Cubic critical, medium range

Percentage Accurate: 31.4% → 95.4%

Time: 19.4s

Alternatives: 10

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.4% 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: 95.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(1.265625 \cdot {c}^{4} + {c}^{4} \cdot 5.0625\right)}{{b}^{7}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+
   (* -0.5 (/ c b))
   (+
    (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
    (*
     -0.16666666666666666
     (/
      (* (pow a 3.0) (+ (* 1.265625 (pow c 4.0)) (* (pow c 4.0) 5.0625)))
      (pow b 7.0)))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow(a, 3.0) * ((1.265625 * pow(c, 4.0)) + (pow(c, 4.0) * 5.0625))) / pow(b, 7.0)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * (((a ** 3.0d0) * ((1.265625d0 * (c ** 4.0d0)) + ((c ** 4.0d0) * 5.0625d0))) / (b ** 7.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow(a, 3.0) * ((1.265625 * Math.pow(c, 4.0)) + (Math.pow(c, 4.0) * 5.0625))) / Math.pow(b, 7.0)))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow(a, 3.0) * ((1.265625 * math.pow(c, 4.0)) + (math.pow(c, 4.0) * 5.0625))) / math.pow(b, 7.0)))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((a ^ 3.0) * Float64(Float64(1.265625 * (c ^ 4.0)) + Float64((c ^ 4.0) * 5.0625))) / (b ^ 7.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * (((a ^ 3.0) * ((1.265625 * (c ^ 4.0)) + ((c ^ 4.0) * 5.0625))) / (b ^ 7.0)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(1.265625 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * 5.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.1%

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

2. Taylor expanded in b around inf 94.4%

   \[\leadsto \frac{\color{blue}{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + \left(-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{{b}^{7}}\right)\right)}}{3 \cdot a} \]

3. Taylor expanded in a around 0 94.9%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}\right)}{{b}^{7}}\right)\right)} \]

4. Final simplification 94.9%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(1.265625 \cdot {c}^{4} + {c}^{4} \cdot 5.0625\right)}{{b}^{7}}\right)\right) \]

Alternative 2: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
end

Wolfram

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

Derivation

1. Initial program 32.1%

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

2. Taylor expanded in b around inf 93.6%

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

3. Final simplification 93.6%

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

Alternative 3: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.086:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{c \cdot \left(a \cdot \frac{-1.5}{b}\right)}{a \cdot 3}\\ \mathbf{elif}\;t_0 \leq -6 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.086)
     t_0
     (if (<= t_0 -3e-6)
       (/ (* c (* a (/ -1.5 b))) (* a 3.0))
       (if (<= t_0 -6e-11) t_0 (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.086) {
		tmp = t_0;
	} else if (t_0 <= -3e-6) {
		tmp = (c * (a * (-1.5 / b))) / (a * 3.0);
	} else if (t_0 <= -6e-11) {
		tmp = t_0;
	} else {
		tmp = (c * -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 <= (-0.086d0)) then
        tmp = t_0
    else if (t_0 <= (-3d-6)) then
        tmp = (c * (a * ((-1.5d0) / b))) / (a * 3.0d0)
    else if (t_0 <= (-6d-11)) then
        tmp = t_0
    else
        tmp = (c * (-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 <= -0.086) {
		tmp = t_0;
	} else if (t_0 <= -3e-6) {
		tmp = (c * (a * (-1.5 / b))) / (a * 3.0);
	} else if (t_0 <= -6e-11) {
		tmp = t_0;
	} else {
		tmp = (c * -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 <= -0.086:
		tmp = t_0
	elif t_0 <= -3e-6:
		tmp = (c * (a * (-1.5 / b))) / (a * 3.0)
	elif t_0 <= -6e-11:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.086)
		tmp = t_0;
	elseif (t_0 <= -3e-6)
		tmp = Float64(Float64(c * Float64(a * Float64(-1.5 / b))) / Float64(a * 3.0));
	elseif (t_0 <= -6e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -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 <= -0.086)
		tmp = t_0;
	elseif (t_0 <= -3e-6)
		tmp = (c * (a * (-1.5 / b))) / (a * 3.0);
	elseif (t_0 <= -6e-11)
		tmp = t_0;
	else
		tmp = (c * -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, -0.086], t$95$0, If[LessEqual[t$95$0, -3e-6], N[(N[(c * N[(a * N[(-1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -6e-11], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.085999999999999993 or -3.0000000000000001e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -6e-11

