Cubic critical, narrow range

Percentage Accurate: 54.8% → 99.1%

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: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((pow(b, 2.0) - pow(-b, 2.0)) - (a * (c * 3.0))) / (b + sqrt((pow(b, 2.0) + (-3.0 * (a * 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)) - (a * (c * 3.0d0))) / (b + sqrt(((b ** 2.0d0) + ((-3.0d0) * (a * 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)) - (a * (c * 3.0))) / (b + Math.sqrt((Math.pow(b, 2.0) + (-3.0 * (a * c)))))) / (a * 3.0);
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.5%

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

   1. expm1-log1p-u 58.5%

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

   2. expm1-undefine 55.5%

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

   3. associate-*l* 55.5%

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

4. Applied egg-rr 55.5%

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

   1. flip-+ 55.2%

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

   2. pow2 55.2%

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

   3. add-sqr-sqrt 56.6%

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

   4. pow2 56.6%

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

   5. expm1-define 58.4%

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

   6. expm1-log1p-u 58.5%

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

   7. pow2 58.5%

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

   8. expm1-define 59.9%

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

   9. expm1-log1p-u 59.9%

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

6. Applied egg-rr 59.9%

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

   1. associate--r- 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l* 99.2%

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

   4. cancel-sign-sub-inv 99.2%

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

   5. metadata-eval 99.2%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 2: 90.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.8)
   (/
    (/
     (- (fma b b (* c (* a -3.0))) (pow b 2.0))
     (+ b (sqrt (fma b b (* -3.0 (* a c))))))
    (* a 3.0))
   (/
    1.0
    (*
     b
     (-
      (fma
       -3.0
       (/ (* (* c (pow a 2.0)) -0.375) (pow b 4.0))
       (* 1.5 (/ a (pow b 2.0))))
      (/ 2.0 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.8) {
		tmp = ((fma(b, b, (c * (a * -3.0))) - pow(b, 2.0)) / (b + sqrt(fma(b, b, (-3.0 * (a * c)))))) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (fma(-3.0, (((c * pow(a, 2.0)) * -0.375) / pow(b, 4.0)), (1.5 * (a / pow(b, 2.0)))) - (2.0 / c)));
	}
	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)) <= -0.8)
		tmp = Float64(Float64(Float64(fma(b, b, Float64(c * Float64(a * -3.0))) - (b ^ 2.0)) / Float64(b + sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))))) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(fma(-3.0, Float64(Float64(Float64(c * (a ^ 2.0)) * -0.375) / (b ^ 4.0)), Float64(1.5 * Float64(a / (b ^ 2.0)))) - Float64(2.0 / c))));
	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], -0.8], N[(N[(N[(N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(-3.0 * N[(N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.80000000000000004

   1. Initial program 84.2%

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

      1. expm1-log1p-u 84.0%

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

      2. expm1-undefine 78.8%

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

      3. associate-*l* 78.8%

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

   4. Applied egg-rr 78.8%

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

      1. flip-+ 78.7%

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

      2. pow2 78.7%

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

      3. add-sqr-sqrt 79.9%

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

      4. pow2 79.9%

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

      5. expm1-define 82.0%

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

      6. expm1-log1p-u 82.2%

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

      7. pow2 82.2%

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

      8. expm1-define 85.6%

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

      9. expm1-log1p-u 85.6%

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

   6. Applied egg-rr 85.6%

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

      1. unpow2 85.6%

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

      2. sqr-neg 85.6%

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

      3. unpow2 85.6%

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

      4. unpow2 85.6%

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

      5. fma-neg 85.4%

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

      6. distribute-lft-neg-in 85.4%

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

      7. metadata-eval 85.4%

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

      8. associate-*r* 85.4%

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

      9. *-commutative 85.4%

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

      10. unpow2 85.4%

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

      11. fma-neg 85.3%

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

      12. distribute-lft-neg-in 85.3%

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

      13. metadata-eval 85.3%

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

      14. associate-*r* 85.3%

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

      15. *-commutative 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in a around 0 85.3%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 52.1%

