Cubic critical, narrow range

Percentage Accurate: 55.1% → 99.1%

Time: 14.9s

Alternatives: 9

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: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

   1. /-rgt-identity 54.9%

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

   2. metadata-eval 54.9%

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

3. Simplified 54.9%

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

5. Taylor expanded in a around inf 54.7%

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

   1. flip-- 54.7%

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

   2. add-sqr-sqrt 55.6%

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

   3. fma-neg 56.7%

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

   4. *-commutative 56.7%

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

   5. fma-define 56.7%

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

   6. pow2 56.7%

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

   7. *-commutative 56.7%

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

   8. fma-define 56.7%

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

7. Applied egg-rr 56.7%

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

   1. fma-undefine 55.6%

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

   2. unsub-neg 55.6%

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

   3. +-commutative 55.6%

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

9. Simplified 55.6%

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

10. Taylor expanded in a around 0 99.2%

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

    1. *-commutative 99.2%

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

    2. associate-*r* 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

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

Alternative 2: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0215:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0215)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (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)) <= -0.0215) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} 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)) <= -0.0215)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	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_] := 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], -0.0215], 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 * 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)) < -0.021499999999999998

   1. Initial program 81.1%

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

      1. /-rgt-identity 81.1%

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

      2. metadata-eval 81.1%

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

   3. Simplified 81.4%

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

   if -0.021499999999999998 < (/.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 45.2%

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

      1. /-rgt-identity 45.2%

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

      2. metadata-eval 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in a around inf 45.1%

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

   6. Taylor expanded in b around inf 91.0%

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

      1. fma-define 91.0%

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

      2. associate-/l* 91.0%

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

      3. unpow2 91.0%

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

      4. unpow2 91.0%

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

      5. times-frac 91.0%

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

      6. unpow2 91.0%

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

   8. Simplified 91.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0215:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.0215:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.0215)
     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 <= -0.0215) {
		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 <= -0.0215)
		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, -0.0215], 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)) < -0.021499999999999998

   1. Initial program 81.1%

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

   if -0.021499999999999998 < (/.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 45.2%

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

      1. /-rgt-identity 45.2%

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

      2. metadata-eval 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in a around inf 45.1%

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

   6. Taylor expanded in b around inf 91.0%

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

      1. fma-define 91.0%

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

      2. associate-/l* 91.0%

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

      3. unpow2 91.0%

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

      4. unpow2 91.0%

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

      5. times-frac 91.0%

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

      6. unpow2 91.0%

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

   8. Simplified 91.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0215:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

   1. /-rgt-identity 54.9%

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

   2. metadata-eval 54.9%

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

3. Simplified 54.9%

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

5. Taylor expanded in a around inf 54.7%

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

   1. flip-- 54.7%

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

   2. add-sqr-sqrt 55.6%

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

   3. fma-neg 56.7%

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

   4. *-commutative 56.7%

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

   5. fma-define 56.7%

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

   6. pow2 56.7%

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

   7. *-commutative 56.7%

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

   8. fma-define 56.7%

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

7. Applied egg-rr 56.7%

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

   1. fma-undefine 55.6%

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

   2. unsub-neg 55.6%

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

   3. +-commutative 55.6%

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

9. Simplified 55.6%

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

10. Taylor expanded in a around 0 99.2%

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

11. Final simplification 99.2%

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

Alternative 5: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.0215:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.0215)
     t_0
     (/ (* c (- (* -0.375 (/ (* a c) (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 <= -0.0215) {
		tmp = t_0;
	} else {
		tmp = (c * ((-0.375 * ((a * c) / 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 <= (-0.0215d0)) then
        tmp = t_0
    else
        tmp = (c * (((-0.375d0) * ((a * c) / (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 <= -0.0215) {
		tmp = t_0;
	} else {
		tmp = (c * ((-0.375 * ((a * c) / 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 <= -0.0215:
		tmp = t_0
	else:
		tmp = (c * ((-0.375 * ((a * c) / 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 <= -0.0215)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (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 <= -0.0215)
		tmp = t_0;
	else
		tmp = (c * ((-0.375 * ((a * c) / (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, -0.0215], t$95$0, N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $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)) < -0.021499999999999998

   1. Initial program 81.1%

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

   if -0.021499999999999998 < (/.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 45.2%

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

   3. Taylor expanded in b around inf 91.0%

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

   4. Taylor expanded in c around 0 90.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0215:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{\left(a \cdot c\right) \cdot -0.375}{{b}^{3}} - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.75)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (* c (- (/ (* (* a c) -0.375) (pow b 3.0)) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.75) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((((a * c) * -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) :: tmp
    if (b <= 1.75d0) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = c * ((((a * c) * (-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 tmp;
	if (b <= 1.75) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((((a * c) * -0.375) / Math.pow(b, 3.0)) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.75:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = c * ((((a * c) * -0.375) / math.pow(b, 3.0)) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.75)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(Float64(Float64(a * c) * -0.375) / (b ^ 3.0)) - Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.75)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = c * ((((a * c) * -0.375) / (b ^ 3.0)) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.75], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(N[(N[(a * c), $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 b < 1.75

   1. Initial program 79.6%

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

   3. Taylor expanded in a around 0 79.5%

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

      1. metadata-eval 79.5%

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

      2. distribute-lft-neg-in 79.5%

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

      3. *-commutative 79.5%

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

      4. associate-*r* 79.6%

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

      5. distribute-rgt-neg-in 79.6%

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

      6. *-commutative 79.6%

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

      7. distribute-lft-neg-in 79.6%

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

      8. metadata-eval 79.6%

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

      9. *-commutative 79.6%

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

   5. Simplified 79.6%

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

   if 1.75 < b

   1. Initial program 49.5%

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

   3. Taylor expanded in b around inf 87.0%

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

   4. Taylor expanded in c around 0 86.9%

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

      1. associate-*r/ 86.9%

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

      2. *-commutative 86.9%

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

      3. associate-*r/ 86.9%

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

      4. metadata-eval 86.9%

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

   6. Simplified 86.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{\left(a \cdot c\right) \cdot -0.375}{{b}^{3}} - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in b around inf 82.7%

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

4. Taylor expanded in c around 0 82.6%

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

   1. associate-*r/ 82.6%

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

   2. *-commutative 82.6%

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

   3. associate-*r/ 82.6%

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

   4. metadata-eval 82.6%

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

6. Simplified 82.6%

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

7. Final simplification 82.6%

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

Alternative 8: 64.7% 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 54.9%

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

3. Taylor expanded in b around inf 65.1%

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

   1. associate-*r/ 65.1%

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

   2. *-commutative 65.1%

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

5. Simplified 65.1%

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

Alternative 9: 64.6% 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 54.9%

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

3. Taylor expanded in b around inf 65.1%

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

   1. associate-*r/ 65.1%

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

   2. *-commutative 65.1%

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

5. Simplified 65.1%

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

   1. associate-/l* 65.1%

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

7. Applied egg-rr 65.1%

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

Reproduce

herbie shell --seed 2024113 
(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)))