Cubic critical, medium range

Percentage Accurate: 31.3% → 99.4%

Time: 13.7s

Alternatives: 8

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: 31.3% 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: 99.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (+ (- (pow (- b) 2.0) (pow b 2.0)) (* c (* a 3.0)))
   (- (- b) (sqrt (- (pow b 2.0) (* 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)) + (c * (a * 3.0))) / (-b - sqrt((pow(b, 2.0) - (a * (c * 3.0)))))) / (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
    code = ((((-b ** 2.0d0) - (b ** 2.0d0)) + (c * (a * 3.0d0))) / (-b - sqrt(((b ** 2.0d0) - (a * (c * 3.0d0)))))) / (a * 3.0d0)
end function

Java

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

Python

def code(a, b, c):
	return (((math.pow(-b, 2.0) - math.pow(b, 2.0)) + (c * (a * 3.0))) / (-b - math.sqrt((math.pow(b, 2.0) - (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(c * Float64(a * 3.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - Float64(a * Float64(c * 3.0)))))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((-b ^ 2.0) - (b ^ 2.0)) + (c * (a * 3.0))) / (-b - sqrt(((b ^ 2.0) - (a * (c * 3.0)))))) / (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[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1%

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

   1. expm1-log1p-u 36.0%

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

   2. associate-*l* 36.0%

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

4. Applied egg-rr 36.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 36.1%

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

   2. pow2 36.1%

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

   3. expm1-log1p-u 36.1%

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

   4. expm1-log1p-u 36.1%

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

   5. add-sqr-sqrt 36.9%

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

   6. pow2 36.9%

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

   7. associate-*r* 36.9%

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

   8. *-commutative 36.9%

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

6. Applied egg-rr 36.9%

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

   1. associate--r- 99.4%

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

   2. associate-*l* 99.2%

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

   3. associate-*l* 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in a around 0 99.2%

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

    1. associate-*r* 99.4%

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

    2. *-commutative 99.4%

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

    3. *-commutative 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \frac{-1}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (* (+ b (sqrt (fma a (* c -3.0) (pow b 2.0)))) (/ -1.0 c))))

C

double code(double a, double b, double c) {
	return 1.0 / ((b + sqrt(fma(a, (c * -3.0), pow(b, 2.0)))) * (-1.0 / c));
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(b + sqrt(fma(a, Float64(c * -3.0), (b ^ 2.0)))) * Float64(-1.0 / c)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1%

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

   1. expm1-log1p-u 36.0%

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

   2. associate-*l* 36.0%

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

4. Applied egg-rr 36.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 36.1%

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

   2. pow2 36.1%

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

   3. expm1-log1p-u 36.1%

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

   4. expm1-log1p-u 36.1%

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

   5. add-sqr-sqrt 36.9%

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

   6. pow2 36.9%

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

   7. associate-*r* 36.9%

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

   8. *-commutative 36.9%

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

6. Applied egg-rr 36.9%

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

   1. associate--r- 99.4%

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

   2. associate-*l* 99.2%

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

   3. associate-*l* 99.2%

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

8. Simplified 99.2%

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

   1. clear-num 99.2%

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

   2. inv-pow 99.2%

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

10. Applied egg-rr 99.2%

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

    1. unpow-1 99.2%

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

    2. associate-/r/ 99.1%

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

    3. fma-udef 99.1%

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

    4. associate-*l* 99.2%

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

    5. +-inverses 99.2%

       \[\leadsto \frac{1}{\frac{a \cdot 3}{\left(a \cdot 3\right) \cdot c + \color{blue}{0}} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)} \]