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

      1. expm1-log1p-u 52.1%

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

      2. expm1-undefine 49.7%

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

      3. associate-*l* 49.7%

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

   4. Applied egg-rr 49.7%

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

      1. clear-num 49.7%

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

      2. inv-pow 49.7%

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

      3. *-commutative 49.7%

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

      4. neg-mul-1 49.7%

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

      5. metadata-eval 49.7%

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

      6. fma-define 49.7%

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

      7. metadata-eval 49.7%

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

      8. pow2 49.7%

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

      9. expm1-define 52.1%

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

      10. expm1-log1p-u 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. unpow-1 52.1%

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

      2. *-commutative 52.1%

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

      3. *-lft-identity 52.1%

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

      4. times-frac 52.1%

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

      5. metadata-eval 52.1%

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

      6. fma-undefine 52.1%

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

      7. *-commutative 52.1%

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

      8. fma-define 52.1%

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

      9. unpow2 52.1%

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

      10. fma-neg 52.2%

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

      11. distribute-lft-neg-in 52.2%

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

      12. metadata-eval 52.2%

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

      13. associate-*r* 52.2%

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

      14. *-commutative 52.2%

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

   8. Simplified 52.2%

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

   9. Taylor expanded in b around inf 91.5%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. fma-define 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\mathsf{fma}\left(-3, \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right)} - 2 \cdot \frac{1}{c}\right)} \]

       2. distribute-rgt-out 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\color{blue}{\left({a}^{2} \cdot c\right) \cdot \left(-0.75 + 0.375\right)}}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       3. *-commutative 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\color{blue}{\left(c \cdot {a}^{2}\right)} \cdot \left(-0.75 + 0.375\right)}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       4. metadata-eval 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot \color{blue}{-0.375}}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       5. associate-*r/ 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \color{blue}{\frac{2 \cdot 1}{c}}\right)} \]

       6. metadata-eval 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \frac{\color{blue}{2}}{c}\right)} \]

   11. Simplified 91.5%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \frac{2}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.8:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.8)
   (* 0.3333333333333333 (/ (fma b -1.0 (sqrt (fma b b (* c (* a -3.0))))) a))
   (/
    1.0
    (*
     b
     (-
      (fma
       -3.0
       (/ (* (* c (pow a 2.0)) -0.375) (pow b 4.0))
       (* 1.5 (/ a (pow b 2.0))))
      (/ 2.0 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.8) {
		tmp = 0.3333333333333333 * (fma(b, -1.0, sqrt(fma(b, b, (c * (a * -3.0))))) / a);
	} else {
		tmp = 1.0 / (b * (fma(-3.0, (((c * pow(a, 2.0)) * -0.375) / pow(b, 4.0)), (1.5 * (a / pow(b, 2.0)))) - (2.0 / c)));
	}
	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)) <= -0.8)
		tmp = Float64(0.3333333333333333 * Float64(fma(b, -1.0, sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))) / a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(fma(-3.0, Float64(Float64(Float64(c * (a ^ 2.0)) * -0.375) / (b ^ 4.0)), Float64(1.5 * Float64(a / (b ^ 2.0)))) - Float64(2.0 / c))));
	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], -0.8], N[(0.3333333333333333 * N[(N[(b * -1.0 + N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(-3.0 * N[(N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.80000000000000004

