Cubic critical, wide range

Percentage Accurate: 17.7% → 99.4%

Time: 7.4s

Alternatives: 4

Speedup: 23.2×

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: 17.7% 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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ -\frac{\frac{t_0}{\left(-b\right) - \sqrt{t_0 + b \cdot b}}}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -3.0))))
   (- (/ (/ t_0 (- (- b) (sqrt (+ t_0 (* b b))))) (* a 3.0)))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	return -((t_0 / (-b - sqrt((t_0 + (b * b))))) / (a * 3.0));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = c * (a * (-3.0d0))
    code = -((t_0 / (-b - sqrt((t_0 + (b * b))))) / (a * 3.0d0))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	return -((t_0 / (-b - Math.sqrt((t_0 + (b * b))))) / (a * 3.0));
}

Python

def code(a, b, c):
	t_0 = c * (a * -3.0)
	return -((t_0 / (-b - math.sqrt((t_0 + (b * b))))) / (a * 3.0))

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * -3.0))
	return Float64(-Float64(Float64(t_0 / Float64(Float64(-b) - sqrt(Float64(t_0 + Float64(b * b))))) / Float64(a * 3.0)))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (a * -3.0);
	tmp = -((t_0 / (-b - sqrt((t_0 + (b * b))))) / (a * 3.0));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, (-N[(N[(t$95$0 / N[((-b) - N[Sqrt[N[(t$95$0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Initial program 17.8%

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

2. Taylor expanded in a around 0 17.7%

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

   1. *-commutative 17.7%

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

   2. *-commutative 17.7%

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

   3. associate-*l* 17.8%

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

4. Simplified 17.8%

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

   1. flip-+ 17.8%

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

   2. add-sqr-sqrt 18.2%

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

   3. associate-*r* 18.3%

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

   4. *-commutative 18.3%

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

   5. associate-*r* 18.3%

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

   6. *-commutative 18.3%

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

6. Applied egg-rr 18.3%

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

   1. sqr-neg 18.3%

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

   2. sub-neg 18.3%

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

   3. associate--r+ 99.2%

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

   4. +-inverses 99.2%

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

   5. distribute-rgt-neg-in 99.2%

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

   6. metadata-eval 99.2%

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

   7. associate-*l* 99.5%

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

   8. sub-neg 99.5%

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

   9. +-commutative 99.5%

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

   10. sqr-neg 99.5%

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

   11. distribute-rgt-neg-out 99.5%

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

   12. unsub-neg 99.5%

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

   13. distribute-rgt-neg-in 99.5%

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

   14. metadata-eval 99.5%

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

   15. associate-*l* 99.5%

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

   16. fma-neg 99.5%

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

   17. distribute-rgt-neg-out 99.5%

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

   18. sqr-neg 99.5%

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

8. Simplified 99.5%

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

   1. fma-udef 99.5%

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

10. Applied egg-rr 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-3.0d0) * (c * -a)) / (-b - sqrt(((c * (a * (-3.0d0))) + (b * b))))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((-3.0 * (c * -a)) / (-b - Math.sqrt(((c * (a * -3.0)) + (b * b))))) / (a * 3.0);
}

Python

def code(a, b, c):
	return ((-3.0 * (c * -a)) / (-b - math.sqrt(((c * (a * -3.0)) + (b * b))))) / (a * 3.0)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((-3.0 * (c * -a)) / (-b - sqrt(((c * (a * -3.0)) + (b * b))))) / (a * 3.0);
end

