Cubic critical

Percentage Accurate: 52.0% → 85.4%

Time: 11.6s

Alternatives: 11

Speedup: 2.2×

• could not determine a ground truth

  1. a = -5.508575635916381e-233
  2. b = 1.0826489245328951e+241
  3. c = 1.824328273400782e-42

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: 52.0% 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: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+148}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 395:\\ \;\;\;\;\frac{-1}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e+148)
   (* b (/ -0.6666666666666666 a))
   (if (<= b 395.0)
     (* (/ -1.0 a) (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) -3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e+148) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 395.0) {
		tmp = (-1.0 / a) * ((sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / -3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e+148)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 395.0)
		tmp = Float64(Float64(-1.0 / a) * Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / -3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e+148], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 395.0], N[(N[(-1.0 / a), $MachinePrecision] * N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.95000000000000001e148

   1. Initial program 34.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 96.2

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

   5. Simplified 96.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 96.4

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

   7. Applied egg-rr 96.4%

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

   if -1.95000000000000001e148 < b < 395

   1. Initial program 81.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 81.9

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

   4. Applied egg-rr 81.9%

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

      1. frac-2neg N/A

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

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. neg-mul-1 N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 82.0%

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

   if 395 < b

   1. Initial program 11.3%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 95.0

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

   5. Simplified 95.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+148}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 395:\\ \;\;\;\;\frac{-1}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+144}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 0.03:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+144)
   (* b (/ -0.6666666666666666 a))
   (if (<= b 0.03)
     (/ (/ (- b (sqrt (fma a (* -3.0 c) (* b b)))) a) -3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e+144) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 0.03) {
		tmp = ((b - sqrt(fma(a, (-3.0 * c), (b * b)))) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e+144)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 0.03)
		tmp = Float64(Float64(Float64(b - sqrt(fma(a, Float64(-3.0 * c), 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, -1.35e+144], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.03], N[(N[(N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $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 < -1.35000000000000008e144

   1. Initial program 36.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. /-lowering-/.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. *-lowering-*.f64 96.3

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

   5. Simplified 96.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 96.5

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

   7. Applied egg-rr 96.5%

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

   if -1.35000000000000008e144 < b < 0.029999999999999999

   1. Initial program 81.6%

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

   4. Applied egg-rr 81.7%

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

   if 0.029999999999999999 < b

   1. Initial program 11.3%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 95.0

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

   5. Simplified 95.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+144}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 0.03:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+152}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 0.0027:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e+152)
   (* b (/ -0.6666666666666666 a))
   (if (<= b 0.0027)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e+152) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 0.0027) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e+152)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 0.0027)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e+152], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0027], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.3000000000000001e152

   1. Initial program 32.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 96.1

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

   5. Simplified 96.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 96.3

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

   7. Applied egg-rr 96.3%

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

   if -3.3000000000000001e152 < b < 0.0027000000000000001

   1. Initial program 82.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 82.1

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

   4. Applied egg-rr 82.1%

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

   if 0.0027000000000000001 < b

   1. Initial program 11.3%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 95.0

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

   5. Simplified 95.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+152}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 0.0027:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 0.046:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e+141)
   (/ b (* a -1.5))
   (if (<= b 0.046)
     (* (- b (sqrt (fma a (* -3.0 c) (* b b)))) (/ -0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e+141) {
		tmp = b / (a * -1.5);
	} else if (b <= 0.046) {
		tmp = (b - sqrt(fma(a, (-3.0 * c), (b * b)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e+141)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 0.046)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(-3.0 * c), 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, -1.8e+141], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.046], N[(N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $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 < -1.8000000000000001e141

   1. Initial program 37.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. /-lowering-/.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. *-lowering-*.f64 96.3

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

   5. Simplified 96.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 96.6

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

   7. Applied egg-rr 96.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-2}{3}}}} \]

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

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

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      7. metadata-eval 96.6

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

   9. Applied egg-rr 96.6%

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

   if -1.8000000000000001e141 < b < 0.045999999999999999

   1. Initial program 81.5%

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

   4. Applied egg-rr 81.5%

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

   if 0.045999999999999999 < b

   1. Initial program 11.3%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 95.0

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

   5. Simplified 95.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 0.046:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -340000:\\ \;\;\;\;\left(-b\right) \cdot \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 0.01:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -340000.0)
   (* (- b) (fma c (/ -0.5 (* b b)) (/ 0.6666666666666666 a)))
   (if (<= b 0.01)
     (* (/ 0.3333333333333333 a) (- (sqrt (* -3.0 (* a c))) b))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -340000.0) {
		tmp = -b * fma(c, (-0.5 / (b * b)), (0.6666666666666666 / a));
	} else if (b <= 0.01) {
		tmp = (0.3333333333333333 / a) * (sqrt((-3.0 * (a * c))) - b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -340000.0)
		tmp = Float64(Float64(-b) * fma(c, Float64(-0.5 / Float64(b * b)), Float64(0.6666666666666666 / a)));
	elseif (b <= 0.01)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -340000.0], N[((-b) * N[(c * N[(-0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.01], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4e5

