Cubic critical

Percentage Accurate: 51.6% → 87.4%

Time: 11.1s

Alternatives: 13

Speedup: 2.2×

• could not determine a ground truth

  1. a = 7.399442277169315e-303
  2. b = 1.9188885890636807e+50
  3. c = -3.903447639198522e-158

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: 51.6% 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: 87.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{if}\;b \leq -1.96 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{a}, b, \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} \cdot \frac{-1}{a}\right)}{-3}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{b \cdot b - b \cdot b}{t\_0} - \frac{c \cdot \left(a \cdot 3\right)}{t\_0}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ b (sqrt (fma b b (* -3.0 (* a c)))))))
   (if (<= b -1.96e+80)
     (/ b (* a -1.5))
     (if (<= b 1.65e-137)
       (/
        (fma (/ 1.0 a) b (* (sqrt (fma a (* -3.0 c) (* b b))) (/ -1.0 a)))
        -3.0)
       (if (<= b 3.4e+39)
         (/ (- (/ (- (* b b) (* b b)) t_0) (/ (* c (* a 3.0)) t_0)) (* a 3.0))
         (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double t_0 = b + sqrt(fma(b, b, (-3.0 * (a * c))));
	double tmp;
	if (b <= -1.96e+80) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.65e-137) {
		tmp = fma((1.0 / a), b, (sqrt(fma(a, (-3.0 * c), (b * b))) * (-1.0 / a))) / -3.0;
	} else if (b <= 3.4e+39) {
		tmp = ((((b * b) - (b * b)) / t_0) - ((c * (a * 3.0)) / t_0)) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b + sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))))
	tmp = 0.0
	if (b <= -1.96e+80)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.65e-137)
		tmp = Float64(fma(Float64(1.0 / a), b, Float64(sqrt(fma(a, Float64(-3.0 * c), Float64(b * b))) * Float64(-1.0 / a))) / -3.0);
	elseif (b <= 3.4e+39)
		tmp = Float64(Float64(Float64(Float64(Float64(b * b) - Float64(b * b)) / t_0) - Float64(Float64(c * Float64(a * 3.0)) / t_0)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b + N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.96e+80], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.65e-137], N[(N[(N[(1.0 / a), $MachinePrecision] * b + N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 3.4e+39], N[(N[(N[(N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.9599999999999999e80

   1. Initial program 46.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 96.7

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

   5. Simplified 96.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 96.7

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

   7. Applied egg-rr 96.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 97.0

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

   9. Applied egg-rr 97.0%

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

   if -1.9599999999999999e80 < b < 1.6500000000000001e-137

   1. Initial program 85.8%

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

   4. Applied egg-rr 85.8%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 85.9

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

   6. Applied egg-rr 85.9%

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

   if 1.6500000000000001e-137 < b < 3.3999999999999999e39

   1. Initial program 50.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 50.1

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

   4. Applied egg-rr 50.1%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. flip-+ N/A

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

      4. rem-square-sqrt N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. sqr-neg N/A

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

      13. associate--r+ N/A

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

   6. Applied egg-rr 86.5%

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

   if 3.3999999999999999e39 < b

   1. Initial program 14.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 97.3

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

   5. Simplified 97.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.96 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{a}, b, \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} \cdot \frac{-1}{a}\right)}{-3}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{b \cdot b - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}} - \frac{c \cdot \left(a \cdot 3\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.75 \cdot 10^{+79}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{a}, b, \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} \cdot \frac{-1}{a}\right)}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.75e+79)
   (/ b (* a -1.5))
   (if (<= b 6.5e-53)
     (/
      (fma (/ 1.0 a) b (* (sqrt (fma a (* -3.0 c) (* b b))) (/ -1.0 a)))
      -3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.75e+79) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.5e-53) {
		tmp = fma((1.0 / a), b, (sqrt(fma(a, (-3.0 * c), (b * b))) * (-1.0 / a))) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.75e+79)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 6.5e-53)
		tmp = Float64(fma(Float64(1.0 / a), b, Float64(sqrt(fma(a, Float64(-3.0 * c), Float64(b * b))) * Float64(-1.0 / a))) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.75e+79], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-53], N[(N[(N[(1.0 / a), $MachinePrecision] * b + N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.75000000000000003e79