   1. Initial program 71.0%

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

   if -0.085999999999999993 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.0000000000000001e-6

   1. Initial program 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in b around inf 41.2%

      \[\leadsto \frac{\color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{3 \cdot a} \]
   5. Step-by-step derivation

      1. associate-*r/ 41.2%

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

      2. associate-*r* 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in b around 0 68.0%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. associate-*r/ 68.0%

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

      2. associate-*r* 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*l/ 68.1%

         \[\leadsto \frac{\color{blue}{\frac{a \cdot -1.5}{b} \cdot c}}{3 \cdot a} \]

      5. *-commutative 68.1%

         \[\leadsto \frac{\color{blue}{c \cdot \frac{a \cdot -1.5}{b}}}{3 \cdot a} \]

      6. *-rgt-identity 68.1%

         \[\leadsto \frac{c \cdot \frac{\color{blue}{\left(a \cdot -1.5\right) \cdot 1}}{b}}{3 \cdot a} \]

      7. associate-*r/ 68.1%

         \[\leadsto \frac{c \cdot \color{blue}{\left(\left(a \cdot -1.5\right) \cdot \frac{1}{b}\right)}}{3 \cdot a} \]

      8. associate-*l* 68.1%

         \[\leadsto \frac{c \cdot \color{blue}{\left(a \cdot \left(-1.5 \cdot \frac{1}{b}\right)\right)}}{3 \cdot a} \]

      9. associate-*r/ 68.1%

         \[\leadsto \frac{c \cdot \left(a \cdot \color{blue}{\frac{-1.5 \cdot 1}{b}}\right)}{3 \cdot a} \]

      10. metadata-eval 68.1%

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

   9. Simplified 68.1%

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

   if -6e-11 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 12.3%

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

   2. Taylor expanded in b around inf 94.7%

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

      1. associate-*r/ 94.7%

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

      2. *-commutative 94.7%

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

   4. Simplified 94.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.086:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{c \cdot \left(a \cdot \frac{-1.5}{b}\right)}{a \cdot 3}\\ \mathbf{elif}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.086:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{elif}\;t_0 \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{c \cdot \left(a \cdot \frac{-1.5}{b}\right)}{a \cdot 3}\\ \mathbf{elif}\;t_0 \leq -6 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.086)
     (* (- b (sqrt (fma b b (* a (* c -3.0))))) (/ -0.3333333333333333 a))
     (if (<= t_0 -3e-6)
       (/ (* c (* a (/ -1.5 b))) (* a 3.0))
       (if (<= t_0 -6e-11) t_0 (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.086) {
		tmp = (b - sqrt(fma(b, b, (a * (c * -3.0))))) * (-0.3333333333333333 / a);
	} else if (t_0 <= -3e-6) {
		tmp = (c * (a * (-1.5 / b))) / (a * 3.0);
	} else if (t_0 <= -6e-11) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.086)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) * Float64(-0.3333333333333333 / a));
	elseif (t_0 <= -3e-6)
		tmp = Float64(Float64(c * Float64(a * Float64(-1.5 / b))) / Float64(a * 3.0));
	elseif (t_0 <= -6e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.086], N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -3e-6], N[(N[(c * N[(a * N[(-1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -6e-11], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.085999999999999993

   1. Initial program 70.4%

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

      1. expm1-log1p-u 70.3%

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

      2. expm1-udef 56.2%

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

      3. associate-*l* 56.2%

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

   3. Applied egg-rr 56.2%

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

      1. frac-2neg 56.2%

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

      2. div-inv 56.2%

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

   5. Applied egg-rr 70.5%

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

      1. associate-*l* 70.5%

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

      2. *-commutative 70.5%

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

      3. *-commutative 70.5%

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

      4. associate-/r* 70.5%

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

      5. metadata-eval 70.5%

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

   7. Simplified 70.5%

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

   if -0.085999999999999993 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.0000000000000001e-6

   1. Initial program 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in b around inf 41.2%

      \[\leadsto \frac{\color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{3 \cdot a} \]
   5. Step-by-step derivation