   1. Initial program 59.9%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. neg-lowering-neg.f64 93.3

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Simplified 93.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   if -3.4e5 < b < 0.0100000000000000002

   1. Initial program 75.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 75.5

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

   4. Applied egg-rr 75.5%

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

      1. clear-num N/A

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

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. associate-/r/ N/A

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

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

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

      6. associate-/r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

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

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in b around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.4

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

   9. Simplified 69.4%

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

   if 0.0100000000000000002 < b

   1. Initial program 11.3%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 95.0

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

   5. Simplified 95.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -340000:\\ \;\;\;\;\left(-b\right) \cdot \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 0.01:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -220000:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b}, b \cdot \frac{-0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 0.0022:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -220000.0)
   (fma c (/ 0.5 b) (* b (/ -0.6666666666666666 a)))
   (if (<= b 0.0022)
     (* (/ 0.3333333333333333 a) (- (sqrt (* -3.0 (* a c))) b))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -220000.0) {
		tmp = fma(c, (0.5 / b), (b * (-0.6666666666666666 / a)));
	} else if (b <= 0.0022) {
		tmp = (0.3333333333333333 / a) * (sqrt((-3.0 * (a * c))) - b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -220000.0)
		tmp = fma(c, Float64(0.5 / b), Float64(b * Float64(-0.6666666666666666 / a)));
	elseif (b <= 0.0022)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -220000.0], N[(c * N[(0.5 / b), $MachinePrecision] + N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0022], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2e5

   1. Initial program 59.9%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. neg-lowering-neg.f64 93.3

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Simplified 93.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \color{blue}{\frac{b}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

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

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{\color{blue}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

      5. associate-*r/ N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right) \]

      6. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{\color{blue}{\frac{1}{2}}}{b}\right) \]

      7. /-lowering-/.f64 74.8

         \[\leadsto c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \color{blue}{\frac{0.5}{b}}\right) \]

   8. Simplified 74.8%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \frac{0.5}{b}\right)} \]

   9. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} + \frac{-2}{3} \cdot \frac{b}{a}} \]

       2. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b}} + \frac{-2}{3} \cdot \frac{b}{a} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{c \cdot \frac{1}{2}}}{b} + \frac{-2}{3} \cdot \frac{b}{a} \]

       4. associate-*r/ N/A

          \[\leadsto \color{blue}{c \cdot \frac{\frac{1}{2}}{b}} + \frac{-2}{3} \cdot \frac{b}{a} \]

       5. metadata-eval N/A

          \[\leadsto c \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{b} + \frac{-2}{3} \cdot \frac{b}{a} \]

       6. associate-*r/ N/A

          \[\leadsto c \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b}\right)} + \frac{-2}{3} \cdot \frac{b}{a} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{1}{2} \cdot \frac{1}{b}, \frac{-2}{3} \cdot \frac{b}{a}\right)} \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}, \frac{-2}{3} \cdot \frac{b}{a}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\color{blue}{\frac{1}{2}}}{b}, \frac{-2}{3} \cdot \frac{b}{a}\right) \]

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

           \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{1}{2}}{b}}, \frac{-2}{3} \cdot \frac{b}{a}\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a}\right) \]

       13. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, \color{blue}{b \cdot \frac{\frac{-2}{3}}{a}}\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{2}{3}\right)}}{a}\right) \]

       15. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2}{3}}{a}\right)\right)}\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2}{3} \cdot 1}}{a}\right)\right)\right) \]

       17. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{2}{3} \cdot \frac{1}{a}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, \color{blue}{b \cdot \left(\mathsf{neg}\left(\frac{2}{3} \cdot \frac{1}{a}\right)\right)}\right) \]

       19. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2}{3}}}{a}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{2}{3}\right)}{a}}\right) \]

       22. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \frac{\color{blue}{\frac{-2}{3}}}{a}\right) \]

       23. /-lowering-/.f64 93.3

           \[\leadsto \mathsf{fma}\left(c, \frac{0.5}{b}, b \cdot \color{blue}{\frac{-0.6666666666666666}{a}}\right) \]

