Cubic critical, medium range

Percentage Accurate: 32.3% → 99.1%

Time: 15.6s

Alternatives: 9

Speedup: 23.2×

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: 32.3% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + t\_0}{\left(-b\right) - \sqrt{{b}^{2} - t\_0}}}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))))
   (/
    (/
     (+ (- (pow (- b) 2.0) (pow b 2.0)) t_0)
     (- (- b) (sqrt (- (pow b 2.0) t_0))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	return (((pow(-b, 2.0) - pow(b, 2.0)) + t_0) / (-b - sqrt((pow(b, 2.0) - t_0)))) / (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 = a * (c * 3.0d0)
    code = ((((-b ** 2.0d0) - (b ** 2.0d0)) + t_0) / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	return (((Math.pow(-b, 2.0) - Math.pow(b, 2.0)) + t_0) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	return (((math.pow(-b, 2.0) - math.pow(b, 2.0)) + t_0) / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	return Float64(Float64(Float64(Float64((Float64(-b) ^ 2.0) - (b ^ 2.0)) + t_0) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (c * 3.0);
	tmp = ((((-b ^ 2.0) - (b ^ 2.0)) + t_0) / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 28.2%

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

   1. add-exp-log 28.1%

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

   2. associate-*l* 28.1%

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

4. Applied egg-rr 28.1%

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

   1. flip-+ 28.1%

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

   2. pow2 28.1%

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

   3. add-sqr-sqrt 28.9%

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

   4. rem-exp-log 29.0%

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

   5. *-commutative 29.0%

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

   6. associate-*r* 29.0%

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

   7. pow2 29.0%

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

   8. rem-exp-log 29.0%

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

   9. *-commutative 29.0%

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

   10. associate-*r* 29.0%

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

   11. pow2 29.0%

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

6. Applied egg-rr 29.0%

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

   1. associate--r- 99.2%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (+ (- (pow (- b) 2.0) (pow b 2.0)) (* a (* c 3.0)))
   (- (- b) (sqrt (fma b b (* a (* c -3.0))))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (((pow(-b, 2.0) - pow(b, 2.0)) + (a * (c * 3.0))) / (-b - sqrt(fma(b, b, (a * (c * -3.0)))))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64((Float64(-b) ^ 2.0) - (b ^ 2.0)) + Float64(a * Float64(c * 3.0))) / Float64(Float64(-b) - sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

   1. add-exp-log 28.1%

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

   2. associate-*l* 28.1%

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

4. Applied egg-rr 28.1%

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

   1. flip-+ 28.1%

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

   2. pow2 28.1%

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

   3. add-sqr-sqrt 28.9%

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

   4. rem-exp-log 29.0%

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

   5. *-commutative 29.0%

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

   6. associate-*r* 29.0%

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

   7. pow2 29.0%

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

   8. rem-exp-log 29.0%

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

   9. *-commutative 29.0%

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

   10. associate-*r* 29.0%

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

   11. pow2 29.0%

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

6. Applied egg-rr 29.0%

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

   1. associate--r- 99.2%

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

8. Simplified 99.2%

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

   1. neg-mul-1 99.2%

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

   2. fma-neg 99.2%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 3\right)}{\color{blue}{\mathsf{fma}\left(-1, b, -\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

10. Applied egg-rr 99.2%

    \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 3\right)}{\color{blue}{\mathsf{fma}\left(-1, b, -\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]
11. Step-by-step derivation

    1. fma-udef 99.2%

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

    2. unsub-neg 99.2%

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

    3. mul-1-neg 99.2%

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

    4. unpow2 99.2%

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

    5. fma-neg 99.1%

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

    6. associate-*r* 99.1%

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

    7. distribute-rgt-neg-in 99.1%

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

    8. metadata-eval 99.1%

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

    9. associate-*r* 99.1%

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

12. Simplified 99.1%

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

13. Final simplification 99.1%

    \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{a \cdot 3} \]
14. Add Preprocessing

Alternative 3: 83.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.002:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.002)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* -3.0 (* a c)))) b) a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.002) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (-3.0 * (a * c)))) - b) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.002)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))) - b) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.002], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2e-3

   1. Initial program 68.8%

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

      1. add-exp-log 68.4%

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

      2. associate-*l* 68.4%

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

   4. Applied egg-rr 68.4%

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

      1. expm1-log1p-u 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - e^{\log \left(3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 23.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - e^{\log \left(3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a}\right)} - 1} \]

   6. Applied egg-rr 23.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a \cdot 3}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 23.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a \cdot 3}\right)\right)} \]

