Cubic critical, medium range

Percentage Accurate: 30.9% → 99.4%

Time: 17.4s

Alternatives: 10

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: 30.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ \frac{\frac{\left({b}^{2} - {\left(-b\right)}^{2}\right) - t\_0}{b + \sqrt{{b}^{2} - t\_0}}}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))))
   (/
    (/
     (- (- (pow b 2.0) (pow (- b) 2.0)) t_0)
     (+ b (sqrt (- (pow b 2.0) t_0))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return (((pow(b, 2.0) - pow(-b, 2.0)) - t_0) / (b + sqrt((pow(b, 2.0) - t_0)))) / (a * 3.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = c * (a * 3.0d0)
    code = ((((b ** 2.0d0) - (-b ** 2.0d0)) - t_0) / (b + sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return (((Math.pow(b, 2.0) - Math.pow(-b, 2.0)) - t_0) / (b + Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
}

Python

def code(a, b, c):
	t_0 = c * (a * 3.0)
	return (((math.pow(b, 2.0) - math.pow(-b, 2.0)) - t_0) / (b + math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 3.0))
	return Float64(Float64(Float64(Float64((b ^ 2.0) - (Float64(-b) ^ 2.0)) - t_0) / Float64(b + sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (a * 3.0);
	tmp = ((((b ^ 2.0) - (-b ^ 2.0)) - t_0) / (b + sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
end

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. neg-sub0 31.1%

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

   2. flip-- 31.2%

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

   3. metadata-eval 31.2%

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

   4. pow2 31.2%

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

   5. add-sqr-sqrt 31.2%

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

   6. sqrt-prod 31.2%

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

   7. sqr-neg 31.2%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 31.2%

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

   14. sqr-neg 31.2%

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

   15. sqrt-prod 31.2%

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

   16. add-sqr-sqrt 31.2%

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

4. Applied egg-rr 31.2%

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

   1. neg-sub0 31.2%

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

6. Simplified 31.2%

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

   1. flip-+ 31.0%

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

8. Applied egg-rr 32.1%

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

   1. associate--r- 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -12:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -12.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -12.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -12.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -12.0], 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[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -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)) < -12

   1. Initial program 79.4%

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

   3. Simplified 79.5%

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

   if -12 < (/.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 24.9%

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

   3. Taylor expanded in a around 0 93.8%

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

   4. Taylor expanded in b around inf 94.2%

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

      1. +-commutative 94.2%

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

      2. fma-define 94.2%

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

      3. associate-/l* 94.2%

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

      4. unpow2 94.2%

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

      5. unpow2 94.2%

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

      6. times-frac 94.2%

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

      7. unpow1 94.2%

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

      8. pow-plus 94.2%

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

      9. metadata-eval 94.2%

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

      10. *-commutative 94.2%

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

   6. Simplified 94.2%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -12:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -12:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -12.0)
     t_0
     (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -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 <= -12.0) {
		tmp = t_0;
	} else {
		tmp = fma(-0.375, (a * pow((c / b), 2.0)), (c * -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 <= -12.0)
		tmp = t_0;
	else
		tmp = Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / 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, -12.0], t$95$0, N[(N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -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)) < -12

   1. Initial program 79.4%

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

   if -12 < (/.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 24.9%

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

   3. Taylor expanded in a around 0 93.8%

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

   4. Taylor expanded in b around inf 94.2%

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

      1. +-commutative 94.2%

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

      2. fma-define 94.2%

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

      3. associate-/l* 94.2%

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

      4. unpow2 94.2%

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

      5. unpow2 94.2%

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

      6. times-frac 94.2%

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

      7. unpow1 94.2%

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

      8. pow-plus 94.2%

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

      9. metadata-eval 94.2%

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

      10. *-commutative 94.2%

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

   6. Simplified 94.2%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -12:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -12:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{\left(c \cdot a\right) \cdot -0.375}{{b}^{3}} - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -12.0)
     t_0
     (* c (- (/ (* (* c a) -0.375) (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 <= -12.0) {
		tmp = t_0;
	} else {
		tmp = c * ((((c * a) * -0.375) / 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 <= (-12.0d0)) then
        tmp = t_0
    else
        tmp = c * ((((c * a) * (-0.375d0)) / (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 <= -12.0) {
		tmp = t_0;
	} else {
		tmp = c * ((((c * a) * -0.375) / 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 <= -12.0:
		tmp = t_0
	else:
		tmp = c * ((((c * a) * -0.375) / 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 <= -12.0)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(Float64(Float64(c * a) * -0.375) / (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 <= -12.0)
		tmp = t_0;
	else
		tmp = c * ((((c * a) * -0.375) / (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, -12.0], t$95$0, N[(c * N[(N[(N[(N[(c * a), $MachinePrecision] * -0.375), $MachinePrecision] / N[Power[b, 3.0], $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)) < -12

