Cubic critical, medium range

Percentage Accurate: 31.7% → 99.3%

Time: 13.0s

Alternatives: 8

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: 31.7% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 29.9%

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

6. Applied rewrites 30.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} + b\right)}{\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\left(a \cdot a\right) \cdot 9} - \frac{b \cdot b}{\left(a \cdot a\right) \cdot 9}}}} \]

7. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 99.2

      \[\leadsto \frac{1}{\frac{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} + b\right)}{\color{blue}{\frac{c}{a}} \cdot -0.3333333333333333}} \]

9. Applied rewrites 99.2%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. clear-num N/A

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

    4. lift-*.f64 N/A

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

    5. associate-/r* N/A

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

11. Applied rewrites 99.4%

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

Alternative 2: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{0.3333333333333333 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{a \cdot \frac{c \cdot -0.3333333333333333}{a}}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (/
   (* 0.3333333333333333 (+ b (sqrt (fma a (* c -3.0) (* b b)))))
   (* a (/ (* c -0.3333333333333333) a)))))

C

double code(double a, double b, double c) {
	return 1.0 / ((0.3333333333333333 * (b + sqrt(fma(a, (c * -3.0), (b * b))))) / (a * ((c * -0.3333333333333333) / a)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(0.3333333333333333 * Float64(b + sqrt(fma(a, Float64(c * -3.0), Float64(b * b))))) / Float64(a * Float64(Float64(c * -0.3333333333333333) / a))))
end

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 31.3%

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

6. Applied rewrites 32.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} + b\right)}{\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\left(a \cdot a\right) \cdot 9} - \frac{b \cdot b}{\left(a \cdot a\right) \cdot 9}}}} \]

7. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 99.1

      \[\leadsto \frac{1}{\frac{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} + b\right)}{\color{blue}{\frac{c}{a}} \cdot -0.3333333333333333}} \]

9. Applied rewrites 99.1%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. associate-*l/ N/A

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

    5. associate-/l/ N/A

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

    6. lower-/.f64 N/A

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

11. Applied rewrites 99.2%

    \[\leadsto \frac{1}{\color{blue}{\frac{0.3333333333333333 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\frac{c \cdot -0.3333333333333333}{a} \cdot a}}} \]

12. Final simplification 99.2%

    \[\leadsto \frac{1}{\frac{0.3333333333333333 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{a \cdot \frac{c \cdot -0.3333333333333333}{a}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(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)))