Cubic critical, wide range

Percentage Accurate: 17.2% → 99.7%

Time: 8.7s

Alternatives: 7

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.2% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.4%

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

   1. add-cbrt-cube 18.4%

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

   2. pow3 18.4%

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

   3. *-commutative 18.4%

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

   4. *-commutative 18.4%

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

3. Applied egg-rr 18.4%

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

   1. flip-+ 18.3%

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

   2. add-sqr-sqrt 18.7%

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

   3. rem-cbrt-cube 18.9%

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

   4. rem-cbrt-cube 18.9%

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

5. Applied egg-rr 18.9%

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

   1. sqr-neg 18.9%

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

   2. unpow2 18.9%

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

   3. unpow2 18.9%

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

   4. associate-+l- 99.5%

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

   5. +-inverses 99.5%

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

   6. +-commutative 99.5%

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

   7. unpow2 99.5%

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

   8. sub-neg 99.5%

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

   9. +-commutative 99.5%

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

   10. distribute-rgt-neg-in 99.5%

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

   11. fma-def 99.5%

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

   12. distribute-rgt-neg-in 99.5%

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

   13. metadata-eval 99.5%

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

   14. unpow2 99.5%

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

7. Simplified 99.5%

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

   1. div-inv 99.3%

      \[\leadsto \frac{\color{blue}{\left(c \cdot \left(a \cdot 3\right) + 0\right) \cdot \frac{1}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}}{3 \cdot a} \]

   2. +-rgt-identity 99.3%

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

9. Applied egg-rr 99.3%

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

    1. associate-*r/ 99.5%

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

    2. *-rgt-identity 99.5%

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

    3. associate-/l* 99.5%

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

    4. associate-/r/ 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 3} \leq -1000000000:\\ \;\;\;\;\frac{\frac{b \cdot b + \left(t_0 - b \cdot b\right)}{\left(-b\right) - t_1}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= (/ (- t_1 b) (* a 3.0)) -1000000000.0)
     (/ (/ (+ (* b b) (- t_0 (* b b))) (- (- b) t_1)) (* a 3.0))
     (+ (* -0.375 (* a (/ (* c c) (pow b 3.0)))) (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 3.0)) <= -1000000000.0) {
		tmp = (((b * b) + (t_0 - (b * b))) / (-b - t_1)) / (a * 3.0);
	} else {
		tmp = (-0.375 * (a * ((c * c) / pow(b, 3.0)))) + (-0.5 * (c / 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) :: t_1
    real(8) :: tmp
    t_0 = c * (a * 3.0d0)
    t_1 = sqrt(((b * b) - t_0))
    if (((t_1 - b) / (a * 3.0d0)) <= (-1000000000.0d0)) then
        tmp = (((b * b) + (t_0 - (b * b))) / (-b - t_1)) / (a * 3.0d0)
    else
        tmp = ((-0.375d0) * (a * ((c * c) / (b ** 3.0d0)))) + ((-0.5d0) * (c / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	double t_1 = Math.sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 3.0)) <= -1000000000.0) {
		tmp = (((b * b) + (t_0 - (b * b))) / (-b - t_1)) / (a * 3.0);
	} else {
		tmp = (-0.375 * (a * ((c * c) / Math.pow(b, 3.0)))) + (-0.5 * (c / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 3.0)
	t_1 = math.sqrt(((b * b) - t_0))
	tmp = 0
	if ((t_1 - b) / (a * 3.0)) <= -1000000000.0:
		tmp = (((b * b) + (t_0 - (b * b))) / (-b - t_1)) / (a * 3.0)
	else:
		tmp = (-0.375 * (a * ((c * c) / math.pow(b, 3.0)))) + (-0.5 * (c / b))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 3.0))
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 3.0)) <= -1000000000.0)
		tmp = Float64(Float64(Float64(Float64(b * b) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - t_1)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))) + Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 3.0);
	t_1 = sqrt(((b * b) - t_0));
	tmp = 0.0;
	if (((t_1 - b) / (a * 3.0)) <= -1000000000.0)
		tmp = (((b * b) + (t_0 - (b * b))) / (-b - t_1)) / (a * 3.0);
	else
		tmp = (-0.375 * (a * ((c * c) / (b ^ 3.0)))) + (-0.5 * (c / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -1000000000.0], N[(N[(N[(N[(b * b), $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / 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 3 a) c)))) (*.f64 3 a)) < -1e9