Wolfram

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

Derivation

1. Initial program 17.8%

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

2. Taylor expanded in a around 0 17.7%

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

   1. *-commutative 17.7%

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

   2. *-commutative 17.7%

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

   3. associate-*l* 17.8%

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

4. Simplified 17.8%

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

   1. flip-+ 17.8%

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

   2. add-sqr-sqrt 18.2%

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

   3. associate-*r* 18.3%

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

   4. *-commutative 18.3%

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

   5. associate-*r* 18.3%

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

   6. *-commutative 18.3%

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

6. Applied egg-rr 18.3%

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

   1. sqr-neg 18.3%

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

   2. sub-neg 18.3%

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

   3. associate--r+ 99.2%

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

   4. +-inverses 99.2%

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

   5. distribute-rgt-neg-in 99.2%

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

   6. metadata-eval 99.2%

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

   7. associate-*l* 99.5%

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

   8. sub-neg 99.5%

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

   9. +-commutative 99.5%

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

   10. sqr-neg 99.5%

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

   11. distribute-rgt-neg-out 99.5%

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

   12. unsub-neg 99.5%

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

   13. distribute-rgt-neg-in 99.5%

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

   14. metadata-eval 99.5%

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

   15. associate-*l* 99.5%

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

   16. fma-neg 99.5%

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

   17. distribute-rgt-neg-out 99.5%

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

   18. sqr-neg 99.5%

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

8. Simplified 99.5%

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

   1. fma-udef 99.5%

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

10. Applied egg-rr 99.5%

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

11. Taylor expanded in c around 0 99.2%

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

12. Final simplification 99.2%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.375 (* a (/ (* c c) (pow b 3.0)))) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	return (-0.375 * (a * ((c * c) / pow(b, 3.0)))) + (-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 = ((-0.375d0) * (a * ((c * c) / (b ** 3.0d0)))) + ((-0.5d0) * (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.375 * (a * ((c * c) / Math.pow(b, 3.0)))) + (-0.5 * (c / b));
}

Python

def code(a, b, c):
	return (-0.375 * (a * ((c * c) / math.pow(b, 3.0)))) + (-0.5 * (c / b))

Julia

function code(a, b, c)
	return Float64(Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))) + Float64(-0.5 * Float64(c / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.375 * (a * ((c * c) / (b ^ 3.0)))) + (-0.5 * (c / b));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

   1. /-rgt-identity 17.8%

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

   2. metadata-eval 17.8%

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

   3. associate-/l* 17.8%

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

   4. associate-*r/ 17.8%

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

   5. *-commutative 17.8%

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

   6. associate-*l/ 17.8%

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

   7. associate-*r/ 17.8%

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

   8. metadata-eval 17.8%

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

   9. metadata-eval 17.8%

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

   10. times-frac 17.8%

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

   11. neg-mul-1 17.8%

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

   12. distribute-rgt-neg-in 17.8%

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

   13. times-frac 17.8%

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

   14. metadata-eval 17.8%

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

   15. neg-mul-1 17.8%

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

3. Simplified 17.8%

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

4. Taylor expanded in b around inf 95.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
5. Step-by-step derivation

   1. +-commutative 95.4%

      \[\leadsto \color{blue}{-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.5 \cdot \frac{c}{b}} \]

   2. fma-def 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{{c}^{2} \cdot a}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)} \]

   3. associate-/l* 95.4%

      \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}}, -0.5 \cdot \frac{c}{b}\right) \]

   4. unpow2 95.4%

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

6. Simplified 95.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, -0.5 \cdot \frac{c}{b}\right)} \]
7. Step-by-step derivation

   1. fma-udef 95.4%

      \[\leadsto \color{blue}{-0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}} + -0.5 \cdot \frac{c}{b}} \]

   2. associate-/r/ 95.4%

      \[\leadsto -0.375 \cdot \color{blue}{\left(\frac{c \cdot c}{{b}^{3}} \cdot a\right)} + -0.5 \cdot \frac{c}{b} \]

8. Applied egg-rr 95.4%

   \[\leadsto \color{blue}{-0.375 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot a\right) + -0.5 \cdot \frac{c}{b}} \]

9. Final simplification 95.4%

   \[\leadsto -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b} \]

Alternative 4: 90.5% accurate, 23.2× 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 17.8%

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

   1. /-rgt-identity 17.8%

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

   2. metadata-eval 17.8%

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

   3. associate-/l* 17.8%

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

   4. associate-*r/ 17.8%

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

   5. *-commutative 17.8%

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

   6. associate-*l/ 17.8%

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

   7. associate-*r/ 17.8%

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

   8. metadata-eval 17.8%

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

   9. metadata-eval 17.8%

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

   10. times-frac 17.8%

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

   11. neg-mul-1 17.8%

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

   12. distribute-rgt-neg-in 17.8%

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

   13. times-frac 17.8%

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

   14. metadata-eval 17.8%

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

   15. neg-mul-1 17.8%

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

3. Simplified 17.8%

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

4. Taylor expanded in b around inf 90.4%

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

5. Final simplification 90.4%

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

Reproduce

herbie shell --seed 2023229 
(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)))