Cubic critical

Percentage Accurate: 52.5% → 85.6%

Time: 12.5s

Alternatives: 13

Speedup: 2.2×

• could not determine a ground truth

  1. a = 2.6573587547775545e-308
  2. b = 1.5530510028790296e+188
  3. c = -9.610058529324851e-17

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: 52.5% 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: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+137}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e+137)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.04e-109)
     (/ (- (sqrt (fma (* a -3.0) c (fma b b 0.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e+137) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.04e-109) {
		tmp = (sqrt(fma((a * -3.0), c, fma(b, b, 0.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e+137)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.04e-109)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, fma(b, b, 0.0))) - 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, -7.2e+137], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.04e-109], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b + 0.0), $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 < -7.1999999999999999e137

   1. Initial program 48.7%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   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 97.9

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

   5. Simplified 97.9%

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

   if -7.1999999999999999e137 < b < 1.03999999999999996e-109

   1. Initial program 86.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. +-lft-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

      15. +-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

      17. mul0-lft N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

      18. accelerator-lowering-fma.f64 86.4

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   4. Applied egg-rr 86.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot -3\right) \cdot c} + \left(b \cdot b + 0\right)} - b}{3 \cdot a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-3 \cdot a\right)} \cdot c + \left(b \cdot b + 0\right)} - b}{3 \cdot a} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot c + \left(b \cdot b + 0\right)} - b}{3 \cdot a} \]

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

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} \cdot c + \left(b \cdot b + 0\right)} - b}{3 \cdot a} \]

      5. +-rgt-identity N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. +-rgt-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, \color{blue}{b \cdot b + 0}\right)} - b}{3 \cdot a} \]

      12. accelerator-lowering-fma.f64 86.4

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   6. Applied egg-rr 86.4%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(a \cdot -3, c, \mathsf{fma}\left(b, b, 0\right)\right)}} - b}{3 \cdot a} \]

   if 1.03999999999999996e-109 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   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 88.1

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

   5. Simplified 88.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+137}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+138)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.04e-109)
     (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+138) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.04e-109) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+138)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.04e-109)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - 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, -1e+138], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.04e-109], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $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 < -1e138

   1. Initial program 48.7%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   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 97.9

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

   5. Simplified 97.9%

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

   if -1e138 < b < 1.03999999999999996e-109

   1. Initial program 86.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. +-lft-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

      15. +-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

      17. mul0-lft N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

      18. accelerator-lowering-fma.f64 86.4

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   4. Applied egg-rr 86.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 86.4

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

   6. Applied egg-rr 86.4%

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

   if 1.03999999999999996e-109 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   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 88.1

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

   5. Simplified 88.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.8e+111)
   (fma 0.5 (/ c b) (* -0.6666666666666666 (/ b a)))
   (if (<= b 1.04e-109)
     (/ (* (- (sqrt (fma b b (* a (* -3.0 c)))) b) 0.3333333333333333) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e+111) {
		tmp = fma(0.5, (c / b), (-0.6666666666666666 * (b / a)));
	} else if (b <= 1.04e-109) {
		tmp = ((sqrt(fma(b, b, (a * (-3.0 * c)))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.8e+111)
		tmp = fma(0.5, Float64(c / b), Float64(-0.6666666666666666 * Float64(b / a)));
	elseif (b <= 1.04e-109)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.8e+111], N[(0.5 * N[(c / b), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.04e-109], N[(N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.8000000000000003e111

   1. Initial program 54.6%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\frac{2}{3}}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

      16. --lowering--.f64 98.0

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

   5. Simplified 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(0 - b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 98.1

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

   8. Simplified 98.1%

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

   if -6.8000000000000003e111 < b < 1.03999999999999996e-109

   1. Initial program 85.7%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. +-lft-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

      15. +-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

      17. mul0-lft N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

      18. accelerator-lowering-fma.f64 85.6

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   4. Applied egg-rr 85.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 85.6

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

   6. Applied egg-rr 85.6%

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

      1. associate-/r* N/A

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

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

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

      3. div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. metadata-eval 85.6

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

   8. Applied egg-rr 85.6%

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

   if 1.03999999999999996e-109 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   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 88.1

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

   5. Simplified 88.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e+117)
   (fma 0.5 (/ c b) (* -0.6666666666666666 (/ b a)))
   (if (<= b 1.04e-109)
     (* (- (sqrt (fma b b (* a (* -3.0 c)))) b) (/ 0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e+117) {
		tmp = fma(0.5, (c / b), (-0.6666666666666666 * (b / a)));
	} else if (b <= 1.04e-109) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e+117)
		tmp = fma(0.5, Float64(c / b), Float64(-0.6666666666666666 * Float64(b / a)));
	elseif (b <= 1.04e-109)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - 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, -8e+117], N[(0.5 * N[(c / b), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.04e-109], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.0000000000000004e117