   1. Initial program 46.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 96.7

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

   5. Simplified 96.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 96.7

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

   7. Applied egg-rr 96.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 97.0

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

   9. Applied egg-rr 97.0%

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

   if -2.75000000000000003e79 < b < 6.4999999999999997e-53

   1. Initial program 83.0%

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

   4. Applied egg-rr 83.1%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 83.3

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

   6. Applied egg-rr 83.3%

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

   if 6.4999999999999997e-53 < b

   1. Initial program 16.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-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 93.1

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

   5. Simplified 93.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.75 \cdot 10^{+79}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{a}, b, \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} \cdot \frac{-1}{a}\right)}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+80)
   (/ b (* a -1.5))
   (if (<= b 1.05e-52)
     (/ (/ (- (sqrt (fma b b (* -3.0 (* a c)))) b) a) 3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+80) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.05e-52) {
		tmp = ((sqrt(fma(b, b, (-3.0 * (a * c)))) - b) / a) / 3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+80)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.05e-52)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))) - b) / a) / 3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+80], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-52], N[(N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.20000000000000003e80

   1. Initial program 46.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 96.7

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

   5. Simplified 96.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 96.7

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

   7. Applied egg-rr 96.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 97.0

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

   9. Applied egg-rr 97.0%

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

   if -2.20000000000000003e80 < b < 1.0499999999999999e-52

   1. Initial program 83.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 83.0

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

   4. Applied egg-rr 83.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

   6. Applied egg-rr 83.1%

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

   if 1.0499999999999999e-52 < b

   1. Initial program 16.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-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 93.1

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

   5. Simplified 93.1%

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

Alternative 4: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+77)
   (/ b (* a -1.5))
   (if (<= b 1.65e-51)
     (* (/ (- (sqrt (fma b b (* -3.0 (* a c)))) b) a) 0.3333333333333333)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+77) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.65e-51) {
		tmp = ((sqrt(fma(b, b, (-3.0 * (a * c)))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+77)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.65e-51)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+77], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.65e-51], N[(N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8e77

   1. Initial program 47.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 96.7

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

   5. Simplified 96.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 96.8

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

   7. Applied egg-rr 96.8%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 97.0

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

   9. Applied egg-rr 97.0%

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

   if -2.8e77 < b < 1.64999999999999986e-51

   1. Initial program 82.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 82.9

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

   4. Applied egg-rr 82.9%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. div-inv N/A

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

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

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

   6. Applied egg-rr 82.9%

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

   if 1.64999999999999986e-51 < b

   1. Initial program 16.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-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 93.1

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

   5. Simplified 93.1%

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

Alternative 5: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.66 \cdot 10^{-51}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+80)
   (/ b (* a -1.5))
   (if (<= b 1.66e-51)
     (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* -3.0 c) (* b b)))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+80) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.66e-51) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (-3.0 * c), (b * b))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+80)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.66e-51)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b)))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+80], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.66e-51], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.20000000000000003e80

   1. Initial program 46.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 96.7

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

   5. Simplified 96.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 96.7

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

   7. Applied egg-rr 96.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 97.0

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

   9. Applied egg-rr 97.0%

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

   if -2.20000000000000003e80 < b < 1.6600000000000001e-51

   1. Initial program 83.0%

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

   4. Applied egg-rr 83.1%

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

   if 1.6600000000000001e-51 < b

   1. Initial program 16.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-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 93.1

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

   5. Simplified 93.1%

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

Alternative 6: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\left(-b\right) \cdot \mathsf{fma}\left(-0.5, \frac{c}{b \cdot b}, \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-91)
   (* (- b) (fma -0.5 (/ c (* b b)) (/ 0.6666666666666666 a)))
   (if (<= b 7.8e-54)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-91) {
		tmp = -b * fma(-0.5, (c / (b * b)), (0.6666666666666666 / a));
	} else if (b <= 7.8e-54) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-91)
		tmp = Float64(Float64(-b) * fma(-0.5, Float64(c / Float64(b * b)), Float64(0.6666666666666666 / a)));
	elseif (b <= 7.8e-54)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - 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, -2.8e-91], N[((-b) * N[(-0.5 * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.8e-54], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $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 < -2.8e-91

   1. Initial program 63.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 64.0

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

   4. Applied egg-rr 64.0%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   6. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. neg-lowering-neg.f64 89.6

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

   7. Simplified 89.6%

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

   if -2.8e-91 < b < 7.8e-54

   1. Initial program 76.4%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 72.9

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

   5. Simplified 72.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 73.0

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

   7. Applied egg-rr 73.0%

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

   if 7.8e-54 < b

   1. Initial program 16.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-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 93.1

