Cubic critical, narrow range

Percentage Accurate: 55.6% → 99.3%

Time: 15.2s

Alternatives: 11

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: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

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

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 (* 3.0 c))))))
  (* 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 * (3.0 * c)))))) / (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 * (3.0d0 * c)))))) / (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 * (3.0 * c)))))) / (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 * (3.0 * c)))))) / (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(3.0 * c)))))) / 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 * (3.0 * c)))))) / (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[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.9%

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

   1. expm1-log1p-u 56.9%

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

   2. associate-*l* 56.9%

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

4. Applied egg-rr 56.9%

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

   1. flip-+ 56.6%

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

   2. pow2 56.6%

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

   3. expm1-log1p-u 56.7%

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

   4. expm1-log1p-u 56.7%

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

   5. add-sqr-sqrt 58.4%

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

   6. pow2 58.4%

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

   7. associate-*r* 58.4%

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

   8. *-commutative 58.4%

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

6. Applied egg-rr 58.4%

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

   1. associate--r- 99.3%

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

   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. pow1 99.2%

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

   2. associate-*r* 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. expm1-log1p-u 56.9%

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

   2. associate-*l* 56.9%

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

4. Applied egg-rr 56.9%

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

   1. flip-+ 56.6%

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

   2. pow2 56.6%

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

   3. expm1-log1p-u 56.7%

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

   4. expm1-log1p-u 56.7%

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

   5. add-sqr-sqrt 58.4%

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

   6. pow2 58.4%

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

   7. associate-*r* 58.4%

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

   8. *-commutative 58.4%

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

6. Applied egg-rr 58.4%

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

   1. associate-+l- 99.3%

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

   2. +-commutative 99.3%

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

   3. fma-def 99.3%

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

   4. neg-mul-1 99.3%

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

   5. unpow-prod-down 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-un-lft-identity 99.3%

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

8. Applied egg-rr 99.3%

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

9. Final simplification 99.3%

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

Alternative 3: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64((Float64(-b) ^ 2.0) - (b ^ 2.0)) + Float64(a * Float64(3.0 * c))) / Float64(Float64(-b) - sqrt(fma(b, b, Float64(a * Float64(3.0 * Float64(-c))))))) / 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[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b + N[(a * N[(3.0 * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.9%

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

   1. expm1-log1p-u 56.9%

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

   2. associate-*l* 56.9%

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

4. Applied egg-rr 56.9%

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

   1. flip-+ 56.6%

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

   2. pow2 56.6%

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

   3. expm1-log1p-u 56.7%

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

   4. expm1-log1p-u 56.7%

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

   5. add-sqr-sqrt 58.4%

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

   6. pow2 58.4%

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

   7. associate-*r* 58.4%

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

   8. *-commutative 58.4%

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

6. Applied egg-rr 58.4%

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

   1. associate--r- 99.3%

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

   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. cancel-sign-sub-inv 99.2%

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

   2. unpow2 99.2%

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

   3. fma-def 99.2%

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

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

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

Alternative 4: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.7)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (+ (* -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 (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.7) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} 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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	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[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], -0.7], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.69999999999999996

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. sqr-neg 86.8%

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

      3. unsub-neg 86.8%

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

      4. div-sub 86.4%

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

      5. --rgt-identity 86.4%

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

      6. div-sub 86.8%

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

   3. Simplified 86.8%

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

   if -0.69999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.4%

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

   3. Taylor expanded in b around inf 87.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -3.8e-5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (* 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)) <= -3.8e-5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} 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)) <= -3.8e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	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], -3.8e-5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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)) < -3.8000000000000002e-5