   1. Initial program 54.6%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\frac{2}{3}}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

      16. --lowering--.f64 98.0

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

   5. Simplified 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(0 - b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 98.1

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

   8. Simplified 98.1%

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

   if -8.0000000000000004e117 < b < 1.03999999999999996e-109

   1. Initial program 85.7%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. +-lft-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

      15. +-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

      17. mul0-lft N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

      18. accelerator-lowering-fma.f64 85.6

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   4. Applied egg-rr 85.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 85.6

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

   6. Applied egg-rr 85.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 85.6

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

   8. Applied egg-rr 85.6%

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

   if 1.03999999999999996e-109 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   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 88.1

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

   5. Simplified 88.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.6e-25)
   (fma 0.5 (/ c b) (* -0.6666666666666666 (/ b a)))
   (if (<= b 1.04e-109)
     (/ (- (sqrt (* (* a -3.0) c)) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-25) {
		tmp = fma(0.5, (c / b), (-0.6666666666666666 * (b / a)));
	} else if (b <= 1.04e-109) {
		tmp = (sqrt(((a * -3.0) * c)) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.6e-25)
		tmp = fma(0.5, Float64(c / b), Float64(-0.6666666666666666 * Float64(b / a)));
	elseif (b <= 1.04e-109)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * -3.0) * c)) - 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, -6.6e-25], N[(0.5 * N[(c / b), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.04e-109], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $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 < -6.5999999999999997e-25

   1. Initial program 67.1%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\frac{2}{3}}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

      16. --lowering--.f64 87.2

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

   5. Simplified 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(0 - b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 87.2

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

   8. Simplified 87.2%

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

   if -6.5999999999999997e-25 < b < 1.03999999999999996e-109

   1. Initial program 84.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. +-lft-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

      15. +-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

      17. mul0-lft N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

      18. accelerator-lowering-fma.f64 84.1

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   4. Applied egg-rr 84.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 84.1

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

   6. Applied egg-rr 84.1%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 75.5

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

   9. Simplified 75.5%

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

   if 1.03999999999999996e-109 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   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 88.1

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

   5. Simplified 88.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e-25)
   (fma 0.5 (/ c b) (* -0.6666666666666666 (/ b a)))
   (if (<= b 8e-113)
     (/ (- (sqrt (* a (* -3.0 c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e-25) {
		tmp = fma(0.5, (c / b), (-0.6666666666666666 * (b / a)));
	} else if (b <= 8e-113) {
		tmp = (sqrt((a * (-3.0 * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e-25)
		tmp = fma(0.5, Float64(c / b), Float64(-0.6666666666666666 * Float64(b / a)));
	elseif (b <= 8e-113)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-3.0 * c))) - 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, -3.4e-25], N[(0.5 * N[(c / b), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-113], N[(N[(N[Sqrt[N[(a * N[(-3.0 * c), $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 < -3.40000000000000002e-25

   1. Initial program 67.1%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\frac{2}{3}}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

      16. --lowering--.f64 87.2

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

   5. Simplified 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(0 - b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 87.2

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

   8. Simplified 87.2%

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

   if -3.40000000000000002e-25 < b < 7.99999999999999983e-113

   1. Initial program 84.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. +-lft-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

      15. +-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

      17. mul0-lft N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

      18. accelerator-lowering-fma.f64 84.1

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   4. Applied egg-rr 84.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 75.5

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

   7. Simplified 75.5%

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

   if 7.99999999999999983e-113 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   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 88.1

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

   5. Simplified 88.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (fma 0.5 (/ c b) (* -0.6666666666666666 (/ b a)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = fma(0.5, (c / b), (-0.6666666666666666 * (b / a)));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = fma(0.5, Float64(c / b), Float64(-0.6666666666666666 * Float64(b / a)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(0.5 * N[(c / b), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 73.1%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\frac{2}{3}}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

      16. --lowering--.f64 63.8

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(0 - b\right)} \]

   5. Simplified 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(0 - b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 65.7

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

   8. Simplified 65.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.5%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   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 67.7

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

   5. Simplified 67.7%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-286}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.2e-286) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2e-286) {
		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 <= 6.2d-286) 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 <= 6.2e-286) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.2e-286:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2e-286)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.2e-286)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2e-286], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.19999999999999964e-286

   1. Initial program 73.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-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 63.3

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

   5. Simplified 63.3%

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

   if 6.19999999999999964e-286 < b

   1. Initial program 33.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 69.7

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

   5. Simplified 69.7%

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

Alternative 9: 67.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-286}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-286) (/ (* b -0.6666666666666666) a) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-286) {
		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 <= 5.2d-286) 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 <= 5.2e-286) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-286:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-286)
		tmp = Float64(Float64(b * -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 <= 5.2e-286)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-286], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1999999999999999e-286