   11. Simplified 93.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{0.5}{b}, b \cdot \frac{-0.6666666666666666}{a}\right)} \]

   if -2.2e5 < b < 0.00220000000000000013

   1. Initial program 75.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 75.5

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

   4. Applied egg-rr 75.5%

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

      1. clear-num N/A

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

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. associate-/r/ N/A

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

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

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

      6. associate-/r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

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

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in b around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.4

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

   9. Simplified 69.4%

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

   if 0.00220000000000000013 < b

   1. Initial program 11.3%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 95.0

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

   5. Simplified 95.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -220000:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b}, b \cdot \frac{-0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 0.0022:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b}, b \cdot \frac{-0.6666666666666666}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310)
   (fma c (/ 0.5 b) (* b (/ -0.6666666666666666 a)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = fma(c, (0.5 / b), (b * (-0.6666666666666666 / a)));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = fma(c, Float64(0.5 / b), Float64(b * Float64(-0.6666666666666666 / a)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(c * N[(0.5 / b), $MachinePrecision] + N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 67.7%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. neg-lowering-neg.f64 65.4

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Simplified 65.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \color{blue}{\frac{b}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

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

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{\color{blue}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

      5. associate-*r/ N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right) \]

      6. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{\color{blue}{\frac{1}{2}}}{b}\right) \]

      7. /-lowering-/.f64 53.7

         \[\leadsto c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \color{blue}{\frac{0.5}{b}}\right) \]

   8. Simplified 53.7%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \frac{0.5}{b}\right)} \]

   9. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} + \frac{-2}{3} \cdot \frac{b}{a}} \]

       2. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b}} + \frac{-2}{3} \cdot \frac{b}{a} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{c \cdot \frac{1}{2}}}{b} + \frac{-2}{3} \cdot \frac{b}{a} \]

       4. associate-*r/ N/A

          \[\leadsto \color{blue}{c \cdot \frac{\frac{1}{2}}{b}} + \frac{-2}{3} \cdot \frac{b}{a} \]

       5. metadata-eval N/A

          \[\leadsto c \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{b} + \frac{-2}{3} \cdot \frac{b}{a} \]

       6. associate-*r/ N/A

          \[\leadsto c \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b}\right)} + \frac{-2}{3} \cdot \frac{b}{a} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{1}{2} \cdot \frac{1}{b}, \frac{-2}{3} \cdot \frac{b}{a}\right)} \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}, \frac{-2}{3} \cdot \frac{b}{a}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\color{blue}{\frac{1}{2}}}{b}, \frac{-2}{3} \cdot \frac{b}{a}\right) \]

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

           \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{1}{2}}{b}}, \frac{-2}{3} \cdot \frac{b}{a}\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a}\right) \]

       13. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, \color{blue}{b \cdot \frac{\frac{-2}{3}}{a}}\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{2}{3}\right)}}{a}\right) \]

       15. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2}{3}}{a}\right)\right)}\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2}{3} \cdot 1}}{a}\right)\right)\right) \]

       17. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{2}{3} \cdot \frac{1}{a}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, \color{blue}{b \cdot \left(\mathsf{neg}\left(\frac{2}{3} \cdot \frac{1}{a}\right)\right)}\right) \]

       19. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2}{3}}}{a}\right)\right)\right) \]

       21. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{2}{3}\right)}{a}}\right) \]

       22. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b}, b \cdot \frac{\color{blue}{\frac{-2}{3}}}{a}\right) \]

       23. /-lowering-/.f64 67.5

           \[\leadsto \mathsf{fma}\left(c, \frac{0.5}{b}, b \cdot \color{blue}{\frac{-0.6666666666666666}{a}}\right) \]

   11. Simplified 67.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{0.5}{b}, b \cdot \frac{-0.6666666666666666}{a}\right)} \]

   if -1.999999999999994e-310 < b

   1. Initial program 31.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 71.5

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

   5. Simplified 71.5%

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

Alternative 8: 67.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-285}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6e-285) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-285) {
		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 <= 1.6d-285) 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 <= 1.6e-285) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.6e-285:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6e-285)
		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 <= 1.6e-285)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6e-285], 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 < 1.60000000000000008e-285

   1. Initial program 68.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. /-lowering-/.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. *-lowering-*.f64 66.5