   1. Initial program 79.4%

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

   if -12 < (/.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 24.9%

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

   3. Taylor expanded in c around 0 96.7%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

   5. Simplified 96.7%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

   6. Taylor expanded in c around 0 94.0%

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

      1. associate-*r/ 94.0%

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

      2. *-commutative 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -12:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{\left(c \cdot a\right) \cdot -0.375}{{b}^{3}} - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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(Float64(c * a) * -3.0))))) / Float64(a * 3.0))
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[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

   1. neg-sub0 31.1%

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

   2. flip-- 31.2%

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

   3. metadata-eval 31.2%

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

   4. pow2 31.2%

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

   5. add-sqr-sqrt 31.2%

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

   6. sqrt-prod 31.2%

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

   7. sqr-neg 31.2%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 31.2%

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

   14. sqr-neg 31.2%

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

   15. sqrt-prod 31.2%

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

   16. add-sqr-sqrt 31.2%

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

4. Applied egg-rr 31.2%

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

   1. neg-sub0 31.2%

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

6. Simplified 31.2%

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

   1. flip-+ 31.0%

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

8. Applied egg-rr 32.1%

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

   1. associate--r- 99.5%

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

   2. unpow2 99.1%

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

   3. unpow2 99.5%

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

   4. difference-of-squares 99.5%

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

   5. +-commutative 99.5%

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

   6. neg-mul-1 99.5%

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

   7. distribute-rgt1-in 99.5%

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

   8. metadata-eval 99.5%

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

   9. mul0-lft 99.5%

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

   10. unpow2 99.5%

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

   11. fmm-def 99.5%

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

   12. associate-*r* 99.5%

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

   13. *-commutative 99.5%

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

   14. distribute-rgt-neg-in 99.5%

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

   15. metadata-eval 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 6: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(a \cdot \left(\frac{\left(c \cdot a\right) \cdot -0.5625}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (-
   (* c (* a (- (/ (* (* c a) -0.5625) (pow b 5.0)) (/ 0.375 (pow b 3.0)))))
   (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((c * (a * ((((c * a) * -0.5625) / pow(b, 5.0)) - (0.375 / 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 * ((c * (a * ((((c * a) * (-0.5625d0)) / (b ** 5.0d0)) - (0.375d0 / (b ** 3.0d0))))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * (a * ((((c * a) * -0.5625) / Math.pow(b, 5.0)) - (0.375 / Math.pow(b, 3.0))))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * ((c * (a * ((((c * a) * -0.5625) / math.pow(b, 5.0)) - (0.375 / math.pow(b, 3.0))))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(Float64(Float64(c * a) * -0.5625) / (b ^ 5.0)) - Float64(0.375 / (b ^ 3.0))))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * (a * ((((c * a) * -0.5625) / (b ^ 5.0)) - (0.375 / (b ^ 3.0))))) - (0.5 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(a * N[(N[(N[(N[(c * a), $MachinePrecision] * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

3. Taylor expanded in c around 0 94.6%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

5. Simplified 94.6%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

6. Taylor expanded in a around 0 93.4%

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

   1. associate-*r/ 93.4%

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

   2. associate-*r/ 93.4%

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

   3. metadata-eval 93.4%

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

8. Simplified 93.4%

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

9. Final simplification 93.4%

   \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\left(c \cdot a\right) \cdot -0.5625}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]
10. Add Preprocessing

Alternative 7: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

3. Taylor expanded in c around 0 94.6%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

5. Simplified 94.6%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

6. Taylor expanded in c around 0 90.2%

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

   1. associate-*r/ 90.2%

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

   2. *-commutative 90.2%

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

8. Simplified 90.2%

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

9. Final simplification 90.2%

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

Alternative 8: 81.6% 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 31.1%

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

3. Taylor expanded in b around inf 81.4%

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

   1. associate-*r/ 81.4%

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

   2. *-commutative 81.4%

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

5. Simplified 81.4%

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

Alternative 9: 81.4% 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 31.1%

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

3. Taylor expanded in b around inf 81.4%

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

   1. associate-*r/ 81.4%

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

   2. *-commutative 81.4%

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

5. Simplified 81.4%

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

6. Taylor expanded in c around 0 81.4%

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

   1. associate-*r/ 81.4%

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

   2. *-commutative 81.4%

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

   3. associate-*r/ 81.1%

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

8. Simplified 81.1%

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

Alternative 10: 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 31.1%

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

   1. neg-sub0 31.1%

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

   2. flip-- 31.2%

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

   3. metadata-eval 31.2%

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

   4. pow2 31.2%

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

   5. add-sqr-sqrt 31.2%

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

   6. sqrt-prod 31.2%

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

   7. sqr-neg 31.2%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 31.2%

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

   14. sqr-neg 31.2%

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

   15. sqrt-prod 31.2%

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

   16. add-sqr-sqrt 31.2%

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

4. Applied egg-rr 31.2%

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

   1. neg-sub0 31.2%

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

6. Simplified 31.2%

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

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
8. 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%

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

9. Simplified 3.2%

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

Reproduce

herbie shell --seed 2024131 
(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)))