Cubic critical

Percentage Accurate: 52.0% → 85.4%

Time: 10.4s

Alternatives: 12

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.6e+100)
   (/ (/ b a) -1.5)
   (if (<= b 9.4e-88)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.6e+100) {
		tmp = (b / a) / -1.5;
	} else if (b <= 9.4e-88) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.6e+100)
		tmp = Float64(Float64(b / a) / -1.5);
	elseif (b <= 9.4e-88)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.6e+100], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], If[LessEqual[b, 9.4e-88], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.60000000000000046e100

   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. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 92.2

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

   7. Applied rewrites 92.2%

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

      1. clear-num N/A

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

      2. associate-*l/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 92.2

         \[\leadsto \frac{1}{a} \cdot \frac{b}{\color{blue}{-1.5}} \]

   9. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b}{-1.5}} \]
   10. Step-by-step derivation

       1. frac-times N/A

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

       2. *-lft-identity N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 92.4

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

   11. Applied rewrites 92.4%

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

   if -9.60000000000000046e100 < b < 9.4e-88

   1. Initial program 80.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 80.3

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

   4. Applied rewrites 80.3%

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

   if 9.4e-88 < b

   1. Initial program 18.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+128)
   (/ (/ b a) -1.5)
   (if (<= b 9.4e-88)
     (/ (* (- (sqrt (fma (* a -3.0) c (* b b))) b) 0.3333333333333333) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+128) {
		tmp = (b / a) / -1.5;
	} else if (b <= 9.4e-88) {
		tmp = ((sqrt(fma((a * -3.0), c, (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+128)
		tmp = Float64(Float64(b / a) / -1.5);
	elseif (b <= 9.4e-88)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+128], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], If[LessEqual[b, 9.4e-88], N[(N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.59999999999999993e128

   1. Initial program 43.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.1

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

   5. Applied rewrites 93.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.0

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

   7. Applied rewrites 93.0%

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

      1. clear-num N/A

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

      2. associate-*l/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 93.1

         \[\leadsto \frac{1}{a} \cdot \frac{b}{\color{blue}{-1.5}} \]

   9. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b}{-1.5}} \]
   10. Step-by-step derivation

       1. frac-times N/A

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

       2. *-lft-identity N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 93.3

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

   11. Applied rewrites 93.3%

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

   if -1.59999999999999993e128 < b < 9.4e-88

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift-neg.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 80.7

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

   8. Applied rewrites 80.7%

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

   if 9.4e-88 < b

   1. Initial program 18.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+128)
   (/ (/ b a) -1.5)
   (if (<= b 9.4e-88)
     (* (/ -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 <= -1.6e+128) {
		tmp = (b / a) / -1.5;
	} else if (b <= 9.4e-88) {
		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 <= -1.6e+128)
		tmp = Float64(Float64(b / a) / -1.5);
	elseif (b <= 9.4e-88)
		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, -1.6e+128], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], If[LessEqual[b, 9.4e-88], 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 < -1.59999999999999993e128

   1. Initial program 43.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.1

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

   5. Applied rewrites 93.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.0

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

   7. Applied rewrites 93.0%

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

      1. clear-num N/A

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

      2. associate-*l/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 93.1

         \[\leadsto \frac{1}{a} \cdot \frac{b}{\color{blue}{-1.5}} \]

   9. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b}{-1.5}} \]
   10. Step-by-step derivation

       1. frac-times N/A

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

       2. *-lft-identity N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 93.3

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

   11. Applied rewrites 93.3%

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

   if -1.59999999999999993e128 < b < 9.4e-88

   1. Initial program 80.8%

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

   4. Applied rewrites 80.7%

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

   if 9.4e-88 < b

   1. Initial program 18.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-86)
   (fma c (/ 0.5 b) (/ (* b -0.6666666666666666) a))
   (if (<= b 9.4e-88)
     (/ (- (sqrt (* a (* -3.0 c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-86) {
		tmp = fma(c, (0.5 / b), ((b * -0.6666666666666666) / a));
	} else if (b <= 9.4e-88) {
		tmp = (sqrt((a * (-3.0 * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-86)
		tmp = fma(c, Float64(0.5 / b), Float64(Float64(b * -0.6666666666666666) / a));
	elseif (b <= 9.4e-88)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-3.0 * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-86], N[(c * N[(0.5 / b), $MachinePrecision] + N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.4e-88], N[(N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.34999999999999996e-86