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

   5. Simplified 66.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 66.7

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

   7. Applied egg-rr 66.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-2}{3}}}} \]

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

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

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      7. metadata-eval 66.7

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

   9. Applied egg-rr 66.7%

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

   if 1.60000000000000008e-285 < b

   1. Initial program 30.4%

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

   3. Taylor expanded in b around 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. /-lowering-/.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. *-lowering-*.f64 72.1

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

   5. Simplified 72.1%

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

Alternative 9: 43.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 600000000:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 600000000.0) (/ b (* a -1.5)) (* c (/ 0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 600000000.0) {
		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 <= 600000000.0d0) 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 <= 600000000.0) {
		tmp = b / (a * -1.5);
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 600000000.0:
		tmp = b / (a * -1.5)
	else:
		tmp = c * (0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 600000000.0)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(c * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 600000000.0)
		tmp = b / (a * -1.5);
	else
		tmp = c * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 600000000.0], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6e8

   1. Initial program 66.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 52.6

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

   5. Simplified 52.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 52.7

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

   7. Applied egg-rr 52.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-2}{3}}}} \]

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

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

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{\frac{-2}{3}}}} \]

      7. metadata-eval 52.7

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

   9. Applied egg-rr 52.7%

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

   if 6e8 < b

   1. Initial program 11.5%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. neg-lowering-neg.f64 2.4

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Simplified 2.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \color{blue}{\frac{b}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

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

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{\color{blue}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

      5. associate-*r/ N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right) \]

      6. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{\color{blue}{\frac{1}{2}}}{b}\right) \]

      7. /-lowering-/.f64 2.2

         \[\leadsto c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \color{blue}{\frac{0.5}{b}}\right) \]

   8. Simplified 2.2%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \frac{0.5}{b}\right)} \]

   9. Taylor expanded in b around 0

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

       1. /-lowering-/.f64 19.7

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

   11. Simplified 19.7%

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

Alternative 10: 43.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1100000000:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1100000000.0) (* b (/ -0.6666666666666666 a)) (* c (/ 0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1100000000.0) {
		tmp = b * (-0.6666666666666666 / 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 <= 1100000000.0d0) then
        tmp = b * ((-0.6666666666666666d0) / 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 <= 1100000000.0) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1100000000.0:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1100000000.0)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1100000000.0)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1100000000.0], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1e9

   1. Initial program 66.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.f64 52.6

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

   5. Simplified 52.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 52.7

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

   7. Applied egg-rr 52.7%

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

   if 1.1e9 < b

   1. Initial program 11.5%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. neg-lowering-neg.f64 2.4

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Simplified 2.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \color{blue}{\frac{b}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

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

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{\color{blue}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

      5. associate-*r/ N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right) \]

      6. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{\color{blue}{\frac{1}{2}}}{b}\right) \]

      7. /-lowering-/.f64 2.2

         \[\leadsto c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \color{blue}{\frac{0.5}{b}}\right) \]

   8. Simplified 2.2%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \frac{0.5}{b}\right)} \]

   9. Taylor expanded in b around 0

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

       1. /-lowering-/.f64 19.7

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

   11. Simplified 19.7%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1100000000:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* c (/ 0.5 b)))

C

double code(double a, double b, double c) {
	return c * (0.5 / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (0.5d0 / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * (0.5 / b);
}

Python

def code(a, b, c):
	return c * (0.5 / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(0.5 / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (0.5 / b);
end

Wolfram

code[a_, b_, c_] := N[(c * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

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

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

      \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   15. neg-lowering-neg.f64 37.6

       \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

5. Simplified 37.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

6. Taylor expanded in c around inf

   \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]
7. Step-by-step derivation

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

      \[\leadsto \color{blue}{c \cdot \left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

      \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{1}{2} \cdot \frac{1}{b}\right)} \]

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

      \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \color{blue}{\frac{b}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

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

      \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{\color{blue}{a \cdot c}}, \frac{1}{2} \cdot \frac{1}{b}\right) \]

   5. associate-*r/ N/A

      \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right) \]

   6. metadata-eval N/A

      \[\leadsto c \cdot \mathsf{fma}\left(\frac{-2}{3}, \frac{b}{a \cdot c}, \frac{\color{blue}{\frac{1}{2}}}{b}\right) \]

   7. /-lowering-/.f64 31.0

      \[\leadsto c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \color{blue}{\frac{0.5}{b}}\right) \]

8. Simplified 31.0%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a \cdot c}, \frac{0.5}{b}\right)} \]

9. Taylor expanded in b around 0

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

    1. /-lowering-/.f64 7.8

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

11. Simplified 7.8%

    \[\leadsto c \cdot \color{blue}{\frac{0.5}{b}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))