    6. +-rgt-identity 99.2%

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

    7. associate-*l* 99.1%

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

    8. *-commutative 99.1%

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

    9. cancel-sign-sub-inv 99.1%

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

    10. *-commutative 99.1%

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

    11. distribute-lft-neg-in 99.1%

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

    12. metadata-eval 99.1%

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

    13. associate-*r* 99.1%

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

    14. +-commutative 99.1%

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

12. Simplified 99.1%

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

13. Taylor expanded in a around 0 99.3%

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

14. Final simplification 99.3%

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

Alternative 3: 94.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + \left(1.125 \cdot \frac{c \cdot {a}^{2}}{{b}^{3}} + 1.5 \cdot \frac{a}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (+
   (* -2.0 (/ b c))
   (+ (* 1.125 (/ (* c (pow a 2.0)) (pow b 3.0))) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + ((1.125 * ((c * pow(a, 2.0)) / pow(b, 3.0))) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + ((1.125d0 * ((c * (a ** 2.0d0)) / (b ** 3.0d0))) + (1.5d0 * (a / b))))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + ((1.125 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 3.0))) + (1.5 * (a / b))));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + ((1.125 * ((c * math.pow(a, 2.0)) / math.pow(b, 3.0))) + (1.5 * (a / b))))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(Float64(1.125 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 3.0))) + Float64(1.5 * Float64(a / b)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + ((1.125 * ((c * (a ^ 2.0)) / (b ^ 3.0))) + (1.5 * (a / b))));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(N[(1.125 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1%

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

   1. expm1-log1p-u 36.0%

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

   2. associate-*l* 36.0%

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

4. Applied egg-rr 36.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 36.1%

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

   2. pow2 36.1%

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

   3. expm1-log1p-u 36.1%

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

   4. expm1-log1p-u 36.1%

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

   5. add-sqr-sqrt 36.9%

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

   6. pow2 36.9%

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

   7. associate-*r* 36.9%

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

   8. *-commutative 36.9%

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

6. Applied egg-rr 36.9%

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

   1. associate--r- 99.4%

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

   2. associate-*l* 99.2%

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

   3. associate-*l* 99.2%

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

8. Simplified 99.2%

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

   1. clear-num 99.2%

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

   2. inv-pow 99.2%

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

10. Applied egg-rr 99.2%

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

    1. unpow-1 99.2%

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

    2. associate-/r/ 99.1%

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

    3. fma-udef 99.1%

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

    4. associate-*l* 99.2%

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

    5. +-inverses 99.2%

       \[\leadsto \frac{1}{\frac{a \cdot 3}{\left(a \cdot 3\right) \cdot c + \color{blue}{0}} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)} \]

    6. +-rgt-identity 99.2%

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

    7. associate-*l* 99.1%

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

    8. *-commutative 99.1%

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

    9. cancel-sign-sub-inv 99.1%

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

    10. *-commutative 99.1%

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

    11. distribute-lft-neg-in 99.1%

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

    12. metadata-eval 99.1%

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

    13. associate-*r* 99.1%

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

    14. +-commutative 99.1%

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

12. Simplified 99.1%

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

13. Taylor expanded in a around 0 92.5%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + \left(1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}} + 1.5 \cdot \frac{a}{b}\right)}} \]

14. Final simplification 92.5%

    \[\leadsto \frac{1}{-2 \cdot \frac{b}{c} + \left(1.125 \cdot \frac{c \cdot {a}^{2}}{{b}^{3}} + 1.5 \cdot \frac{a}{b}\right)} \]
15. Add Preprocessing

Alternative 4: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma 1.5 (/ a b) (* -2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(1.5, (a / b), (-2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(1.5, Float64(a / b), Float64(-2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(1.5 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1%

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

   1. expm1-log1p-u 36.0%

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

   2. associate-*l* 36.0%

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

4. Applied egg-rr 36.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. clear-num 36.0%

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

   2. inv-pow 36.0%

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

   3. *-commutative 36.0%

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

   4. neg-mul-1 36.0%

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

   5. fma-def 36.0%

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

   6. pow2 36.0%

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

   7. expm1-log1p-u 36.1%

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

   8. associate-*r* 36.1%

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

   9. *-commutative 36.1%

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

6. Applied egg-rr 36.1%

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

   1. unpow-1 36.1%

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

   2. *-commutative 36.1%

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

   3. *-lft-identity 36.1%

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

   4. times-frac 36.1%

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

   5. metadata-eval 36.1%

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

   6. fma-udef 36.1%

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

   7. *-commutative 36.1%

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

   8. fma-def 36.1%

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

   9. unpow2 36.1%

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

   10. fma-neg 36.2%

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

   11. *-commutative 36.2%

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

   12. distribute-rgt-neg-in 36.2%

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

   13. distribute-rgt-neg-in 36.2%

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

   14. metadata-eval 36.2%

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

8. Simplified 36.2%

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

9. Taylor expanded in b around inf 89.0%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
10. Step-by-step derivation

