Cubic critical, medium range

Percentage Accurate: 31.8% → 99.7%

Time: 10.5s

Alternatives: 6

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.8% 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: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 34.6%

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

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 34.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 34.5

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 35.7%

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

7. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.8

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

9. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 34.6%

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

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 34.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 34.5

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 35.7%

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

7. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.8

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

9. Applied rewrites 99.8%

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

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-*.f64 99.8

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

11. Applied rewrites 99.8%

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

Alternative 3: 90.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 34.6%

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

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 34.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 34.5

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 35.7%

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

7. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.8

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in c around 0

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

    1. *-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. associate-/l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-*.f64 90.7

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

12. Applied rewrites 90.7%

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

13. Final simplification 90.7%

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

Alternative 4: 90.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 34.6%

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

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 34.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 34.5

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 35.7%

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

7. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.8

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 90.7

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

12. Applied rewrites 90.7%

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

13. Final simplification 90.7%

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

Alternative 5: 81.0% 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 34.6%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 79.8

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

5. Applied rewrites 79.8%

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

6. Final simplification 79.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} \]
7. Add Preprocessing

Alternative 6: 80.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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) / b) * c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.6%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

5. Applied rewrites 93.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 79.6%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024284 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))