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

   5. Simplified 93.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\left(-b\right) \cdot \mathsf{fma}\left(-0.5, \frac{c}{b \cdot b}, \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-93}:\\ \;\;\;\;\left(-b\right) \cdot \mathsf{fma}\left(-0.5, \frac{c}{b \cdot b}, \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{a \cdot \left(-3 \cdot c\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-93)
   (* (- b) (fma -0.5 (/ c (* b b)) (/ 0.6666666666666666 a)))
   (if (<= b 8.6e-54)
     (/ (* 0.3333333333333333 (- (sqrt (* a (* -3.0 c))) b)) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-93) {
		tmp = -b * fma(-0.5, (c / (b * b)), (0.6666666666666666 / a));
	} else if (b <= 8.6e-54) {
		tmp = (0.3333333333333333 * (sqrt((a * (-3.0 * c))) - b)) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-93)
		tmp = Float64(Float64(-b) * fma(-0.5, Float64(c / Float64(b * b)), Float64(0.6666666666666666 / a)));
	elseif (b <= 8.6e-54)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(Float64(a * Float64(-3.0 * c))) - b)) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-93], N[((-b) * N[(-0.5 * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-54], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.49999999999999968e-93

   1. Initial program 63.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 64.0

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

   4. Applied egg-rr 64.0%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   6. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. neg-lowering-neg.f64 89.6

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

   7. Simplified 89.6%

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

   if -5.49999999999999968e-93 < b < 8.5999999999999999e-54

   1. Initial program 76.4%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 72.9

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

   5. Simplified 72.9%

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

      1. frac-2neg N/A

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

      2. neg-mul-1 N/A

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

      3. *-commutative N/A

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

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

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

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

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

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

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

      10. div-inv N/A

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

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

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

   7. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right) \cdot 0.3333333333333333\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 72.9

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

   9. Applied egg-rr 72.9%

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

   if 8.5999999999999999e-54 < b

   1. Initial program 16.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-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 93.1

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

   5. Simplified 93.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-93}:\\ \;\;\;\;\left(-b\right) \cdot \mathsf{fma}\left(-0.5, \frac{c}{b \cdot b}, \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{a \cdot \left(-3 \cdot c\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-91}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{a \cdot \left(-3 \cdot c\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e-91)
   (/ b (* a -1.5))
   (if (<= b 5.3e-49)
     (/ (* 0.3333333333333333 (- (sqrt (* a (* -3.0 c))) b)) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e-91) {
		tmp = b / (a * -1.5);
	} else if (b <= 5.3e-49) {
		tmp = (0.3333333333333333 * (sqrt((a * (-3.0 * c))) - 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 <= (-1.75d-91)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 5.3d-49) then
        tmp = (0.3333333333333333d0 * (sqrt((a * ((-3.0d0) * c))) - 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 <= -1.75e-91) {
		tmp = b / (a * -1.5);
	} else if (b <= 5.3e-49) {
		tmp = (0.3333333333333333 * (Math.sqrt((a * (-3.0 * c))) - b)) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.75e-91:
		tmp = b / (a * -1.5)
	elif b <= 5.3e-49:
		tmp = (0.3333333333333333 * (math.sqrt((a * (-3.0 * c))) - b)) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.75e-91)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 5.3e-49)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(Float64(a * Float64(-3.0 * c))) - b)) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.75e-91)
		tmp = b / (a * -1.5);
	elseif (b <= 5.3e-49)
		tmp = (0.3333333333333333 * (sqrt((a * (-3.0 * c))) - b)) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.75e-91], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.3e-49], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7499999999999999e-91

   1. Initial program 63.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 89.2

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

   5. Simplified 89.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 89.2

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

   7. Applied egg-rr 89.2%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 89.3

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

   9. Applied egg-rr 89.3%

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

   if -1.7499999999999999e-91 < b < 5.3000000000000003e-49

   1. Initial program 76.4%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 72.9

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

   5. Simplified 72.9%

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

      1. frac-2neg N/A

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

      2. neg-mul-1 N/A

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

      3. *-commutative N/A

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

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

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

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

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

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

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

      10. div-inv N/A

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

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

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

   7. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right) \cdot 0.3333333333333333\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 72.9

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

   9. Applied egg-rr 72.9%

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

   if 5.3000000000000003e-49 < b

   1. Initial program 16.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-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 93.1