   1. Initial program 73.9%

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

      1. +-commutative 73.9%

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

      2. sqr-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. div-sub 73.2%

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

      5. --rgt-identity 73.2%

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

      6. div-sub 73.9%

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

   3. Simplified 74.1%

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

   if -3.8000000000000002e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.4%

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

   3. Taylor expanded in b around inf 75.9%

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

      1. associate-*r/ 75.9%

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

   5. Simplified 75.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. expm1-log1p-u 56.9%

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

   2. associate-*l* 56.9%

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

4. Applied egg-rr 56.9%

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

   1. flip-+ 56.6%

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

   2. pow2 56.6%

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

   3. expm1-log1p-u 56.7%

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

   4. expm1-log1p-u 56.7%

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

   5. add-sqr-sqrt 58.4%

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

   6. pow2 58.4%

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

   7. associate-*r* 58.4%

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

   8. *-commutative 58.4%

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

6. Applied egg-rr 58.4%

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

   1. associate--r- 99.3%

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

   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. expm1-log1p-u 88.0%

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

   2. expm1-udef 62.2%

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

10. Applied egg-rr 62.2%

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

    1. expm1-def 88.0%

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

    2. expm1-log1p 99.2%

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

    3. fma-udef 99.2%

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

    4. associate-*l* 99.3%

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

    5. +-inverses 99.3%

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

    6. +-rgt-identity 99.3%

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

    7. associate-*l* 99.2%

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

    8. *-commutative 99.2%

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

    9. *-commutative 99.2%

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

    10. associate-*r* 99.2%

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

    11. cancel-sign-sub-inv 99.2%

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

    12. metadata-eval 99.2%

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

    13. +-commutative 99.2%

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

    14. fma-def 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

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

Alternative 7: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -3.8e-5)
   (/ (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) (* a 3.0))
   (/ (* 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)) <= -3.8e-5) {
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (a * 3.0);
	} 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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-3.8d-5)) then
        tmp = (sqrt(((b * b) - (a * (3.0d0 * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -3.8e-5) {
		tmp = (Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -3.8e-5:
		tmp = (math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (a * 3.0)
	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)) <= -3.8e-5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -3.8e-5)
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = 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], -3.8e-5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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)) < -3.8000000000000002e-5

   1. Initial program 73.9%

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

   3. Taylor expanded in a around 0 73.9%

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

      1. associate-*r* 73.9%

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

      2. *-commutative 73.9%

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

      3. associate-*l* 73.9%

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

   5. Simplified 73.9%

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

   if -3.8000000000000002e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.4%

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

   3. Taylor expanded in b around inf 75.9%

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

      1. associate-*r/ 75.9%

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

   5. Simplified 75.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. expm1-log1p-u 56.9%

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

   2. associate-*l* 56.9%

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

4. Applied egg-rr 56.9%

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

   1. flip-+ 56.6%

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

   2. pow2 56.6%

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

   3. expm1-log1p-u 56.7%

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

   4. expm1-log1p-u 56.7%

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

   5. add-sqr-sqrt 58.4%

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

   6. pow2 58.4%

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

   7. associate-*r* 58.4%

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

   8. *-commutative 58.4%

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

6. Applied egg-rr 58.4%

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

   1. unpow2 58.4%

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

   2. sqr-neg 58.4%

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

   3. unpow2 58.4%

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

   4. unpow2 58.4%

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

   5. fma-neg 58.1%

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

   6. *-commutative 58.1%

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

   7. distribute-rgt-neg-in 58.1%

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

   8. distribute-rgt-neg-in 58.1%

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

   9. metadata-eval 58.1%

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

   10. unpow2 58.1%

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

   11. fma-neg 58.1%

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

   12. *-commutative 58.1%

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

   13. distribute-rgt-neg-in 58.1%

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

   14. distribute-rgt-neg-in 58.1%

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

   15. metadata-eval 58.1%

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

8. Simplified 58.1%

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

   1. sub-neg 58.1%

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

10. Applied egg-rr 58.1%

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

    1. sub-neg 58.1%

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

    2. fma-udef 58.4%

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

    3. unpow2 58.4%

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

    4. *-commutative 58.4%

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

    5. *-commutative 58.4%

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

    6. associate-*r* 58.4%

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

    7. metadata-eval 58.4%

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

    8. cancel-sign-sub-inv 58.4%

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

    9. associate-*r* 58.4%

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

    10. *-commutative 58.4%

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

    11. sub-neg 58.4%

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

    12. associate--r+ 99.4%

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

    13. +-inverses 99.4%

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

    14. *-commutative 99.4%

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

    15. associate-*r* 99.2%

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

    16. distribute-lft-neg-in 99.2%

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

    17. metadata-eval 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

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

Alternative 9: 64.3% 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 56.9%

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

3. Taylor expanded in b around inf 63.4%

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

   1. associate-*r/ 63.4%

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

   2. associate-/l* 63.4%

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

5. Simplified 63.4%

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

   1. associate-/r/ 63.4%

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

7. Applied egg-rr 63.4%

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

8. Final simplification 63.4%

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

Alternative 10: 64.3% 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 56.9%

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

3. Taylor expanded in b around inf 63.4%

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

   1. associate-*r/ 63.4%

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

5. Simplified 63.4%

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

6. Final simplification 63.4%

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

Alternative 11: 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 56.9%

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

   1. expm1-log1p-u 56.9%

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

   2. associate-*l* 56.9%

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

4. Applied egg-rr 56.9%

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

   1. expm1-log1p-u 46.8%

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

   2. expm1-udef 45.2%

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

6. Applied egg-rr 45.2%

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

   1. expm1-def 46.8%

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

   2. expm1-log1p 56.9%

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

   3. *-lft-identity 56.9%

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

   4. associate-*l/ 56.9%

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

   5. *-commutative 56.9%

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

   6. associate-/r* 56.9%

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

   7. metadata-eval 56.9%

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

   8. fma-udef 56.9%

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

   9. *-commutative 56.9%

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

   10. fma-def 56.9%

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

   11. unpow2 56.9%

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

   12. fma-neg 57.0%

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

   13. *-commutative 57.0%

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

   14. distribute-rgt-neg-in 57.0%

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

   15. distribute-rgt-neg-in 57.0%

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

   16. metadata-eval 57.0%

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

8. Simplified 57.0%

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

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 2024029 
(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)))