   1. Initial program 87.5%

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

      1. add-cbrt-cube 87.3%

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

      2. pow3 87.4%

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

      3. *-commutative 87.4%

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

      4. *-commutative 87.4%

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

   3. Applied egg-rr 87.4%

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

      1. flip-+ 87.0%

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

      2. add-sqr-sqrt 87.8%

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

      3. rem-cbrt-cube 89.2%

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

      4. rem-cbrt-cube 89.2%

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

   5. Applied egg-rr 89.2%

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

   if -1e9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 14.7%

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

      1. /-rgt-identity 14.7%

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

      2. metadata-eval 14.7%

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

      3. associate-/l* 14.7%

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

      4. associate-*r/ 14.7%

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

      5. *-commutative 14.7%

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

      6. associate-*l/ 14.7%

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

      7. associate-*r/ 14.7%

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

      8. metadata-eval 14.7%

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

      9. metadata-eval 14.7%

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

      10. times-frac 14.7%

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

      11. neg-mul-1 14.7%

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

      12. distribute-rgt-neg-in 14.7%

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

      13. times-frac 14.7%

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

      14. metadata-eval 14.7%

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

      15. neg-mul-1 14.7%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in b around inf 96.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 96.9%

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

      2. fma-def 96.9%

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

      3. associate-/l* 96.9%

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

      4. unpow2 96.9%

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

   6. Simplified 96.9%

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

      1. fma-udef 96.9%

         \[\leadsto \color{blue}{-0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}} + -0.5 \cdot \frac{c}{b}} \]

      2. associate-/r/ 96.9%

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

   8. Applied egg-rr 96.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1000000000:\\ \;\;\;\;\frac{\frac{b \cdot b + \left(c \cdot \left(a \cdot 3\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -1000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -1000000000.0)
     t_0
     (+ (* -0.375 (* a (/ (* c c) (pow b 3.0)))) (* -0.5 (/ c 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 <= -1000000000.0) {
		tmp = t_0;
	} else {
		tmp = (-0.375 * (a * ((c * c) / pow(b, 3.0)))) + (-0.5 * (c / 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 <= (-1000000000.0d0)) then
        tmp = t_0
    else
        tmp = ((-0.375d0) * (a * ((c * c) / (b ** 3.0d0)))) + ((-0.5d0) * (c / 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 <= -1000000000.0) {
		tmp = t_0;
	} else {
		tmp = (-0.375 * (a * ((c * c) / Math.pow(b, 3.0)))) + (-0.5 * (c / 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 <= -1000000000.0:
		tmp = t_0
	else:
		tmp = (-0.375 * (a * ((c * c) / math.pow(b, 3.0)))) + (-0.5 * (c / 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 <= -1000000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))) + Float64(-0.5 * Float64(c / 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 <= -1000000000.0)
		tmp = t_0;
	else
		tmp = (-0.375 * (a * ((c * c) / (b ^ 3.0)))) + (-0.5 * (c / 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, -1000000000.0], t$95$0, N[(N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / 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 3 a) c)))) (*.f64 3 a)) < -1e9