      1. associate-*r/ 41.2%

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

      2. associate-*r* 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in b around 0 68.0%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. associate-*r/ 68.0%

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

      2. associate-*r* 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*l/ 68.1%

         \[\leadsto \frac{\color{blue}{\frac{a \cdot -1.5}{b} \cdot c}}{3 \cdot a} \]

      5. *-commutative 68.1%

         \[\leadsto \frac{\color{blue}{c \cdot \frac{a \cdot -1.5}{b}}}{3 \cdot a} \]

      6. *-rgt-identity 68.1%

         \[\leadsto \frac{c \cdot \frac{\color{blue}{\left(a \cdot -1.5\right) \cdot 1}}{b}}{3 \cdot a} \]

      7. associate-*r/ 68.1%

         \[\leadsto \frac{c \cdot \color{blue}{\left(\left(a \cdot -1.5\right) \cdot \frac{1}{b}\right)}}{3 \cdot a} \]

      8. associate-*l* 68.1%

         \[\leadsto \frac{c \cdot \color{blue}{\left(a \cdot \left(-1.5 \cdot \frac{1}{b}\right)\right)}}{3 \cdot a} \]

      9. associate-*r/ 68.1%

         \[\leadsto \frac{c \cdot \left(a \cdot \color{blue}{\frac{-1.5 \cdot 1}{b}}\right)}{3 \cdot a} \]

      10. metadata-eval 68.1%

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

   9. Simplified 68.1%

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

   if -3.0000000000000001e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -6e-11

   1. Initial program 72.1%

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

   if -6e-11 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 12.3%

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

   2. Taylor expanded in b around inf 94.7%

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

      1. associate-*r/ 94.7%

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

      2. *-commutative 94.7%

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

   4. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.086:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{elif}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{c \cdot \left(a \cdot \frac{-1.5}{b}\right)}{a \cdot 3}\\ \mathbf{elif}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.086:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{elif}\;t_0 \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{\frac{0.037037037037037035}{a}}{{a}^{2}}}\\ \mathbf{elif}\;t_0 \leq -6 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.086)
     (* (- b (sqrt (fma b b (* a (* c -3.0))))) (/ -0.3333333333333333 a))
     (if (<= t_0 -3e-6)
       (*
        (* c (* a (/ -1.5 b)))
        (cbrt (/ (/ 0.037037037037037035 a) (pow a 2.0))))
       (if (<= t_0 -6e-11) t_0 (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.086) {
		tmp = (b - sqrt(fma(b, b, (a * (c * -3.0))))) * (-0.3333333333333333 / a);
	} else if (t_0 <= -3e-6) {
		tmp = (c * (a * (-1.5 / b))) * cbrt(((0.037037037037037035 / a) / pow(a, 2.0)));
	} else if (t_0 <= -6e-11) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.086)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) * Float64(-0.3333333333333333 / a));
	elseif (t_0 <= -3e-6)
		tmp = Float64(Float64(c * Float64(a * Float64(-1.5 / b))) * cbrt(Float64(Float64(0.037037037037037035 / a) / (a ^ 2.0))));
	elseif (t_0 <= -6e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.086], N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -3e-6], N[(N[(c * N[(a * N[(-1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(0.037037037037037035 / a), $MachinePrecision] / N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -6e-11], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.085999999999999993

   1. Initial program 70.4%

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

      1. expm1-log1p-u 70.3%

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

      2. expm1-udef 56.2%

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

      3. associate-*l* 56.2%

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

   3. Applied egg-rr 56.2%

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

      1. frac-2neg 56.2%

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

      2. div-inv 56.2%

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

   5. Applied egg-rr 70.5%

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

      1. associate-*l* 70.5%

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

      2. *-commutative 70.5%

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

      3. *-commutative 70.5%

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

      4. associate-/r* 70.5%

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

      5. metadata-eval 70.5%

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

   7. Simplified 70.5%

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

   if -0.085999999999999993 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.0000000000000001e-6

   1. Initial program 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in b around inf 41.2%

      \[\leadsto \frac{\color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{3 \cdot a} \]
   5. Step-by-step derivation

      1. associate-*r/ 41.2%

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

      2. associate-*r* 41.2%

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

   6. Simplified 41.2%

      \[\leadsto \frac{\color{blue}{\left(b + \frac{\left(-1.5 \cdot a\right) \cdot c}{b}\right)} - b}{3 \cdot a} \]
   7. Step-by-step derivation

