Cubic critical, narrow range

Percentage Accurate: 55.9% → 99.3%

Time: 30.3s

Alternatives: 7

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. add-sqr-sqrt 55.8%

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

   2. pow2 55.8%

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

   3. associate-*l* 55.8%

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

4. Applied egg-rr 55.8%

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

   1. flip-+ 55.7%

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

   2. pow2 55.8%

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

   3. add-sqr-sqrt 57.2%

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

   4. pow2 57.2%

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

   5. unpow2 57.2%

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

   6. add-sqr-sqrt 57.2%

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

   7. pow2 57.2%

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

   8. unpow2 57.2%

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

   9. add-sqr-sqrt 57.2%

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

6. Applied egg-rr 57.2%

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

   1. associate--r- 99.1%

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

   2. neg-mul-1 99.1%

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

   3. unpow-prod-down 99.1%

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

   4. metadata-eval 99.1%

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

   5. *-un-lft-identity 99.1%

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

   6. associate-*r* 99.3%

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

   7. *-commutative 99.3%

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

8. Applied egg-rr 99.3%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = 3.0 * (a * c);
	return (t_0 / (-b - sqrt((pow(b, 2.0) - t_0)))) / (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
    real(8) :: t_0
    t_0 = 3.0d0 * (a * c)
    code = (t_0 / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. add-sqr-sqrt 55.8%

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

   2. pow2 55.8%

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

   3. associate-*l* 55.8%

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

4. Applied egg-rr 55.8%

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

   1. flip-+ 55.7%

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

   2. pow2 55.8%

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

   3. add-sqr-sqrt 57.2%

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

   4. pow2 57.2%

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

   5. unpow2 57.2%

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

   6. add-sqr-sqrt 57.2%

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

   7. pow2 57.2%

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

   8. unpow2 57.2%

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

   9. add-sqr-sqrt 57.2%

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

6. Applied egg-rr 57.2%

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

7. Taylor expanded in b around 0 99.1%

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

Alternative 3: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.55)
   (/ (- (sqrt (fma b b (* -3.0 (* a c)))) b) (* 3.0 a))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.55) {
		tmp = (sqrt(fma(b, b, (-3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.55)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.55], N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 2.5499999999999998

   1. Initial program 82.7%

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

      1. /-rgt-identity 82.7%

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

      2. metadata-eval 82.7%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in a around 0 82.8%

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

   if 2.5499999999999998 < b

   1. Initial program 49.3%

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

   3. Taylor expanded in a around 0 88.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.5)
   (/ (- (sqrt (fma b b (* -3.0 (* a c)))) b) (* 3.0 a))
   (* c (- (* -0.375 (/ (* a c) (pow b 3.0))) (* 0.5 (/ 1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5) {
		tmp = (sqrt(fma(b, b, (-3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * ((a * c) / pow(b, 3.0))) - (0.5 * (1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 3.0))) - Float64(0.5 * Float64(1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.5], N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5

   1. Initial program 82.7%

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

      1. /-rgt-identity 82.7%

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

      2. metadata-eval 82.7%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in a around 0 82.8%

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

   if 2.5 < b

   1. Initial program 49.3%

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

   3. Taylor expanded in c around 0 88.3%

      \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.5)
   (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))
   (* c (- (* -0.375 (/ (* a c) (pow b 3.0))) (* 0.5 (/ 1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5) {
		tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * ((a * c) / pow(b, 3.0))) - (0.5 * (1.0 / 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) :: tmp
    if (b <= 2.5d0) then
        tmp = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
    else
        tmp = c * (((-0.375d0) * ((a * c) / (b ** 3.0d0))) - (0.5d0 * (1.0d0 / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5) {
		tmp = (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * ((a * c) / Math.pow(b, 3.0))) - (0.5 * (1.0 / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.5:
		tmp = (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)
	else:
		tmp = c * ((-0.375 * ((a * c) / math.pow(b, 3.0))) - (0.5 * (1.0 / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.5)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 3.0))) - Float64(0.5 * Float64(1.0 / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.5)
		tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	else
		tmp = c * ((-0.375 * ((a * c) / (b ^ 3.0))) - (0.5 * (1.0 / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.5], 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], N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5

   1. Initial program 82.7%

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

   if 2.5 < b

   1. Initial program 49.3%

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

   3. Taylor expanded in c around 0 88.3%

      \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (* -0.375 (/ (* a c) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((-0.375 * ((a * c) / pow(b, 3.0))) - (0.5 * (1.0 / 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 * (((-0.375d0) * ((a * c) / (b ** 3.0d0))) - (0.5d0 * (1.0d0 / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.375 * ((a * c) / Math.pow(b, 3.0))) - (0.5 * (1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((-0.375 * ((a * c) / math.pow(b, 3.0))) - (0.5 * (1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 3.0))) - Float64(0.5 * Float64(1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.375 * ((a * c) / (b ^ 3.0))) - (0.5 * (1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.9%

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

3. Taylor expanded in c around 0 82.5%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
4. Add Preprocessing

Alternative 7: 64.0% 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 55.9%

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

3. Taylor expanded in b around inf 64.2%

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

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))