Cubic critical, narrow range

Percentage Accurate: 55.3% → 98.9%

Time: 18.2s

Alternatives: 11

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((b ** 2.0d0) - (-b ** 2.0d0)) - (c * (a * 3.0d0))) / (b + sqrt(((b ** 2.0d0) + ((-3.0d0) * (c * a)))))) / (sqrt((a * 3.0d0)) ** 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.0%

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

   1. add-sqr-sqrt 58.0%

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

   2. pow2 58.0%

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

4. Applied egg-rr 58.0%

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

   1. log1p-expm1-u 50.3%

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

   2. neg-mul-1 50.3%

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

   3. fma-define 50.3%

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

   4. pow2 50.3%

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

   5. *-commutative 50.3%

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

   6. *-commutative 50.3%

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

6. Applied egg-rr 50.3%

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

   1. log1p-expm1-u 58.0%

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

   2. fma-undefine 58.0%

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

   3. neg-mul-1 58.0%

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

   4. pow2 58.0%

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

   5. *-commutative 58.0%

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

   6. *-commutative 58.0%

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

   7. flip-+ 57.8%

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

   8. pow2 57.8%

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

   9. add-sqr-sqrt 59.1%

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

   10. *-commutative 59.1%

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

   11. pow2 59.1%

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

   12. *-commutative 59.1%

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

8. Applied egg-rr 59.1%

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

   1. associate--r- 99.0%

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

   2. associate-*r* 99.0%

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

   3. *-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. metadata-eval 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 2: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0068:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0068)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* -3.0 (* c a)))) b) a))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))
      (* -0.375 (/ (pow c 2.0) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0068) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (-3.0 * (c * a)))) - b) / a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * (pow(c, 2.0) / pow(b, 3.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0068)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(c * a)))) - b) / a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.0068], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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)) < -0.00679999999999999962

   1. Initial program 81.3%

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

      1. add-log-exp 66.1%

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

      2. *-commutative 66.1%

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

      3. exp-prod 65.1%

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

   4. Applied egg-rr 65.1%

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

      1. pow-exp 66.1%

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

      2. add-log-exp 81.3%

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

      3. *-un-lft-identity 81.3%

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

      4. neg-mul-1 81.3%

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

      5. *-commutative 81.3%

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

      6. *-commutative 81.3%

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

      7. pow2 81.3%

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

      8. fma-undefine 81.3%

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

   6. Applied egg-rr 81.3%

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

      1. associate-*r/ 81.3%

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

      2. *-commutative 81.3%

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

      3. times-frac 81.3%

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

      4. metadata-eval 81.3%

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

   8. Simplified 81.4%

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

   if -0.00679999999999999962 < (/.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 46.6%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in a around 0 92.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0068:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0068:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0068)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* -3.0 (* c a)))) b) a))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (*
      (pow c 3.0)
      (- (* -0.5625 (/ a (pow b 5.0))) (/ 0.375 (* c (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0068) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (-3.0 * (c * a)))) - b) / a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * (pow(c, 3.0) * ((-0.5625 * (a / pow(b, 5.0))) - (0.375 / (c * pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0068)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(c * a)))) - b) / a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64((c ^ 3.0) * Float64(Float64(-0.5625 * Float64(a / (b ^ 5.0))) - Float64(0.375 / Float64(c * (b ^ 3.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.0068], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(-0.5625 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $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)) < -0.00679999999999999962

   1. Initial program 81.3%

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

      1. add-log-exp 66.1%

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

      2. *-commutative 66.1%

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

      3. exp-prod 65.1%

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

   4. Applied egg-rr 65.1%

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

      1. pow-exp 66.1%

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

      2. add-log-exp 81.3%

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

      3. *-un-lft-identity 81.3%

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

      4. neg-mul-1 81.3%

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

      5. *-commutative 81.3%

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

      6. *-commutative 81.3%

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

      7. pow2 81.3%

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

      8. fma-undefine 81.3%

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

   6. Applied egg-rr 81.3%

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

      1. associate-*r/ 81.3%

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

      2. *-commutative 81.3%

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

      3. times-frac 81.3%

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

      4. metadata-eval 81.3%

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

   8. Simplified 81.4%

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

   if -0.00679999999999999962 < (/.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 46.6%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in a around 0 92.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]

   6. Taylor expanded in c around inf 92.8%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \color{blue}{\left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 92.8%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \color{blue}{\frac{0.375 \cdot 1}{{b}^{3} \cdot c}}\right)\right) \]

      2. metadata-eval 92.8%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \frac{\color{blue}{0.375}}{{b}^{3} \cdot c}\right)\right) \]

      3. *-commutative 92.8%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \frac{0.375}{\color{blue}{c \cdot {b}^{3}}}\right)\right) \]