    1. +-commutative 89.0%

       \[\leadsto \frac{1}{\color{blue}{1.5 \cdot \frac{a}{b} + -2 \cdot \frac{b}{c}}} \]

    2. fma-def 89.0%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}} \]

11. Simplified 89.0%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}} \]

12. Final simplification 89.0%

    \[\leadsto \frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)} \]
13. Add Preprocessing

Alternative 5: 91.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1%

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

   1. expm1-log1p-u 36.0%

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

   2. associate-*l* 36.0%

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

4. Applied egg-rr 36.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. clear-num 36.0%

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

   2. inv-pow 36.0%

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

   3. *-commutative 36.0%

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

   4. neg-mul-1 36.0%

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

   5. fma-def 36.0%

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

   6. pow2 36.0%

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

   7. expm1-log1p-u 36.1%

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

   8. associate-*r* 36.1%

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

   9. *-commutative 36.1%

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

6. Applied egg-rr 36.1%

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

   1. unpow-1 36.1%

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

   2. *-commutative 36.1%

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

   3. *-lft-identity 36.1%

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

   4. times-frac 36.1%

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

   5. metadata-eval 36.1%

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

   6. fma-udef 36.1%

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

   7. *-commutative 36.1%

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

   8. fma-def 36.1%

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

   9. unpow2 36.1%

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

   10. fma-neg 36.2%

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

   11. *-commutative 36.2%

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

   12. distribute-rgt-neg-in 36.2%

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

   13. distribute-rgt-neg-in 36.2%

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

   14. metadata-eval 36.2%

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

8. Simplified 36.2%

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

9. Taylor expanded in b around inf 89.0%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]

10. Final simplification 89.0%

    \[\leadsto \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \]
11. Add Preprocessing

Alternative 6: 81.2% 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 36.1%

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

3. Taylor expanded in b around inf 77.7%

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

   1. associate-*r/ 77.7%

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

   2. associate-/l* 77.5%

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

5. Simplified 77.5%

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

   1. associate-/r/ 77.5%

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

7. Applied egg-rr 77.5%

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

8. Final simplification 77.5%

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

Alternative 7: 81.4% 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 36.1%

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

3. Taylor expanded in b around inf 77.7%

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

   1. associate-*r/ 77.7%

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

5. Simplified 77.7%

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

6. Final simplification 77.7%

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

Alternative 8: 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 36.1%

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

   1. expm1-log1p-u 36.0%

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

   2. associate-*l* 36.0%

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

4. Applied egg-rr 36.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. clear-num 36.0%

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

   2. inv-pow 36.0%

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

   3. *-commutative 36.0%

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

   4. neg-mul-1 36.0%

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

   5. fma-def 36.0%

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

   6. pow2 36.0%

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

   7. expm1-log1p-u 36.1%

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

   8. associate-*r* 36.1%

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

   9. *-commutative 36.1%

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

6. Applied egg-rr 36.1%

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

   1. unpow-1 36.1%

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

   2. *-commutative 36.1%

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

   3. *-lft-identity 36.1%

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

   4. times-frac 36.1%

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

   5. metadata-eval 36.1%

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

   6. fma-udef 36.1%

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

   7. *-commutative 36.1%

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

   8. fma-def 36.1%

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

   9. unpow2 36.1%

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

   10. fma-neg 36.2%

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

   11. *-commutative 36.2%

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

   12. distribute-rgt-neg-in 36.2%

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

   13. distribute-rgt-neg-in 36.2%

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

   14. metadata-eval 36.2%

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

8. Simplified 36.2%

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

9. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

11. Simplified 3.2%

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

12. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2024021 
(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)))