 56.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 99.7

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\frac{b}{-1.5}}{\color{blue}{a}} \]

   if -3.69999999999999987e118 < b < 5.99999999999999956e-128

   1. Initial program 87.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 87.3

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 87.2

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

   4. Applied rewrites 87.2%

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

   if 5.99999999999999956e-128 < b

   1. Initial program 23.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+168)
   (/ b (* a -1.5))
   (if (<= b 6e-128)
     (* (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) a) 0.3333333333333333)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+168) {
		tmp = b / (a * -1.5);
	} else if (b <= 6e-128) {
		tmp = ((sqrt(fma(b, b, (a * (c * -3.0)))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+168)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 6e-128)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+168], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-128], N[(N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999983e168

   1. Initial program 42.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 99.7

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.8%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]

   if -2.49999999999999983e168 < b < 5.99999999999999956e-128

   1. Initial program 88.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a} \cdot 0.3333333333333333} \]

   if 5.99999999999999956e-128 < b

   1. Initial program 23.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+168)
   (/ b (* a -1.5))
   (if (<= b 6e-128)
     (* (- b (sqrt (fma a (* c -3.0) (* b b)))) (/ -0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+168) {
		tmp = b / (a * -1.5);
	} else if (b <= 6e-128) {
		tmp = (b - sqrt(fma(a, (c * -3.0), (b * b)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+168)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 6e-128)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+168], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-128], N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999983e168

   1. Initial program 42.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 99.7

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.8%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]

   if -2.49999999999999983e168 < b < 5.99999999999999956e-128

   1. Initial program 88.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

   if 5.99999999999999956e-128 < b

   1. Initial program 23.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.4e-13)
   (/ (/ (- b (- b)) a) -3.0)
   (if (<= b 6e-128)
     (* 0.3333333333333333 (/ (- (sqrt (* a (* c -3.0))) b) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-13) {
		tmp = ((b - -b) / a) / -3.0;
	} else if (b <= 6e-128) {
		tmp = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.4d-13)) then
        tmp = ((b - -b) / a) / (-3.0d0)
    else if (b <= 6d-128) then
        tmp = 0.3333333333333333d0 * ((sqrt((a * (c * (-3.0d0)))) - b) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-13) {
		tmp = ((b - -b) / a) / -3.0;
	} else if (b <= 6e-128) {
		tmp = 0.3333333333333333 * ((Math.sqrt((a * (c * -3.0))) - b) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.4e-13:
		tmp = ((b - -b) / a) / -3.0
	elif b <= 6e-128:
		tmp = 0.3333333333333333 * ((math.sqrt((a * (c * -3.0))) - b) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.4e-13)
		tmp = Float64(Float64(Float64(b - Float64(-b)) / a) / -3.0);
	elseif (b <= 6e-128)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.4e-13)
		tmp = ((b - -b) / a) / -3.0;
	elseif (b <= 6e-128)
		tmp = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.4e-13], N[(N[(N[(b - (-b)), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 6e-128], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.39999999999999977e-13

   1. Initial program 71.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{b - \color{blue}{-1 \cdot b}}{a}}{-3} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{b - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{a}}{-3} \]

      2. lower-neg.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if -7.39999999999999977e-13 < b < 5.99999999999999956e-128

   1. Initial program 82.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\frac{\sqrt{a \cdot \left(-3 \cdot c\right)} - b}{a} \cdot 0.3333333333333333} \]

   if 5.99999999999999956e-128 < b

   1. Initial program 23.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.4e-13)
   (/ (/ (- b (- b)) a) -3.0)
   (if (<= b 6e-128)
     (* (- (sqrt (* a (* c -3.0))) b) (/ 0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-13) {
		tmp = ((b - -b) / a) / -3.0;
	} else if (b <= 6e-128) {
		tmp = (sqrt((a * (c * -3.0))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.4d-13)) then
        tmp = ((b - -b) / a) / (-3.0d0)
    else if (b <= 6d-128) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-13) {
		tmp = ((b - -b) / a) / -3.0;
	} else if (b <= 6e-128) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.4e-13:
		tmp = ((b - -b) / a) / -3.0
	elif b <= 6e-128:
		tmp = (math.sqrt((a * (c * -3.0))) - b) * (0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.4e-13)
		tmp = Float64(Float64(Float64(b - Float64(-b)) / a) / -3.0);
	elseif (b <= 6e-128)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.4e-13)
		tmp = ((b - -b) / a) / -3.0;
	elseif (b <= 6e-128)
		tmp = (sqrt((a * (c * -3.0))) - b) * (0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.4e-13], N[(N[(N[(b - (-b)), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 6e-128], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.39999999999999977e-13

