Cubic critical, wide range

Percentage Accurate: 17.9% → 99.4%

Time: 13.9s

Alternatives: 6

Speedup: 2.9×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.9% 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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in c around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 17.7

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

5. Applied rewrites 17.7%

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

7. Applied rewrites 18.0%

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

8. Taylor expanded in b around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.2

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

10. Applied rewrites 99.2%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. associate-/r* N/A

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

12. Applied rewrites 99.4%

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

13. Final simplification 99.4%

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

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(3 \cdot a\right)}{a \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ (* c (* 3.0 a)) (* a (- (- b) (sqrt (fma c (* a -3.0) (* b b))))))
  0.3333333333333333))

C

double code(double a, double b, double c) {
	return ((c * (3.0 * a)) / (a * (-b - sqrt(fma(c, (a * -3.0), (b * b)))))) * 0.3333333333333333;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(3.0 * a)) / Float64(a * Float64(Float64(-b) - sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))))) * 0.3333333333333333)
end

Wolfram

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

Derivation

1. Initial program 17.9%

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

3. Taylor expanded in c around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 17.9

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

5. Applied rewrites 17.9%

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

7. Applied rewrites 18.3%

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

8. Taylor expanded in b around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.2

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

10. Applied rewrites 99.2%

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

12. Applied rewrites 99.1%

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

13. Final simplification 99.1%

    \[\leadsto \frac{c \cdot \left(3 \cdot a\right)}{a \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \cdot 0.3333333333333333 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024225 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))