   1. Initial program 73.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-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 63.3

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

   5. Simplified 63.3%

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

   if 5.1999999999999999e-286 < b

   1. Initial program 33.3%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. +-lft-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

      15. +-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

      17. mul0-lft N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

      18. accelerator-lowering-fma.f64 33.3

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   4. Applied egg-rr 33.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 33.4

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

   6. Applied egg-rr 33.4%

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

   7. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 69.4

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

   9. Simplified 69.4%

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

Alternative 10: 67.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-286}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-286) (* -0.6666666666666666 (/ b a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-286) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.2d-286) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-286) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-286:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-286)
		tmp = Float64(-0.6666666666666666 * Float64(b / 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 <= 5.2e-286)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-286], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1999999999999999e-286

   1. Initial program 73.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. +-lft-identity N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot \color{blue}{\left(0 + b\right)}\right)}}{3 \cdot a} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot \color{blue}{\left(b + 0\right)}\right)}}{3 \cdot a} \]

      11. distribute-rgt-out N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, \color{blue}{b \cdot b + 0 \cdot b}\right)}}{3 \cdot a} \]

      12. mul0-lft N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b + \color{blue}{0}\right)}}{3 \cdot a} \]

      13. accelerator-lowering-fma.f64 73.1

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)}}{3 \cdot a} \]

   4. Applied egg-rr 73.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot -3, c, \mathsf{fma}\left(b, b, 0\right)\right)}}}{3 \cdot a} \]

   5. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

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

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

      3. /-lowering-/.f64 63.3

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

   7. Simplified 63.3%

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

   if 5.1999999999999999e-286 < b

   1. Initial program 33.3%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. +-lft-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

      15. +-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

      17. mul0-lft N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

      18. accelerator-lowering-fma.f64 33.3

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   4. Applied egg-rr 33.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 33.4

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

   6. Applied egg-rr 33.4%

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

   7. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 69.4

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

   9. Simplified 69.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-286}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-286}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-286) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-286) {
		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 <= 5.2d-286) 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 <= 5.2e-286) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-286:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-286)
		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 <= 5.2e-286)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-286], 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 < 5.1999999999999999e-286

   1. Initial program 73.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-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 63.3

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

   5. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   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 63.3

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

   7. Applied egg-rr 63.3%

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

   if 5.1999999999999999e-286 < b

   1. Initial program 33.3%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. +-lft-identity N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

      15. +-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

      16. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

      17. mul0-lft N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

      18. accelerator-lowering-fma.f64 33.3

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

   4. Applied egg-rr 33.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 33.4

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

   6. Applied egg-rr 33.4%

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

   7. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 69.4

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

   9. Simplified 69.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-286}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.3% 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 53.1%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. metadata-eval N/A

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

   14. +-lft-identity N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(0 + b\right)}\right)} - b}{3 \cdot a} \]

   15. +-commutative N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot \color{blue}{\left(b + 0\right)}\right)} - b}{3 \cdot a} \]

   16. distribute-rgt-out N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{b \cdot b + 0 \cdot b}\right)} - b}{3 \cdot a} \]

   17. mul0-lft N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b + \color{blue}{0}\right)} - b}{3 \cdot a} \]

   18. accelerator-lowering-fma.f64 53.0

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}{3 \cdot a} \]

4. Applied egg-rr 53.0%

   \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}{3 \cdot a} \]
5. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. +-commutative N/A

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

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

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

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

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

   5. *-lowering-*.f64 53.1

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

6. Applied egg-rr 53.1%

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

7. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 36.1

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

9. Simplified 36.1%

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

Alternative 13: 2.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \cdot 0.6666666666666666 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ b a) 0.6666666666666666))

C

double code(double a, double b, double c) {
	return (b / a) * 0.6666666666666666;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (b / a) * 0.6666666666666666d0
end function

Java

public static double code(double a, double b, double c) {
	return (b / a) * 0.6666666666666666;
}

Python

def code(a, b, c):
	return (b / a) * 0.6666666666666666

Julia

function code(a, b, c)
	return Float64(Float64(b / a) * 0.6666666666666666)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (b / a) * 0.6666666666666666;
end

Wolfram

code[a_, b_, c_] := N[(N[(b / a), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in b around -inf

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

   1. associate-*r/ N/A

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

   2. /-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 32.9

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

5. Simplified 32.9%

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

   1. frac-2neg N/A

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

7. Applied egg-rr 2.8%

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

8. Final simplification 2.8%

   \[\leadsto \frac{b}{a} \cdot 0.6666666666666666 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))