      1. div-sub 40.7%

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

      2. sub-neg 40.7%

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

      3. div-inv 40.5%

         \[\leadsto \color{blue}{\left(b + \frac{\left(-1.5 \cdot a\right) \cdot c}{b}\right) \cdot \frac{1}{3 \cdot a}} + \left(-\frac{b}{3 \cdot a}\right) \]

      4. associate-*l* 40.5%

         \[\leadsto \left(b + \frac{\color{blue}{-1.5 \cdot \left(a \cdot c\right)}}{b}\right) \cdot \frac{1}{3 \cdot a} + \left(-\frac{b}{3 \cdot a}\right) \]

      5. associate-*r/ 40.5%

         \[\leadsto \left(b + \color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}\right) \cdot \frac{1}{3 \cdot a} + \left(-\frac{b}{3 \cdot a}\right) \]

      6. associate-*l/ 40.5%

         \[\leadsto \left(b + -1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a} + \left(-\frac{b}{3 \cdot a}\right) \]

      7. *-commutative 40.5%

         \[\leadsto \left(b + -1.5 \cdot \color{blue}{\left(c \cdot \frac{a}{b}\right)}\right) \cdot \frac{1}{3 \cdot a} + \left(-\frac{b}{3 \cdot a}\right) \]

      8. metadata-eval 40.5%

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

      9. associate-/r* 40.6%

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

      10. metadata-eval 40.6%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{\frac{\color{blue}{1}}{3}}{a} + \left(-\frac{b}{3 \cdot a}\right) \]

      11. metadata-eval 40.6%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} + \left(-\frac{b}{3 \cdot a}\right) \]

      12. div-inv 40.9%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-\color{blue}{b \cdot \frac{1}{3 \cdot a}}\right) \]

      13. metadata-eval 40.9%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{\color{blue}{\frac{2}{2}}}{3 \cdot a}\right) \]

      14. associate-/r* 40.8%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \color{blue}{\frac{\frac{\frac{2}{2}}{3}}{a}}\right) \]

      15. metadata-eval 40.8%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{\frac{\color{blue}{1}}{3}}{a}\right) \]

      16. metadata-eval 40.8%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{\color{blue}{0.3333333333333333}}{a}\right) \]

   8. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{0.3333333333333333}{a}\right)} \]
   9. Step-by-step derivation

      1. sub-neg 40.8%

         \[\leadsto \color{blue}{\left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} - b \cdot \frac{0.3333333333333333}{a}} \]

      2. distribute-rgt-out-- 41.2%

         \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) - b\right)} \]

      3. +-commutative 41.2%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b\right)} - b\right) \]

      4. associate-+r- 68.0%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + \left(b - b\right)\right)} \]

      5. *-commutative 68.0%

         \[\leadsto \color{blue}{\left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + \left(b - b\right)\right) \cdot \frac{0.3333333333333333}{a}} \]

      6. +-inverses 68.0%

         \[\leadsto \left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + \color{blue}{0}\right) \cdot \frac{0.3333333333333333}{a} \]

      7. +-rgt-identity 68.0%

         \[\leadsto \color{blue}{\left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

      8. *-commutative 68.0%

         \[\leadsto \left(-1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right) \cdot \frac{0.3333333333333333}{a} \]

      9. associate-*l* 68.0%

         \[\leadsto \color{blue}{\left(\left(-1.5 \cdot \frac{a}{b}\right) \cdot c\right)} \cdot \frac{0.3333333333333333}{a} \]

      10. associate-*r/ 68.0%

          \[\leadsto \left(\color{blue}{\frac{-1.5 \cdot a}{b}} \cdot c\right) \cdot \frac{0.3333333333333333}{a} \]

      11. *-commutative 68.0%

          \[\leadsto \left(\frac{\color{blue}{a \cdot -1.5}}{b} \cdot c\right) \cdot \frac{0.3333333333333333}{a} \]

      12. *-rgt-identity 68.0%

          \[\leadsto \left(\frac{\color{blue}{\left(a \cdot -1.5\right) \cdot 1}}{b} \cdot c\right) \cdot \frac{0.3333333333333333}{a} \]

