Cubic critical, wide range

Percentage Accurate: 17.6% → 99.9%

Time: 13.9s

Alternatives: 3

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: 17.6% 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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 14.6%

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

4. Applied rewrites 15.0%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. sub-div N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

6. Applied rewrites 15.1%

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

7. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.9

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

9. Applied rewrites 99.9%

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

Alternative 2: 90.5% 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 14.6%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 92.9

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

5. Applied rewrites 92.9%

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

Alternative 3: 3.3% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 14.6%

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

4. Applied rewrites 15.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

6. Applied rewrites 16.0%

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

7. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt-out N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \left(b \cdot \color{blue}{0}\right)}{a} \]

   4. mul0-rgt N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \color{blue}{0}}{a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   6. lower-/.f64 3.3

      \[\leadsto \color{blue}{\frac{0}{a}} \]

9. Applied rewrites 3.3%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

10. Taylor expanded in a around 0

    \[\leadsto 0 \]
11. Step-by-step derivation

12. Applied rewrites 3.3%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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)))