   1. Initial program 87.5%

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

   if -1e9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 14.7%

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

      1. /-rgt-identity 14.7%

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

      2. metadata-eval 14.7%

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

      3. associate-/l* 14.7%

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

      4. associate-*r/ 14.7%

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

      5. *-commutative 14.7%

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

      6. associate-*l/ 14.7%

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

      7. associate-*r/ 14.7%

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

      8. metadata-eval 14.7%

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

      9. metadata-eval 14.7%

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

      10. times-frac 14.7%

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

      11. neg-mul-1 14.7%

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

      12. distribute-rgt-neg-in 14.7%

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

      13. times-frac 14.7%

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

      14. metadata-eval 14.7%

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

      15. neg-mul-1 14.7%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in b around inf 96.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 96.9%

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

      2. fma-def 96.9%

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

      3. associate-/l* 96.9%

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

      4. unpow2 96.9%

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

   6. Simplified 96.9%

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

      1. fma-udef 96.9%

         \[\leadsto \color{blue}{-0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}} + -0.5 \cdot \frac{c}{b}} \]

      2. associate-/r/ 96.9%

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

   8. Applied egg-rr 96.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1000000000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a} \cdot \frac{c}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ 1.0 a) (/ c (/ (- (- b) (sqrt (fma c (* a -3.0) (* b b)))) a))))

C

double code(double a, double b, double c) {
	return (1.0 / a) * (c / ((-b - sqrt(fma(c, (a * -3.0), (b * b)))) / a));
}

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / a) * Float64(c / Float64(Float64(Float64(-b) - sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))) / a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / a), $MachinePrecision] * N[(c / N[(N[((-b) - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

   1. add-cbrt-cube 18.4%

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

   2. pow3 18.4%

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

   3. *-commutative 18.4%

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

   4. *-commutative 18.4%

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

3. Applied egg-rr 18.4%

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

   1. flip-+ 18.3%

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

   2. add-sqr-sqrt 18.7%

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

   3. rem-cbrt-cube 18.9%

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

   4. rem-cbrt-cube 18.9%

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

5. Applied egg-rr 18.9%

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

   1. sqr-neg 18.9%

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

   2. unpow2 18.9%

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

   3. unpow2 18.9%

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

   4. associate-+l- 99.5%

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

   5. +-inverses 99.5%

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

   6. +-commutative 99.5%

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

   7. unpow2 99.5%

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

   8. sub-neg 99.5%

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

   9. +-commutative 99.5%

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

   10. distribute-rgt-neg-in 99.5%

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

   11. fma-def 99.5%

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

   12. distribute-rgt-neg-in 99.5%

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

   13. metadata-eval 99.5%

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

   14. unpow2 99.5%

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

7. Simplified 99.5%

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

   1. div-inv 99.3%

      \[\leadsto \frac{\color{blue}{\left(c \cdot \left(a \cdot 3\right) + 0\right) \cdot \frac{1}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}}{3 \cdot a} \]