      13. associate-*r/ 68.0%

          \[\leadsto \left(\color{blue}{\left(\left(a \cdot -1.5\right) \cdot \frac{1}{b}\right)} \cdot c\right) \cdot \frac{0.3333333333333333}{a} \]

      14. *-commutative 68.0%

          \[\leadsto \color{blue}{\left(c \cdot \left(\left(a \cdot -1.5\right) \cdot \frac{1}{b}\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

      15. associate-*l* 68.0%

          \[\leadsto \left(c \cdot \color{blue}{\left(a \cdot \left(-1.5 \cdot \frac{1}{b}\right)\right)}\right) \cdot \frac{0.3333333333333333}{a} \]

      16. associate-*r/ 68.0%

          \[\leadsto \left(c \cdot \left(a \cdot \color{blue}{\frac{-1.5 \cdot 1}{b}}\right)\right) \cdot \frac{0.3333333333333333}{a} \]

      17. metadata-eval 68.0%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{\color{blue}{-1.5}}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} \]

   10. Simplified 68.0%

       \[\leadsto \color{blue}{\left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \frac{0.3333333333333333}{a}} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{0.3333333333333333}{a} \cdot \frac{0.3333333333333333}{a}\right) \cdot \frac{0.3333333333333333}{a}}} \]

       2. *-commutative 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\frac{0.3333333333333333}{a} \cdot \frac{0.3333333333333333}{a}\right)}} \]

       3. frac-times 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \color{blue}{\frac{0.3333333333333333 \cdot 0.3333333333333333}{a \cdot a}}} \]

       4. metadata-eval 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{\color{blue}{0.1111111111111111}}{a \cdot a}} \]

       5. pow1 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{\color{blue}{{a}^{1}} \cdot a}} \]

       6. pow1 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{{a}^{1} \cdot \color{blue}{{a}^{1}}}} \]

       7. pow-sqr 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{\color{blue}{{a}^{\left(2 \cdot 1\right)}}}} \]

       8. metadata-eval 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{{a}^{\color{blue}{2}}}} \]

   12. Applied egg-rr 68.1%

       \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \color{blue}{\sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{{a}^{2}}}} \]
   13. Step-by-step derivation

       1. associate-*r/ 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{\frac{0.3333333333333333}{a} \cdot 0.1111111111111111}{{a}^{2}}}} \]

       2. associate-*l/ 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{\color{blue}{\frac{0.3333333333333333 \cdot 0.1111111111111111}{a}}}{{a}^{2}}} \]

       3. metadata-eval 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{\frac{\color{blue}{0.037037037037037035}}{a}}{{a}^{2}}} \]

   14. Simplified 68.1%

       \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \color{blue}{\sqrt[3]{\frac{\frac{0.037037037037037035}{a}}{{a}^{2}}}} \]

   if -3.0000000000000001e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -6e-11

   1. Initial program 72.1%

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

   if -6e-11 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 12.3%

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

   2. Taylor expanded in b around inf 94.7%

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

      1. associate-*r/ 94.7%

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

      2. *-commutative 94.7%

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

   4. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.086:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{elif}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{\frac{0.037037037037037035}{a}}{{a}^{2}}}\\ \mathbf{elif}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.086:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{elif}\;t_0 \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{\frac{0.037037037037037035}{a}}{{a}^{2}}}\\ \mathbf{elif}\;t_0 \leq -6 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.086)
     (* (- b (sqrt (fma b b (* c (* a -3.0))))) (/ 1.0 (* a -3.0)))
     (if (<= t_0 -3e-6)
       (*
        (* c (* a (/ -1.5 b)))
        (cbrt (/ (/ 0.037037037037037035 a) (pow a 2.0))))
       (if (<= t_0 -6e-11) t_0 (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.086) {
		tmp = (b - sqrt(fma(b, b, (c * (a * -3.0))))) * (1.0 / (a * -3.0));
	} else if (t_0 <= -3e-6) {
		tmp = (c * (a * (-1.5 / b))) * cbrt(((0.037037037037037035 / a) / pow(a, 2.0)));
	} else if (t_0 <= -6e-11) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.086)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))) * Float64(1.0 / Float64(a * -3.0)));
	elseif (t_0 <= -3e-6)
		tmp = Float64(Float64(c * Float64(a * Float64(-1.5 / b))) * cbrt(Float64(Float64(0.037037037037037035 / a) / (a ^ 2.0))));
	elseif (t_0 <= -6e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.086], N[(N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -3e-6], N[(N[(c * N[(a * N[(-1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(0.037037037037037035 / a), $MachinePrecision] / N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -6e-11], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.085999999999999993