   1. Initial program 84.2%

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

      1. expm1-log1p-u 84.0%

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

      2. expm1-undefine 78.8%

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

      3. associate-*l* 78.8%

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

   4. Applied egg-rr 78.8%

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

      1. *-un-lft-identity 78.8%

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

      2. neg-mul-1 78.8%

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

      3. metadata-eval 78.8%

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

      4. fma-define 78.8%

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

      5. metadata-eval 78.8%

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

      6. pow2 78.8%

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

      7. expm1-define 84.0%

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

      8. expm1-log1p-u 84.2%

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

      9. *-commutative 84.2%

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

   6. Applied egg-rr 84.2%

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

      1. associate-*r/ 84.2%

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

      2. *-commutative 84.2%

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

      3. times-frac 84.1%

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

      4. metadata-eval 84.1%

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

      5. fma-undefine 84.1%

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

      6. *-commutative 84.1%

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

      7. fma-define 84.1%

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

      8. unpow2 84.1%

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

      9. fma-neg 84.3%

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

      10. distribute-lft-neg-in 84.3%

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

      11. metadata-eval 84.3%

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

      12. associate-*r* 84.4%

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

      13. *-commutative 84.4%

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

   8. Simplified 84.4%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 52.1%

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

      1. expm1-log1p-u 52.1%

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

      2. expm1-undefine 49.7%

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

      3. associate-*l* 49.7%

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

   4. Applied egg-rr 49.7%

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

      1. clear-num 49.7%

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

      2. inv-pow 49.7%

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

      3. *-commutative 49.7%

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

      4. neg-mul-1 49.7%

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

      5. metadata-eval 49.7%

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

      6. fma-define 49.7%

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

      7. metadata-eval 49.7%

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

      8. pow2 49.7%

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

      9. expm1-define 52.1%

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

      10. expm1-log1p-u 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. unpow-1 52.1%

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

      2. *-commutative 52.1%

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

      3. *-lft-identity 52.1%

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

      4. times-frac 52.1%

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

      5. metadata-eval 52.1%

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

      6. fma-undefine 52.1%

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

      7. *-commutative 52.1%

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

      8. fma-define 52.1%

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

      9. unpow2 52.1%

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

      10. fma-neg 52.2%

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

      11. distribute-lft-neg-in 52.2%

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

      12. metadata-eval 52.2%

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

      13. associate-*r* 52.2%

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

      14. *-commutative 52.2%

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

   8. Simplified 52.2%

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

   9. Taylor expanded in b around inf 91.5%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. fma-define 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\mathsf{fma}\left(-3, \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right)} - 2 \cdot \frac{1}{c}\right)} \]

       2. distribute-rgt-out 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\color{blue}{\left({a}^{2} \cdot c\right) \cdot \left(-0.75 + 0.375\right)}}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       3. *-commutative 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\color{blue}{\left(c \cdot {a}^{2}\right)} \cdot \left(-0.75 + 0.375\right)}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       4. metadata-eval 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot \color{blue}{-0.375}}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       5. associate-*r/ 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \color{blue}{\frac{2 \cdot 1}{c}}\right)} \]

       6. metadata-eval 91.5%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \frac{\color{blue}{2}}{c}\right)} \]

   11. Simplified 91.5%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \frac{2}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.8:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.8:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.8)
   (* 0.3333333333333333 (/ (fma b -1.0 (sqrt (fma b b (* c (* a -3.0))))) a))
   (*
    c
    (+
     (*
      c
      (+
       (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))
       (* -0.375 (/ a (pow b 3.0)))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.8) {
		tmp = 0.3333333333333333 * (fma(b, -1.0, sqrt(fma(b, b, (c * (a * -3.0))))) / a);
	} else {
		tmp = c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.8)
		tmp = Float64(0.3333333333333333 * Float64(fma(b, -1.0, sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))) / a));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.8], N[(0.3333333333333333 * N[(N[(b * -1.0 + N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.80000000000000004