   2. +-rgt-identity 99.3%

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

9. Applied egg-rr 99.3%

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

    1. associate-*r/ 99.5%

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

    2. *-rgt-identity 99.5%

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

    3. associate-/l* 99.5%

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

    4. associate-/r/ 99.8%

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

11. Simplified 99.8%

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

    1. expm1-log1p-u 83.0%

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

    2. expm1-udef 20.8%

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

    3. times-frac 20.8%

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

13. Applied egg-rr 20.8%

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

    1. expm1-def 82.7%

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

    2. expm1-log1p 99.4%

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

    3. *-commutative 99.4%

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

    4. associate-/l/ 99.3%

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

    5. times-frac 99.1%

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

    6. *-commutative 99.1%

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

    7. associate-*r* 99.2%

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

    8. *-commutative 99.2%

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

    9. associate-*l* 99.1%

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

    10. times-frac 99.2%

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

    11. *-commutative 99.2%

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

    12. associate-/r* 99.3%

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

    13. metadata-eval 99.3%

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

    14. associate-/l* 99.4%

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

15. Simplified 99.4%

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

16. Final simplification 99.4%

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

Alternative 5: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.4%

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

   1. add-cbrt-cube 18.4%

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

   2. pow3 18.4%

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

   3. *-commutative 18.4%

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

   4. *-commutative 18.4%

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

3. Applied egg-rr 18.4%

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

   1. flip-+ 18.3%

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

   2. add-sqr-sqrt 18.7%

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

   3. rem-cbrt-cube 18.9%

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

   4. rem-cbrt-cube 18.9%

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

5. Applied egg-rr 18.9%

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

   1. sqr-neg 18.9%

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

   2. unpow2 18.9%

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

   3. unpow2 18.9%

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

   4. associate-+l- 99.5%

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

   5. +-inverses 99.5%

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

   6. +-commutative 99.5%

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

   7. unpow2 99.5%

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

   8. sub-neg 99.5%

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

   9. +-commutative 99.5%

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

   10. distribute-rgt-neg-in 99.5%

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

   11. fma-def 99.5%

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

   12. distribute-rgt-neg-in 99.5%

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

   13. metadata-eval 99.5%

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

   14. unpow2 99.5%

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

7. Simplified 99.5%

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

   1. div-inv 99.4%

      \[\leadsto \color{blue}{\frac{c \cdot \left(a \cdot 3\right) + 0}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \cdot \frac{1}{3 \cdot a}} \]

   2. +-rgt-identity 99.4%

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

   3. *-commutative 99.4%

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

9. Applied egg-rr 99.4%

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

    1. associate-*l/ 99.5%

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

    2. associate-*r/ 99.8%

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

    3. *-rgt-identity 99.8%

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

    4. associate-*r* 99.4%

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

    5. *-commutative 99.4%

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

    6. *-commutative 99.4%

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

    7. times-frac 99.7%

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

    8. metadata-eval 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 6: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-0.375 * (a * ((c * c) / pow(b, 3.0)))) + (-0.5 * (c / b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.375d0) * (a * ((c * c) / (b ** 3.0d0)))) + ((-0.5d0) * (c / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.4%

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

   1. /-rgt-identity 18.4%

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

   2. metadata-eval 18.4%

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

   3. associate-/l* 18.4%

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

   4. associate-*r/ 18.4%

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

   5. *-commutative 18.4%

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

   6. associate-*l/ 18.4%

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

   7. associate-*r/ 18.4%

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

   8. metadata-eval 18.4%

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

   9. metadata-eval 18.4%

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

   10. times-frac 18.4%

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

   11. neg-mul-1 18.4%

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

   12. distribute-rgt-neg-in 18.4%

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

   13. times-frac 18.4%

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

   14. metadata-eval 18.4%

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

   15. neg-mul-1 18.4%

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

3. Simplified 18.5%

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

4. Taylor expanded in b around inf 94.5%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
5. Step-by-step derivation

   1. +-commutative 94.5%

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

   2. fma-def 94.5%

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

   3. associate-/l* 94.5%

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

   4. unpow2 94.5%

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

6. Simplified 94.5%

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

   1. fma-udef 94.5%

      \[\leadsto \color{blue}{-0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}} + -0.5 \cdot \frac{c}{b}} \]

   2. associate-/r/ 94.5%

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

8. Applied egg-rr 94.5%

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

9. Final simplification 94.5%

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

Alternative 7: 90.8% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.4%

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

   1. /-rgt-identity 18.4%

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

   2. metadata-eval 18.4%

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

   3. associate-/l* 18.4%

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

   4. associate-*r/ 18.4%

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

   5. *-commutative 18.4%

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

   6. associate-*l/ 18.4%

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

   7. associate-*r/ 18.4%

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

   8. metadata-eval 18.4%

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

   9. metadata-eval 18.4%

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

   10. times-frac 18.4%

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

   11. neg-mul-1 18.4%

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

   12. distribute-rgt-neg-in 18.4%

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

   13. times-frac 18.4%

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

   14. metadata-eval 18.4%

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

   15. neg-mul-1 18.4%

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

3. Simplified 18.5%

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

4. Taylor expanded in b around inf 89.7%

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

5. Final simplification 89.7%

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

Reproduce

herbie shell --seed 2023230 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))