   1. Initial program 70.4%

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

      1. expm1-log1p-u 70.3%

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

      2. expm1-udef 56.2%

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

      3. associate-*l* 56.2%

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

   3. Applied egg-rr 56.2%

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

      1. frac-2neg 56.2%

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

      2. div-inv 56.2%

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

   5. Applied egg-rr 70.5%

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

   if -0.085999999999999993 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.0000000000000001e-6

   1. Initial program 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in b around inf 41.2%

      \[\leadsto \frac{\color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{3 \cdot a} \]
   5. Step-by-step derivation

      1. associate-*r/ 41.2%

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

      2. associate-*r* 41.2%

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

   6. Simplified 41.2%

      \[\leadsto \frac{\color{blue}{\left(b + \frac{\left(-1.5 \cdot a\right) \cdot c}{b}\right)} - b}{3 \cdot a} \]
   7. Step-by-step derivation

      1. div-sub 40.7%

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

      2. sub-neg 40.7%

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

      3. div-inv 40.5%

         \[\leadsto \color{blue}{\left(b + \frac{\left(-1.5 \cdot a\right) \cdot c}{b}\right) \cdot \frac{1}{3 \cdot a}} + \left(-\frac{b}{3 \cdot a}\right) \]

      4. associate-*l* 40.5%

         \[\leadsto \left(b + \frac{\color{blue}{-1.5 \cdot \left(a \cdot c\right)}}{b}\right) \cdot \frac{1}{3 \cdot a} + \left(-\frac{b}{3 \cdot a}\right) \]

      5. associate-*r/ 40.5%

         \[\leadsto \left(b + \color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}\right) \cdot \frac{1}{3 \cdot a} + \left(-\frac{b}{3 \cdot a}\right) \]

      6. associate-*l/ 40.5%

         \[\leadsto \left(b + -1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a} + \left(-\frac{b}{3 \cdot a}\right) \]

      7. *-commutative 40.5%

         \[\leadsto \left(b + -1.5 \cdot \color{blue}{\left(c \cdot \frac{a}{b}\right)}\right) \cdot \frac{1}{3 \cdot a} + \left(-\frac{b}{3 \cdot a}\right) \]

      8. metadata-eval 40.5%

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

      9. associate-/r* 40.6%

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

      10. metadata-eval 40.6%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{\frac{\color{blue}{1}}{3}}{a} + \left(-\frac{b}{3 \cdot a}\right) \]

      11. metadata-eval 40.6%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} + \left(-\frac{b}{3 \cdot a}\right) \]

      12. div-inv 40.9%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-\color{blue}{b \cdot \frac{1}{3 \cdot a}}\right) \]

      13. metadata-eval 40.9%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{\color{blue}{\frac{2}{2}}}{3 \cdot a}\right) \]

      14. associate-/r* 40.8%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \color{blue}{\frac{\frac{\frac{2}{2}}{3}}{a}}\right) \]

      15. metadata-eval 40.8%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{\frac{\color{blue}{1}}{3}}{a}\right) \]

      16. metadata-eval 40.8%

          \[\leadsto \left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{\color{blue}{0.3333333333333333}}{a}\right) \]

   8. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{0.3333333333333333}{a}\right)} \]
   9. Step-by-step derivation

      1. sub-neg 40.8%

         \[\leadsto \color{blue}{\left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} - b \cdot \frac{0.3333333333333333}{a}} \]

      2. distribute-rgt-out-- 41.2%

         \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\left(b + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right) - b\right)} \]

      3. +-commutative 41.2%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b\right)} - b\right) \]

      4. associate-+r- 68.0%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + \left(b - b\right)\right)} \]

      5. *-commutative 68.0%

         \[\leadsto \color{blue}{\left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + \left(b - b\right)\right) \cdot \frac{0.3333333333333333}{a}} \]

      6. +-inverses 68.0%

         \[\leadsto \left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + \color{blue}{0}\right) \cdot \frac{0.3333333333333333}{a} \]