   8. Simplified 92.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0068:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0 \cdot \left(b + b\right) - c \cdot \left(a \cdot 3\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}}{{\left(\sqrt{a \cdot 3}\right)}^{2}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return (((0.0 * (b + b)) - (c * (a * 3.0))) / (b + sqrt(fma(b, b, (-3.0 * (c * a)))))) / pow(sqrt((a * 3.0)), 2.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(0.0 * Float64(b + b)) - Float64(c * Float64(a * 3.0))) / Float64(b + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a)))))) / (sqrt(Float64(a * 3.0)) ^ 2.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(0.0 * N[(b + b), $MachinePrecision]), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sqrt[N[(a * 3.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.0%

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

   1. add-sqr-sqrt 58.0%

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

   2. pow2 58.0%

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

4. Applied egg-rr 58.0%

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

   1. log1p-expm1-u 50.3%

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

   2. neg-mul-1 50.3%

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

   3. fma-define 50.3%

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

   4. pow2 50.3%

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

   5. *-commutative 50.3%

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

   6. *-commutative 50.3%

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

6. Applied egg-rr 50.3%

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

   1. log1p-expm1-u 58.0%

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

   2. fma-undefine 58.0%

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

   3. neg-mul-1 58.0%

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

   4. pow2 58.0%

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

   5. *-commutative 58.0%

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

   6. *-commutative 58.0%

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

   7. flip-+ 57.8%

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

   8. pow2 57.8%

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

   9. add-sqr-sqrt 59.1%

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

   10. *-commutative 59.1%

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

   11. pow2 59.1%

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

   12. *-commutative 59.1%

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

8. Applied egg-rr 59.1%

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

   1. associate--r- 99.0%

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

   2. unpow2 99.0%

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

   3. unpow2 99.0%

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

   4. difference-of-squares 99.0%

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

   5. neg-mul-1 99.0%

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

   6. distribute-lft1-in 99.0%

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

   7. metadata-eval 99.0%

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

   8. mul0-lft 99.0%

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

   9. associate-*r* 99.0%

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

   10. *-commutative 99.0%

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

   11. *-commutative 99.0%

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

   12. unpow2 99.0%

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

   13. *-commutative 99.0%

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

   14. *-commutative 99.0%

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

   15. associate-*r* 99.0%

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

   16. fmm-def 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

    \[\leadsto \frac{\frac{0 \cdot \left(b + b\right) - c \cdot \left(a \cdot 3\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}}{{\left(\sqrt{a \cdot 3}\right)}^{2}} \]
12. Add Preprocessing

Alternative 5: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0034:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0034)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* -3.0 (* c a)))) b) a))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0034) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (-3.0 * (c * a)))) - b) / a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0034)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(c * a)))) - b) / a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.0034], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.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)) < -0.00339999999999999981

   1. Initial program 81.0%

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

      1. add-log-exp 63.9%

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

      2. *-commutative 63.9%

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

      3. exp-prod 63.0%

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

   4. Applied egg-rr 63.0%

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

      1. pow-exp 63.9%

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

      2. add-log-exp 81.0%

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

      3. *-un-lft-identity 81.0%

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

      4. neg-mul-1 81.0%

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

      5. *-commutative 81.0%

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

      6. *-commutative 81.0%

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

      7. pow2 81.0%

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

      8. fma-undefine 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. associate-*r/ 81.0%

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

      2. *-commutative 81.0%

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

      3. times-frac 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.1%

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

   if -0.00339999999999999981 < (/.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 45.5%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in a around 0 90.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0034:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0034:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0034)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* -3.0 (* c a)))) b) a))
   (/ (fma -0.5 c (* -0.375 (* a (pow (/ c b) 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0034) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (-3.0 * (c * a)))) - b) / a);
	} else {
		tmp = fma(-0.5, c, (-0.375 * (a * pow((c / b), 2.0)))) / 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)) <= -0.0034)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(c * a)))) - b) / a));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / 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], -0.0034], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00339999999999999981

   1. Initial program 81.0%

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

      1. add-log-exp 63.9%

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

      2. *-commutative 63.9%

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

      3. exp-prod 63.0%

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

   4. Applied egg-rr 63.0%

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

      1. pow-exp 63.9%

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

      2. add-log-exp 81.0%

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

      3. *-un-lft-identity 81.0%

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

      4. neg-mul-1 81.0%

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

      5. *-commutative 81.0%

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

      6. *-commutative 81.0%

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

      7. pow2 81.0%

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

      8. fma-undefine 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. associate-*r/ 81.0%

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

      2. *-commutative 81.0%

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

      3. times-frac 81.0%

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

      4. metadata-eval 81.0%

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

   8. Simplified 81.1%

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

   if -0.00339999999999999981 < (/.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 45.5%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in a around 0 89.6%

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

   6. Taylor expanded in b around inf 90.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   7. Step-by-step derivation

      1. fma-define 90.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      2. associate-*r/ 90.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right)}{b} \]

      3. unpow2 90.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)\right)}{b} \]

      4. unpow2 90.0%

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

      5. times-frac 90.0%

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

      6. unpow1 90.0%

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

      7. pow-plus 90.0%

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

      8. metadata-eval 90.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right)\right)}{b} \]