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

   5. Simplified 93.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-91}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{a \cdot \left(-3 \cdot c\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-94}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-54}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e-94)
   (/ b (* a -1.5))
   (if (<= b 6.2e-54)
     (* 0.3333333333333333 (/ (- (sqrt (* -3.0 (* a c))) b) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e-94) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.2e-54) {
		tmp = 0.3333333333333333 * ((sqrt((-3.0 * (a * c))) - 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 <= (-7d-94)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 6.2d-54) then
        tmp = 0.3333333333333333d0 * ((sqrt(((-3.0d0) * (a * c))) - 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 <= -7e-94) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.2e-54) {
		tmp = 0.3333333333333333 * ((Math.sqrt((-3.0 * (a * c))) - b) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7e-94:
		tmp = b / (a * -1.5)
	elif b <= 6.2e-54:
		tmp = 0.3333333333333333 * ((math.sqrt((-3.0 * (a * c))) - b) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e-94)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 6.2e-54)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7e-94)
		tmp = b / (a * -1.5);
	elseif (b <= 6.2e-54)
		tmp = 0.3333333333333333 * ((sqrt((-3.0 * (a * c))) - b) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e-94], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-54], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.99999999999999996e-94

   1. Initial program 63.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 89.2

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

   5. Simplified 89.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 89.2

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

   7. Applied egg-rr 89.2%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 89.3

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

   9. Applied egg-rr 89.3%

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

   if -6.99999999999999996e-94 < b < 6.20000000000000008e-54

   1. Initial program 76.4%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 72.9

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

   5. Simplified 72.9%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

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

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

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

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

      16. metadata-eval 72.9

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

   7. Applied egg-rr 72.9%

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

   if 6.20000000000000008e-54 < b

   1. Initial program 16.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-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 93.1

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

   5. Simplified 93.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-94}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-54}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-53}:\\ \;\;\;\;\left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-91)
   (/ b (* a -1.5))
   (if (<= b 2.95e-53)
     (* (- (sqrt (* -3.0 (* a c))) b) (/ 0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-91) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.95e-53) {
		tmp = (sqrt((-3.0 * (a * c))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.8d-91)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.95d-53) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-91) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.95e-53) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e-91:
		tmp = b / (a * -1.5)
	elif b <= 2.95e-53:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) * (0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-91)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.95e-53)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e-91)
		tmp = b / (a * -1.5);
	elseif (b <= 2.95e-53)
		tmp = (sqrt((-3.0 * (a * c))) - b) * (0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-91], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.95e-53], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $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 < -2.8e-91

   1. Initial program 63.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 89.2

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

   5. Simplified 89.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 89.2

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

   7. Applied egg-rr 89.2%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 89.3

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

   9. Applied egg-rr 89.3%

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

   if -2.8e-91 < b < 2.95e-53

   1. Initial program 76.4%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 72.9

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

   5. Simplified 72.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. unsub-neg N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

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

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

      16. *-lowering-*.f64 72.8

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

   7. Applied egg-rr 72.8%

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

   if 2.95e-53 < b

   1. Initial program 16.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-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 93.1

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

   5. Simplified 93.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-53}:\\ \;\;\;\;\left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-309) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-309)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-309:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-309)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-309)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-309], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.000000000000002e-309

   1. Initial program 66.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 75.0

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

   5. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   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 75.0

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

   7. Applied egg-rr 75.0%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 75.1

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

   9. Applied egg-rr 75.1%

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

   if -1.000000000000002e-309 < b

   1. Initial program 34.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 71.4

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

   5. Simplified 71.4%

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

Alternative 12: 35.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a \cdot -1.5} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b (* a -1.5)))

C

double code(double a, double b, double c) {
	return b / (a * -1.5);
}

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 * (-1.5d0))
end function

Java

public static double code(double a, double b, double c) {
	return b / (a * -1.5);
}

Python

def code(a, b, c):
	return b / (a * -1.5)

Julia

function code(a, b, c)
	return Float64(b / Float64(a * -1.5))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / (a * -1.5);
end

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in b around -inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 38.9

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

5. Simplified 38.9%

   \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
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 38.9

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

7. Applied egg-rr 38.9%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. *-lowering-*.f64 38.9

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

9. Applied egg-rr 38.9%

   \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]
10. Add Preprocessing

Alternative 13: 35.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in b around -inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 38.9

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

5. Simplified 38.9%

   \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
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 38.9

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

7. Applied egg-rr 38.9%

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

8. Final simplification 38.9%

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

Reproduce

herbie shell --seed 2024205 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))