      7. +-rgt-identity 68.0%

         \[\leadsto \color{blue}{\left(-1.5 \cdot \left(c \cdot \frac{a}{b}\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

      8. *-commutative 68.0%

         \[\leadsto \left(-1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right) \cdot \frac{0.3333333333333333}{a} \]

      9. associate-*l* 68.0%

         \[\leadsto \color{blue}{\left(\left(-1.5 \cdot \frac{a}{b}\right) \cdot c\right)} \cdot \frac{0.3333333333333333}{a} \]

      10. associate-*r/ 68.0%

          \[\leadsto \left(\color{blue}{\frac{-1.5 \cdot a}{b}} \cdot c\right) \cdot \frac{0.3333333333333333}{a} \]

      11. *-commutative 68.0%

          \[\leadsto \left(\frac{\color{blue}{a \cdot -1.5}}{b} \cdot c\right) \cdot \frac{0.3333333333333333}{a} \]

      12. *-rgt-identity 68.0%

          \[\leadsto \left(\frac{\color{blue}{\left(a \cdot -1.5\right) \cdot 1}}{b} \cdot c\right) \cdot \frac{0.3333333333333333}{a} \]

      13. associate-*r/ 68.0%

          \[\leadsto \left(\color{blue}{\left(\left(a \cdot -1.5\right) \cdot \frac{1}{b}\right)} \cdot c\right) \cdot \frac{0.3333333333333333}{a} \]

      14. *-commutative 68.0%

          \[\leadsto \color{blue}{\left(c \cdot \left(\left(a \cdot -1.5\right) \cdot \frac{1}{b}\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

      15. associate-*l* 68.0%

          \[\leadsto \left(c \cdot \color{blue}{\left(a \cdot \left(-1.5 \cdot \frac{1}{b}\right)\right)}\right) \cdot \frac{0.3333333333333333}{a} \]

      16. associate-*r/ 68.0%

          \[\leadsto \left(c \cdot \left(a \cdot \color{blue}{\frac{-1.5 \cdot 1}{b}}\right)\right) \cdot \frac{0.3333333333333333}{a} \]

      17. metadata-eval 68.0%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{\color{blue}{-1.5}}{b}\right)\right) \cdot \frac{0.3333333333333333}{a} \]

   10. Simplified 68.0%

       \[\leadsto \color{blue}{\left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \frac{0.3333333333333333}{a}} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{0.3333333333333333}{a} \cdot \frac{0.3333333333333333}{a}\right) \cdot \frac{0.3333333333333333}{a}}} \]

       2. *-commutative 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\frac{0.3333333333333333}{a} \cdot \frac{0.3333333333333333}{a}\right)}} \]

       3. frac-times 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \color{blue}{\frac{0.3333333333333333 \cdot 0.3333333333333333}{a \cdot a}}} \]

       4. metadata-eval 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{\color{blue}{0.1111111111111111}}{a \cdot a}} \]

       5. pow1 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{\color{blue}{{a}^{1}} \cdot a}} \]

       6. pow1 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{{a}^{1} \cdot \color{blue}{{a}^{1}}}} \]

       7. pow-sqr 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{\color{blue}{{a}^{\left(2 \cdot 1\right)}}}} \]

       8. metadata-eval 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{{a}^{\color{blue}{2}}}} \]

   12. Applied egg-rr 68.1%

       \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \color{blue}{\sqrt[3]{\frac{0.3333333333333333}{a} \cdot \frac{0.1111111111111111}{{a}^{2}}}} \]
   13. Step-by-step derivation

       1. associate-*r/ 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{\frac{0.3333333333333333}{a} \cdot 0.1111111111111111}{{a}^{2}}}} \]

       2. associate-*l/ 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{\color{blue}{\frac{0.3333333333333333 \cdot 0.1111111111111111}{a}}}{{a}^{2}}} \]

       3. metadata-eval 68.1%

          \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{\frac{\color{blue}{0.037037037037037035}}{a}}{{a}^{2}}} \]

   14. Simplified 68.1%

       \[\leadsto \left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \color{blue}{\sqrt[3]{\frac{\frac{0.037037037037037035}{a}}{{a}^{2}}}} \]

   if -3.0000000000000001e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -6e-11