   1. Initial program 84.2%

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

      1. expm1-log1p-u 84.0%

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

      2. expm1-undefine 78.8%

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

      3. associate-*l* 78.8%

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

   4. Applied egg-rr 78.8%

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

      1. *-un-lft-identity 78.8%

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

      2. neg-mul-1 78.8%

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

      3. metadata-eval 78.8%

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

      4. fma-define 78.8%

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

      5. metadata-eval 78.8%

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

      6. pow2 78.8%

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

      7. expm1-define 84.0%

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

      8. expm1-log1p-u 84.2%

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

      9. *-commutative 84.2%

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

   6. Applied egg-rr 84.2%

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

      1. associate-*r/ 84.2%

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

      2. *-commutative 84.2%

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

      3. times-frac 84.1%

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

      4. metadata-eval 84.1%

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

      5. fma-undefine 84.1%

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

      6. *-commutative 84.1%

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

      7. fma-define 84.1%

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

      8. unpow2 84.1%

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

      9. fma-neg 84.3%

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

      10. distribute-lft-neg-in 84.3%

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

      11. metadata-eval 84.3%

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

      12. associate-*r* 84.4%

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

      13. *-commutative 84.4%

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

   8. Simplified 84.4%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 52.1%

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

   3. Taylor expanded in c around 0 91.2%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.8:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0008:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} + 2 \cdot \frac{-1}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0008)
   (* 0.3333333333333333 (/ (fma b -1.0 (sqrt (fma b b (* c (* a -3.0))))) a))
   (/ 1.0 (* b (+ (* 1.5 (/ a (pow b 2.0))) (* 2.0 (/ -1.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0008) {
		tmp = 0.3333333333333333 * (fma(b, -1.0, sqrt(fma(b, b, (c * (a * -3.0))))) / a);
	} else {
		tmp = 1.0 / (b * ((1.5 * (a / pow(b, 2.0))) + (2.0 * (-1.0 / c))));
	}
	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)) <= -0.0008)
		tmp = Float64(0.3333333333333333 * Float64(fma(b, -1.0, sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))) / a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(1.5 * Float64(a / (b ^ 2.0))) + Float64(2.0 * Float64(-1.0 / c)))));
	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], -0.0008], N[(0.3333333333333333 * N[(N[(b * -1.0 + N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / c), $MachinePrecision]), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -8.00000000000000038e-4

   1. Initial program 79.7%

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

      1. expm1-log1p-u 79.6%

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

      2. expm1-undefine 73.3%

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

      3. associate-*l* 73.3%

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

   4. Applied egg-rr 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. neg-mul-1 73.3%

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

      3. metadata-eval 73.3%

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

      4. fma-define 73.3%

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

      5. metadata-eval 73.3%

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

      6. pow2 73.3%

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

      7. expm1-define 79.6%

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

      8. expm1-log1p-u 79.7%

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

      9. *-commutative 79.7%

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

   6. Applied egg-rr 79.7%

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

      1. associate-*r/ 79.7%

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

      2. *-commutative 79.7%

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

      3. times-frac 79.7%

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

      4. metadata-eval 79.7%

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

      5. fma-undefine 79.7%

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

      6. *-commutative 79.7%

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

      7. fma-define 79.7%

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

      8. unpow2 79.7%

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

      9. fma-neg 79.9%

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

      10. distribute-lft-neg-in 79.9%

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

      11. metadata-eval 79.9%

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

      12. associate-*r* 79.9%

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

      13. *-commutative 79.9%

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

   8. Simplified 79.9%

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

   if -8.00000000000000038e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.0%

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

      1. expm1-log1p-u 46.0%

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

      2. expm1-undefine 45.0%

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

      3. associate-*l* 45.0%

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

   4. Applied egg-rr 45.0%

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

      1. clear-num 45.0%

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

      2. inv-pow 45.0%

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

      3. *-commutative 45.0%

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

      4. neg-mul-1 45.0%

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

      5. metadata-eval 45.0%

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

      6. fma-define 45.0%

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

      7. metadata-eval 45.0%

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

      8. pow2 45.0%

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

      9. expm1-define 46.0%

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

      10. expm1-log1p-u 46.0%

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

   6. Applied egg-rr 46.0%

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

      1. unpow-1 46.0%

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

      2. *-commutative 46.0%

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

      3. *-lft-identity 46.0%

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

      4. times-frac 46.0%

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

      5. metadata-eval 46.0%

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

      6. fma-undefine 46.0%

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

      7. *-commutative 46.0%

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

      8. fma-define 46.0%

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

      9. unpow2 46.0%

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

      10. fma-neg 46.0%

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

      11. distribute-lft-neg-in 46.0%

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

      12. metadata-eval 46.0%

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

      13. associate-*r* 46.0%

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

      14. *-commutative 46.0%

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

   8. Simplified 46.0%

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

   9. Taylor expanded in b around inf 89.8%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0008:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} + 2 \cdot \frac{-1}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((pow(b, 2.0) - pow(-b, 2.0)) - (3.0 * (a * c))) / (b + sqrt((pow(b, 2.0) + (-3.0 * (a * 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)) - (3.0d0 * (a * c))) / (b + sqrt(((b ** 2.0d0) + ((-3.0d0) * (a * 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)) - (3.0 * (a * c))) / (b + Math.sqrt((Math.pow(b, 2.0) + (-3.0 * (a * c)))))) / (a * 3.0);
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.5%