   1. Initial program 71.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{b - \color{blue}{-1 \cdot b}}{a}}{-3} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{b - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{a}}{-3} \]

      2. lower-neg.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if -7.39999999999999977e-13 < b < 5.99999999999999956e-128

   1. Initial program 82.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 72.3

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \left(\left(-b\right) + \sqrt{a \cdot \left(c \cdot -3\right)}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 72.3

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)} \]

   7. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(-3 \cdot c\right)} - b\right)} \]

   if 5.99999999999999956e-128 < b

   1. Initial program 23.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-128}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.85e-299) (/ (/ (- b (- b)) a) -3.0) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-299) {
		tmp = ((b - -b) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / 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 <= 1.85d-299) then
        tmp = ((b - -b) / a) / (-3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-299) {
		tmp = ((b - -b) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.85e-299:
		tmp = ((b - -b) / a) / -3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e-299)
		tmp = Float64(Float64(Float64(b - Float64(-b)) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.85e-299)
		tmp = ((b - -b) / a) / -3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.85e-299], N[(N[(N[(b - (-b)), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.85000000000000007e-299

   1. Initial program 78.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{b - \color{blue}{-1 \cdot b}}{a}}{-3} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{b - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{a}}{-3} \]

      2. lower-neg.f64 68.0

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

   7. Applied rewrites 68.0%

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

   if 1.85000000000000007e-299 < b

   1. Initial program 32.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 71.1

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.85e-299) (/ (/ b -1.5) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-299) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c * -0.5) / 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 <= 1.85d-299) then
        tmp = (b / (-1.5d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-299) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.85e-299:
		tmp = (b / -1.5) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e-299)
		tmp = Float64(Float64(b / -1.5) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.85e-299)
		tmp = (b / -1.5) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.85e-299], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.85000000000000007e-299

   1. Initial program 78.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 68.0

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.0%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.0%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

       \[\leadsto \frac{\frac{b}{-1.5}}{\color{blue}{a}} \]

   if 1.85000000000000007e-299 < b

   1. Initial program 32.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 71.1

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-299}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.85e-299) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-299) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / 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 <= 1.85d-299) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-299) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.85e-299:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e-299)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.85e-299)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.85e-299], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.85000000000000007e-299

   1. Initial program 78.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 68.0

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.0%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.0%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]

   if 1.85000000000000007e-299 < b

   1. Initial program 32.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 71.1

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 43.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -1e-309) (/ b (* a -1.5)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		tmp = b / (a * -1.5);
	} else {
		tmp = 0.0;
	}
	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 <= (-1d-309)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		tmp = b / (a * -1.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-309:
		tmp = b / (a * -1.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-309)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-309)
		tmp = b / (a * -1.5);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-309], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.000000000000002e-309

   1. Initial program 78.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 70.0

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.0%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]
   8. Step-by-step derivation

   9. Applied rewrites 70.0%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]

   if -1.000000000000002e-309 < b

   1. Initial program 33.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 33.6

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

   4. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. sub-div N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. sub-neg N/A

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

   6. Applied rewrites 29.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -0.3333333333333333, \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{a \cdot 3}\right)} \]

   7. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{b}{a} + \frac{1}{3} \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt 20.1

         \[\leadsto \color{blue}{0} \]

   9. Applied rewrites 20.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 43.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-309) (* b (/ -0.6666666666666666 a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0;
	}
	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 <= (-1d-309)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-309:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-309)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-309)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-309], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.000000000000002e-309

   1. Initial program 78.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 70.0

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.0%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]

   if -1.000000000000002e-309 < b

   1. Initial program 33.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 33.6

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

   4. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. sub-div N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. sub-neg N/A

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

   6. Applied rewrites 29.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -0.3333333333333333, \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{a \cdot 3}\right)} \]

   7. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{b}{a} + \frac{1}{3} \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt 20.1

         \[\leadsto \color{blue}{0} \]

   9. Applied rewrites 20.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 11.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 55.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 55.6

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

4. Applied rewrites 55.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-fma.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. sub-neg N/A

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

   14. sub-div N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. sub-neg N/A

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

6. Applied rewrites 53.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -0.3333333333333333, \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{a \cdot 3}\right)} \]

7. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{b}{a} + \frac{1}{3} \cdot \frac{b}{a}} \]
8. Step-by-step derivation

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 11.4

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 11.4%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))