   1. Initial program 68.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 80.2

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

   8. Applied rewrites 80.2%

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

   if -1.34999999999999996e-86 < b < 9.4e-88

   1. Initial program 74.1%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if 9.4e-88 < b

   1. Initial program 18.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-86)
   (fma c (/ 0.5 b) (/ (* b -0.6666666666666666) a))
   (if (<= b 9.4e-88)
     (/ (* 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 <= -1.35e-86) {
		tmp = fma(c, (0.5 / b), ((b * -0.6666666666666666) / a));
	} else if (b <= 9.4e-88) {
		tmp = (0.3333333333333333 * (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 <= -1.35e-86)
		tmp = fma(c, Float64(0.5 / b), Float64(Float64(b * -0.6666666666666666) / a));
	elseif (b <= 9.4e-88)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b)) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-86], N[(c * N[(0.5 / b), $MachinePrecision] + N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.4e-88], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(-3.0 * N[(a * 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 < -1.34999999999999996e-86

   1. Initial program 68.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 80.2

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

   8. Applied rewrites 80.2%

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

   if -1.34999999999999996e-86 < b < 9.4e-88

   1. Initial program 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 74.1

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

   4. Applied rewrites 74.1%

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

      1. lift-neg.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 73.7%

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

   7. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

   if 9.4e-88 < b

   1. Initial program 18.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{-3 \cdot \left(a \cdot c\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 69.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 59.1

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

   8. Applied rewrites 59.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 32.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

Alternative 7: 68.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-286}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.25e-286) (/ (/ b a) -1.5) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.25e-286) {
		tmp = (b / a) / -1.5;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.25d-286) then
        tmp = (b / a) / (-1.5d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.25e-286) {
		tmp = (b / a) / -1.5;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.25e-286:
		tmp = (b / a) / -1.5
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.25e-286)
		tmp = Float64(Float64(b / a) / -1.5);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.25e-286)
		tmp = (b / a) / -1.5;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.25e-286], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25000000000000009e-286

   1. Initial program 69.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   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. lower-*.f64 N/A

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

      4. lower-/.f64 57.3

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

   7. Applied rewrites 57.3%

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

      1. clear-num N/A

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

      2. associate-*l/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 57.3

         \[\leadsto \frac{1}{a} \cdot \frac{b}{\color{blue}{-1.5}} \]

   9. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b}{-1.5}} \]
   10. Step-by-step derivation

       1. frac-times N/A

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

       2. *-lft-identity N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 57.4

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

   11. Applied rewrites 57.4%

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

   if 1.25000000000000009e-286 < b

   1. Initial program 32.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

Alternative 8: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-286}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.25e-286) (* (/ b a) -0.6666666666666666) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.25e-286) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 <= 1.25d-286) then
        tmp = (b / a) * (-0.6666666666666666d0)
    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 <= 1.25e-286) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.25e-286:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.25e-286)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	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 <= 1.25e-286)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.25e-286], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25000000000000009e-286

   1. Initial program 69.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if 1.25000000000000009e-286 < b

   1. Initial program 32.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

Alternative 9: 42.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15000000000000:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15000000000000.0) (* (/ b a) -0.6666666666666666) (* c (/ 0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15000000000000.0) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 <= 15000000000000.0d0) then
        tmp = (b / a) * (-0.6666666666666666d0)
    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 <= 15000000000000.0) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 15000000000000.0:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = c * (0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15000000000000.0)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(c * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 15000000000000.0)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = c * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15000000000000.0], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(c * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e13

   1. Initial program 67.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f64 42.9

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

   7. Applied rewrites 42.9%

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

   if 1.5e13 < b

   1. Initial program 12.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 2.2

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

   8. Applied rewrites 2.2%

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

   9. Taylor expanded in b around 0

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

       1. lower-/.f64 26.3

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

   11. Applied rewrites 26.3%

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

Alternative 10: 42.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15000000000000:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15000000000000.0) (* b (/ -0.6666666666666666 a)) (* c (/ 0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15000000000000.0) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 15000000000000.0d0) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * (0.5d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 15000000000000.0) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 15000000000000.0:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15000000000000.0)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 15000000000000.0)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15000000000000.0], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e13

   1. Initial program 67.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   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. lower-*.f64 N/A

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

      4. lower-/.f64 42.9

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

   7. Applied rewrites 42.9%

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

   if 1.5e13 < b

   1. Initial program 12.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 2.2

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

   8. Applied rewrites 2.2%

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

   9. Taylor expanded in b around 0

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

       1. lower-/.f64 26.3

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

   11. Applied rewrites 26.3%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15000000000000:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 11.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. lower-neg.f64 29.9

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

5. Applied rewrites 29.9%

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

6. Taylor expanded in c around inf

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 23.3

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

8. Applied rewrites 23.3%

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

9. Taylor expanded in b around 0

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

    1. lower-/.f64 10.2

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

11. Applied rewrites 10.2%

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

Alternative 12: 2.6% 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 50.9%

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

4. Applied rewrites 33.7%

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

5. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 2.3

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

7. Applied rewrites 2.3%

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

8. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 2.8

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

10. Applied rewrites 2.8%

    \[\leadsto \color{blue}{b \cdot \frac{0.6666666666666666}{a}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))