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

   1. expm1-log1p-u 58.5%

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

   2. expm1-undefine 55.5%

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

   3. associate-*l* 55.5%

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

4. Applied egg-rr 55.5%

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

   1. flip-+ 55.2%

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

   2. pow2 55.2%

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

   3. add-sqr-sqrt 56.6%

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

   4. pow2 56.6%

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

   5. expm1-define 58.4%

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

   6. expm1-log1p-u 58.5%

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

   7. pow2 58.5%

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

   8. expm1-define 59.9%

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

   9. expm1-log1p-u 59.9%

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

6. Applied egg-rr 59.9%

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

   1. associate--r- 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l* 99.2%

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

   4. cancel-sign-sub-inv 99.2%

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

   5. metadata-eval 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in a around 0 99.2%

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

    1. *-commutative 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 7: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0008:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} + 2 \cdot \frac{-1}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0008)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ 1.0 (* b (+ (* 1.5 (/ a (pow b 2.0))) (* 2.0 (/ -1.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0008) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * ((1.5 * (a / pow(b, 2.0))) + (2.0 * (-1.0 / c))));
	}
	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)) <= -0.0008)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(1.5 * Float64(a / (b ^ 2.0))) + Float64(2.0 * Float64(-1.0 / c)))));
	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], -0.0008], 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[(1.0 / N[(b * N[(N[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / c), $MachinePrecision]), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -8.00000000000000038e-4

   1. Initial program 79.7%

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

      1. /-rgt-identity 79.7%

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

      2. metadata-eval 79.7%

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

   3. Simplified 79.9%

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

   if -8.00000000000000038e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.0%

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

      1. expm1-log1p-u 46.0%

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

      2. expm1-undefine 45.0%

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

      3. associate-*l* 45.0%

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

   4. Applied egg-rr 45.0%

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

      1. clear-num 45.0%

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

      2. inv-pow 45.0%

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

      3. *-commutative 45.0%

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

      4. neg-mul-1 45.0%

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

      5. metadata-eval 45.0%

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

      6. fma-define 45.0%

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

      7. metadata-eval 45.0%

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

      8. pow2 45.0%

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

      9. expm1-define 46.0%

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

      10. expm1-log1p-u 46.0%

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

   6. Applied egg-rr 46.0%

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

      1. unpow-1 46.0%

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

      2. *-commutative 46.0%

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

      3. *-lft-identity 46.0%

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

      4. times-frac 46.0%

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

      5. metadata-eval 46.0%

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

      6. fma-undefine 46.0%

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

      7. *-commutative 46.0%

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

      8. fma-define 46.0%

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

      9. unpow2 46.0%

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

      10. fma-neg 46.0%

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

      11. distribute-lft-neg-in 46.0%

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

      12. metadata-eval 46.0%

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

      13. associate-*r* 46.0%

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

      14. *-commutative 46.0%

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

   8. Simplified 46.0%

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

   9. Taylor expanded in b around inf 89.8%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0008:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} + 2 \cdot \frac{-1}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.0008:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} + 2 \cdot \frac{-1}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.0008)
     t_0
     (/ 1.0 (* b (+ (* 1.5 (/ a (pow b 2.0))) (* 2.0 (/ -1.0 c))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0008) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (b * ((1.5 * (a / pow(b, 2.0))) + (2.0 * (-1.0 / c))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.0008d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (b * ((1.5d0 * (a / (b ** 2.0d0))) + (2.0d0 * ((-1.0d0) / c))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0008) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (b * ((1.5 * (a / Math.pow(b, 2.0))) + (2.0 * (-1.0 / c))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -0.0008:
		tmp = t_0
	else:
		tmp = 1.0 / (b * ((1.5 * (a / math.pow(b, 2.0))) + (2.0 * (-1.0 / c))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.0008)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(1.5 * Float64(a / (b ^ 2.0))) + Float64(2.0 * Float64(-1.0 / c)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -0.0008)
		tmp = t_0;
	else
		tmp = 1.0 / (b * ((1.5 * (a / (b ^ 2.0))) + (2.0 * (-1.0 / c))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0008], t$95$0, N[(1.0 / N[(b * N[(N[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / c), $MachinePrecision]), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -8.00000000000000038e-4

