Cubic critical, narrow range

Percentage Accurate: 54.9% → 99.3%

Time: 7.7s

Alternatives: 6

Speedup: 2.9×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 54.9% 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: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 54.3

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 54.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 54.3

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

4. Applied rewrites 54.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. clear-num N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. associate-/l/ N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 55.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. +-inverses N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. metadata-eval N/A

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

   16. lower-*.f64 99.4

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

    \[\leadsto \frac{-3 \cdot a}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} + b\right) \cdot \left(3 \cdot a\right)} \cdot c \]
12. Add Preprocessing

Alternative 2: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0057:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0057)
   (* (- (sqrt (fma b b (* (* -3.0 a) c))) b) (/ 0.3333333333333333 a))
   (/ 1.0 (/ (fma 1.5 (* (/ c b) a) (* -2.0 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0057) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / (fma(1.5, ((c / b) * a), (-2.0 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0057)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(1.0 / Float64(fma(1.5, Float64(Float64(c / b) * a), Float64(-2.0 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0057], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0057000000000000002

   1. Initial program 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 79.6%

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

   if -0.0057000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 42.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 42.6

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 90.5

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(1.5, a \cdot \frac{c}{b}, \color{blue}{-2 \cdot b}\right)}{c}} \]

   7. Applied rewrites 90.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(1.5, a \cdot \frac{c}{b}, -2 \cdot b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0057:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0057:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0057)
   (* (- (sqrt (fma b b (* (* -3.0 a) c))) b) (/ 0.3333333333333333 a))
   (/ 1.0 (fma 1.5 (/ a b) (* (/ b c) -2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0057) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / fma(1.5, (a / b), ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0057)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(1.0 / fma(1.5, Float64(a / b), Float64(Float64(b / c) * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0057], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.5 * N[(a / b), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0057000000000000002

   1. Initial program 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 79.6%

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

   if -0.0057000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 42.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 42.6

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.5

         \[\leadsto \frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \color{blue}{\frac{b}{c}}\right)} \]

   7. Applied rewrites 90.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0057:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 54.3

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 54.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 54.3

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

4. Applied rewrites 54.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. clear-num N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. associate-/l/ N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 55.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. +-inverses N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. metadata-eval N/A

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

   16. lower-*.f64 99.4

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-rgt-identity N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.2

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.2

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

10. Applied rewrites 99.2%

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

11. Final simplification 99.2%

    \[\leadsto \frac{a \cdot c}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} + b\right) \cdot \left(3 \cdot a\right)} \cdot -3 \]
12. Add Preprocessing

Alternative 5: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{b}{c} \cdot -2\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma 1.5 (/ a b) (* (/ b c) -2.0))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(1.5, (a / b), ((b / c) * -2.0));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(1.5, Float64(a / b), Float64(Float64(b / c) * -2.0)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(1.5 * N[(a / b), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 54.3

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 54.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 54.3

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

4. Applied rewrites 54.3%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 82.0

      \[\leadsto \frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \color{blue}{\frac{b}{c}}\right)} \]

7. Applied rewrites 82.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}} \]

8. Final simplification 82.0%

   \[\leadsto \frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{b}{c} \cdot -2\right)} \]
9. Add Preprocessing

Alternative 6: 64.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -0.5 * (c / 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 = (-0.5d0) * (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.9

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

5. Applied rewrites 64.9%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))