Cubic critical, medium range

Percentage Accurate: 31.5% → 99.2%

Time: 15.3s

Alternatives: 8

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.5% 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: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

   1. expm1-log1p-u 32.2%

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

   2. expm1-undefine 27.2%

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

   3. associate-*l* 27.2%

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

4. Applied egg-rr 27.2%

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

   1. flip-+ 27.1%

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

6. Applied egg-rr 33.0%

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

   1. associate--r- 99.1%

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

   2. unpow2 99.1%

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

   3. unpow2 99.1%

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

   4. difference-of-squares 99.1%

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

   5. neg-mul-1 99.1%

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

   6. distribute-lft1-in 99.1%

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

   7. metadata-eval 99.1%

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

   8. mul0-lft 99.1%

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

   9. *-commutative 99.1%

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

   10. associate-*l* 99.1%

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

   11. *-commutative 99.1%

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

   12. unpow2 99.1%

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

   13. fma-neg 99.1%

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

   14. *-commutative 99.1%

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

   15. distribute-rgt-neg-in 99.1%

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

   16. metadata-eval 99.1%

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

   17. rem-square-sqrt 0.0%

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

   18. unpow2 0.0%

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

   19. associate-*r* 0.0%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 2: 90.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5

   1. Initial program 78.8%

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

      1. /-rgt-identity 78.8%

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

      2. metadata-eval 78.8%

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

   3. Simplified 78.9%

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

   if -5 < (/.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 26.3%

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

   3. Taylor expanded in a around 0 93.9%

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

4. Final simplification 92.2%

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

Alternative 3: 90.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) 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 (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -5.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - 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

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], -5.0], 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[(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 (/.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)) < -5

   1. Initial program 78.8%

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

      1. /-rgt-identity 78.8%

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

      2. metadata-eval 78.8%

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

   3. Simplified 78.9%

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

   if -5 < (/.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 26.3%

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

   3. Taylor expanded in c around 0 93.7%

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

   4. Taylor expanded in a around inf 93.6%

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

   5. Taylor expanded in c around 0 93.7%

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

      1. associate-/l* 93.7%

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

      2. associate-*r/ 93.7%

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

      3. metadata-eval 93.7%

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

   7. Simplified 93.7%

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

4. Final simplification 92.0%

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

Alternative 4: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -5.0)
     t_0
     (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 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 <= -5.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-5.0d0)) then
        tmp = t_0
    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 t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = t_0;
	} else {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (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 <= -5.0:
		tmp = t_0
	else:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (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 <= -5.0)
		tmp = t_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)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -5.0)
		tmp = t_0;
	else
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (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, -5.0], t$95$0, 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 (/.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)) < -5

   1. Initial program 78.8%

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

   if -5 < (/.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 26.3%

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

   3. Taylor expanded in c around 0 93.7%

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

   4. Taylor expanded in a around inf 93.6%

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

   5. Taylor expanded in c around 0 93.7%

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

      1. associate-/l* 93.7%

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

      2. associate-*r/ 93.7%

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

      3. metadata-eval 93.7%

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

   7. Simplified 93.7%

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

4. Final simplification 92.0%

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

Alternative 5: 90.4% 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 32.2%

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

3. Taylor expanded in c around 0 90.2%

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

4. Taylor expanded in a around inf 90.2%

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

5. Taylor expanded in c around 0 90.2%

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

   1. associate-/l* 90.2%

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

   2. associate-*r/ 90.2%

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

   3. metadata-eval 90.2%

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

7. Simplified 90.2%

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

Alternative 6: 81.1% 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 32.2%

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

3. Taylor expanded in b around inf 80.7%

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

   1. associate-*r/ 80.7%

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

   2. *-commutative 80.7%

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

5. Simplified 80.7%

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

Alternative 7: 80.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

3. Taylor expanded in c around 0 90.2%

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

4. Taylor expanded in a around 0 80.5%

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

Alternative 8: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

3. Taylor expanded in b around 0 32.2%

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

   1. cancel-sign-sub-inv 32.2%

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

   2. metadata-eval 32.2%

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

   3. +-commutative 32.2%

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

   4. *-commutative 32.2%

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

   5. associate-*r* 32.2%

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

   6. fma-define 32.2%

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

5. Simplified 32.2%

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

   1. div-inv 32.2%

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

   2. neg-mul-1 32.2%

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

   3. fma-define 32.2%

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

   4. *-commutative 32.2%

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

7. Applied egg-rr 32.2%

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

8. Taylor expanded in a around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

10. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))