   8. Simplified 90.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0034:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.0034:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.0034)
     t_0
     (/ (fma -0.5 c (* -0.375 (* a (pow (/ c b) 2.0)))) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0034) {
		tmp = t_0;
	} else {
		tmp = fma(-0.5, c, (-0.375 * (a * pow((c / b), 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.0034)
		tmp = t_0;
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b);
	end
	return 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, -0.0034], t$95$0, N[(N[(-0.5 * c + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00339999999999999981

   1. Initial program 81.0%

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

   if -0.00339999999999999981 < (/.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 45.5%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in a around 0 89.6%

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

   6. Taylor expanded in b around inf 90.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   7. Step-by-step derivation

      1. fma-define 90.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      2. associate-*r/ 90.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right)}{b} \]

      3. unpow2 90.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)\right)}{b} \]

      4. unpow2 90.0%

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

      5. times-frac 90.0%

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

      6. unpow1 90.0%

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

      7. pow-plus 90.0%

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

      8. metadata-eval 90.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right)\right)}{b} \]

   8. Simplified 90.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0034:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.0034:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.0034)
     t_0
     (/ (* c (- (* -0.375 (/ (* c a) (pow b 2.0))) 0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0034) {
		tmp = t_0;
	} else {
		tmp = (c * ((-0.375 * ((c * a) / pow(b, 2.0))) - 0.5)) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.0034d0)) then
        tmp = t_0
    else
        tmp = (c * (((-0.375d0) * ((c * a) / (b ** 2.0d0))) - 0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0034) {
		tmp = t_0;
	} else {
		tmp = (c * ((-0.375 * ((c * a) / Math.pow(b, 2.0))) - 0.5)) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -0.0034:
		tmp = t_0
	else:
		tmp = (c * ((-0.375 * ((c * a) / math.pow(b, 2.0))) - 0.5)) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.0034)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(c * a) / (b ^ 2.0))) - 0.5)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -0.0034)
		tmp = t_0;
	else
		tmp = (c * ((-0.375 * ((c * a) / (b ^ 2.0))) - 0.5)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0034], t$95$0, N[(N[(c * N[(N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00339999999999999981

   1. Initial program 81.0%

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

   if -0.00339999999999999981 < (/.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 45.5%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in b around inf 95.1%

      \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(\left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}}{b} \]

      2. *-un-lft-identity 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\color{blue}{1 \cdot \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)} + -0.5 \cdot c\right)}{b} \]

      3. fma-define 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{fma}\left(1, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}, -0.5 \cdot c\right)}}{b} \]

   7. Applied egg-rr 95.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{6}}, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right), -0.5 \cdot c\right)}}{b} \]
   8. Step-by-step derivation

      1. fma-undefine 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(1 \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{6}}, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right) + -0.5 \cdot c\right)}}{b} \]

      2. *-lft-identity 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{6}}, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)} + -0.5 \cdot c\right)}{b} \]

      3. times-frac 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}}, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right) + -0.5 \cdot c\right)}{b} \]

      4. *-commutative 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\color{blue}{\left(c \cdot a\right)}}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, \frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right) + -0.5 \cdot c\right)}{b} \]

      5. associate-*r/ 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, \color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}\right) + -0.5 \cdot c\right)}{b} \]

      6. associate-*r/ 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right) + -0.5 \cdot c\right)}{b} \]

      7. unpow2 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, -0.375 \cdot \left(a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)\right) + -0.5 \cdot c\right)}{b} \]

      8. unpow2 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, -0.375 \cdot \left(a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)\right) + -0.5 \cdot c\right)}{b} \]

      9. times-frac 95.1%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, -0.375 \cdot \left(a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)\right) + -0.5 \cdot c\right)}{b} \]

      10. unpow1 95.1%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, -0.375 \cdot \left(a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right)\right)\right) + -0.5 \cdot c\right)}{b} \]

      11. pow-plus 95.1%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, -0.375 \cdot \left(a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}\right)\right) + -0.5 \cdot c\right)}{b} \]

      12. metadata-eval 95.1%

          \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right)\right) + -0.5 \cdot c\right)}{b} \]

   9. Simplified 95.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{6}}, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right) + -0.5 \cdot c\right)}}{b} \]

   10. Taylor expanded in c around 0 89.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0034:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 19.5)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 19.5) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 19.5d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = c * (((-0.375d0) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 19.5) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 19.5:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 19.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 19.5)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 19.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 19.5

   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. sqr-neg 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-*l* 79.8%

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

   3. Simplified 79.8%

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

   if 19.5 < b

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in c around 0 86.1%

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

      1. associate-/l* 86.1%

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

      2. associate-*r/ 86.1%

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

      3. metadata-eval 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.3% 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 58.0%

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

3. Simplified 58.0%

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

5. Taylor expanded in c around 0 79.4%

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

   1. associate-/l* 79.4%

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

   2. associate-*r/ 79.4%

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

   3. metadata-eval 79.4%

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

7. Simplified 79.4%

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

Alternative 11: 64.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.0%

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

3. Simplified 58.0%

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

5. Taylor expanded in b around inf 62.3%

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

Reproduce

herbie shell --seed 2024165 
(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)))