   1. Initial program 79.7%

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

   if -8.00000000000000038e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.0%

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

      1. expm1-log1p-u 46.0%

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

      2. expm1-undefine 45.0%

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

      3. associate-*l* 45.0%

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

   4. Applied egg-rr 45.0%

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

      1. clear-num 45.0%

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

      2. inv-pow 45.0%

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

      3. *-commutative 45.0%

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

      4. neg-mul-1 45.0%

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

      5. metadata-eval 45.0%

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

      6. fma-define 45.0%

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

      7. metadata-eval 45.0%

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

      8. pow2 45.0%

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

      9. expm1-define 46.0%

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

      10. expm1-log1p-u 46.0%

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

   6. Applied egg-rr 46.0%

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

      1. unpow-1 46.0%

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

      2. *-commutative 46.0%

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

      3. *-lft-identity 46.0%

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

      4. times-frac 46.0%

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

      5. metadata-eval 46.0%

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

      6. fma-undefine 46.0%

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

      7. *-commutative 46.0%

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

      8. fma-define 46.0%

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

      9. unpow2 46.0%

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

      10. fma-neg 46.0%

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

      11. distribute-lft-neg-in 46.0%

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

      12. metadata-eval 46.0%

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

      13. associate-*r* 46.0%

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

      14. *-commutative 46.0%

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

   8. Simplified 46.0%

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

   9. Taylor expanded in b around inf 89.8%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0008:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} + 2 \cdot \frac{-1}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / (b * ((1.5d0 * (a / (b ** 2.0d0))) + (2.0d0 * ((-1.0d0) / c))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.5%

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

   1. expm1-log1p-u 58.5%

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

   2. expm1-undefine 55.5%

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

   3. associate-*l* 55.5%

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

4. Applied egg-rr 55.5%

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

   1. clear-num 55.5%

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

   2. inv-pow 55.5%

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

   3. *-commutative 55.5%

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

   4. neg-mul-1 55.5%

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

   5. metadata-eval 55.5%

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

   6. fma-define 55.5%

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

   7. metadata-eval 55.5%

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

   8. pow2 55.5%

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

   9. expm1-define 58.5%

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

   10. expm1-log1p-u 58.5%

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

6. Applied egg-rr 58.5%

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

   1. unpow-1 58.5%

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

   2. *-commutative 58.5%

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

   3. *-lft-identity 58.5%

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

   4. times-frac 58.5%

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

   5. metadata-eval 58.5%

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

   6. fma-undefine 58.5%

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

   7. *-commutative 58.5%

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

   8. fma-define 58.5%

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

   9. unpow2 58.5%

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

   10. fma-neg 58.6%

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

   11. distribute-lft-neg-in 58.6%

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

   12. metadata-eval 58.6%

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

   13. associate-*r* 58.6%

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

   14. *-commutative 58.6%

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

8. Simplified 58.6%

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

9. Taylor expanded in b around inf 80.1%

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

10. Final simplification 80.1%

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

Alternative 10: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in b around inf 79.5%

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

4. Taylor expanded in c around 0 79.5%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
5. Step-by-step derivation

   1. associate-/l* 79.5%

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

   2. associate-*r/ 79.5%

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

   3. metadata-eval 79.5%

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

6. Simplified 79.5%

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

Alternative 11: 64.9% 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 58.5%

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

3. Taylor expanded in b around inf 61.9%

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

   1. associate-*r/ 61.9%

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

   2. *-commutative 61.9%

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

5. Simplified 61.9%

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

Reproduce

herbie shell --seed 2024086 
(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)))