      2. expm1-log1p 68.8%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a \cdot 3}} \]

      3. *-lft-identity 68.8%

         \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}{a \cdot 3} \]

      4. *-commutative 68.8%

         \[\leadsto \frac{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{\color{blue}{3 \cdot a}} \]

      5. times-frac 68.8%

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a}} \]

      6. metadata-eval 68.8%

         \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a} \]

      7. fma-udef 68.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot b + \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{a} \]

      8. neg-mul-1 68.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\left(-b\right)} + \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}{a} \]

      9. +-commutative 68.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)} + \left(-b\right)}}{a} \]

      10. unsub-neg 68.8%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)} - b}}{a} \]

      11. unpow2 68.8%

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

      12. fma-neg 69.0%

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

      13. associate-*r* 69.0%

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

      14. distribute-rgt-neg-in 69.0%

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

      15. metadata-eval 69.0%

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

   8. Simplified 69.0%

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

   if -2e-3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.3%

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

   3. Taylor expanded in b around inf 91.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 91.0%

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

   5. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.002:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.002) t_0 (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.002d0)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -0.002:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -0.002)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.002], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2e-3

   1. Initial program 68.8%

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

   if -2e-3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.3%

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

   3. Taylor expanded in b around inf 91.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 91.0%

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

   5. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.002:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{3 \cdot \left(a \cdot c\right) + 0 \cdot \left(b + b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (+ (* 3.0 (* a c)) (* 0.0 (+ b b)))
   (- (- b) (sqrt (fma b b (* -3.0 (* a c))))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (((3.0 * (a * c)) + (0.0 * (b + b))) / (-b - sqrt(fma(b, b, (-3.0 * (a * c)))))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(3.0 * Float64(a * c)) + Float64(0.0 * Float64(b + b))) / Float64(Float64(-b) - sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(0.0 * N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

   1. add-exp-log 28.1%

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

   2. associate-*l* 28.1%

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

4. Applied egg-rr 28.1%

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

   1. flip-+ 28.1%

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

   2. pow2 28.1%

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

   3. add-sqr-sqrt 28.9%

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

   4. rem-exp-log 29.0%

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

   5. *-commutative 29.0%

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

   6. associate-*r* 29.0%

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

   7. pow2 29.0%

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

   8. rem-exp-log 29.0%

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

   9. *-commutative 29.0%

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

   10. associate-*r* 29.0%

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

   11. pow2 29.0%

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

6. Applied egg-rr 29.0%

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

   1. associate--r- 99.2%

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

   2. +-commutative 99.2%

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

   3. associate-*r* 99.1%

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

   4. *-commutative 99.1%

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

   5. unpow2 99.1%

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

   6. unpow2 99.1%

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

   7. difference-of-squares 99.1%

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

   8. neg-mul-1 99.1%

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

   9. distribute-lft1-in 99.1%

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

   10. metadata-eval 99.1%

       \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right) + \left(\color{blue}{0} \cdot b\right) \cdot \left(\left(-b\right) - b\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{3 \cdot a} \]

   11. mul0-lft 99.1%

       \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right) + \color{blue}{0} \cdot \left(\left(-b\right) - b\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{3 \cdot a} \]

   12. unpow2 99.1%

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

   13. fma-neg 99.1%

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

   14. associate-*r* 99.1%

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

   15. distribute-rgt-neg-in 99.1%

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

   16. metadata-eval 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

   \[\leadsto \frac{\frac{3 \cdot \left(a \cdot c\right) + 0 \cdot \left(b + b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 3} \]
10. Add Preprocessing

Alternative 6: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 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 = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in b around inf 92.0%

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

4. Final simplification 92.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in b around inf 83.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 83.8%

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

   2. associate-/l* 83.5%

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

5. Simplified 83.5%

   \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]
6. Step-by-step derivation

   1. associate-/r/ 83.5%

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

7. Applied egg-rr 83.5%

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

8. Final simplification 83.5%

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

Alternative 8: 80.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in b around inf 83.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 83.8%

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

5. Simplified 83.8%

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

6. Final simplification 83.8%

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

Alternative 9: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

   1. add-sqr-sqrt 28.2%

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

   2. difference-of-squares 28.3%

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

   3. associate-*l* 28.3%

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

   4. associate-*l* 28.3%

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

4. Applied egg-rr 28.3%

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

   1. associate-*r* 28.3%

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

   2. *-commutative 28.3%

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

   3. associate-*r* 28.3%

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

   4. *-commutative 28.3%

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

6. Simplified 28.3%

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

7. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

   2. distribute-lft1-in 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

10. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024030 
(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)))