   1. Initial program 72.1%

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

   if -6e-11 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 12.3%

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

   2. Taylor expanded in b around inf 94.7%

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

      1. associate-*r/ 94.7%

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

      2. *-commutative 94.7%

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

   4. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.086:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{elif}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\left(c \cdot \left(a \cdot \frac{-1.5}{b}\right)\right) \cdot \sqrt[3]{\frac{\frac{0.037037037037037035}{a}}{{a}^{2}}}\\ \mathbf{elif}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, c \cdot \frac{a}{b}, -1.125 \cdot \frac{\frac{\left(a \cdot c\right) \cdot \left(a \cdot \frac{c}{b}\right)}{b}}{b}\right)\right)}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -1.6875
   (/ (pow (* a c) 3.0) (pow b 5.0))
   (fma -1.5 (* c (/ a b)) (* -1.125 (/ (/ (* (* a c) (* a (/ c b))) b) b))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return fma(-1.6875, (pow((a * c), 3.0) / pow(b, 5.0)), fma(-1.5, (c * (a / b)), (-1.125 * ((((a * c) * (a * (c / b))) / b) / b)))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(fma(-1.6875, Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0)), fma(-1.5, Float64(c * Float64(a / b)), Float64(-1.125 * Float64(Float64(Float64(Float64(a * c) * Float64(a * Float64(c / b))) / b) / b)))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(-1.6875 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[(N[(N[(a * c), $MachinePrecision] * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.1%

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

2. Taylor expanded in b around inf 93.1%

   \[\leadsto \frac{\color{blue}{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
3. Step-by-step derivation

   1. fma-def 93.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]

   2. cube-prod 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{\color{blue}{{\left(a \cdot c\right)}^{3}}}{{b}^{5}}, -1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]

   3. fma-def 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \color{blue}{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}\right)}{3 \cdot a} \]

   4. associate-/l* 93.0%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \color{blue}{\frac{a}{\frac{b}{c}}}, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)\right)}{3 \cdot a} \]

   5. associate-/r/ 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \color{blue}{\frac{a}{b} \cdot c}, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)\right)}{3 \cdot a} \]

   6. associate-/l* 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \color{blue}{\frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}\right)\right)}{3 \cdot a} \]

   7. associate-/r/ 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \color{blue}{\left(\frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}\right)}\right)\right)}{3 \cdot a} \]

4. Simplified 93.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}\right)\right)\right)}}{3 \cdot a} \]
5. Step-by-step derivation

   1. associate-*l/ 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)\right)}{3 \cdot a} \]

   2. unpow3 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{\color{blue}{\left(b \cdot b\right) \cdot b}}\right)\right)}{3 \cdot a} \]

   3. associate-/r* 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \color{blue}{\frac{\frac{{a}^{2} \cdot {c}^{2}}{b \cdot b}}{b}}\right)\right)}{3 \cdot a} \]

   4. unpow2 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\frac{{a}^{2} \cdot \color{blue}{\left(c \cdot c\right)}}{b \cdot b}}{b}\right)\right)}{3 \cdot a} \]

   5. unpow2 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\frac{\color{blue}{\left(a \cdot a\right)} \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\right)\right)}{3 \cdot a} \]

   6. swap-sqr 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{b \cdot b}}{b}\right)\right)}{3 \cdot a} \]

   7. frac-times 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\color{blue}{\frac{a \cdot c}{b} \cdot \frac{a \cdot c}{b}}}{b}\right)\right)}{3 \cdot a} \]

   8. associate-*l/ 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot \frac{a \cdot c}{b}}{b}\right)\right)}{3 \cdot a} \]

   9. associate-*l/ 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\left(\frac{a}{b} \cdot c\right) \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}}{b}\right)\right)}{3 \cdot a} \]

   10. pow1 93.1%

       \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\color{blue}{{\left(\frac{a}{b} \cdot c\right)}^{1}} \cdot \left(\frac{a}{b} \cdot c\right)}{b}\right)\right)}{3 \cdot a} \]

   11. metadata-eval 93.1%

       \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{{\left(\frac{a}{b} \cdot c\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \left(\frac{a}{b} \cdot c\right)}{b}\right)\right)}{3 \cdot a} \]

   12. pow1 93.1%

       \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{{\left(\frac{a}{b} \cdot c\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(\frac{a}{b} \cdot c\right)}^{1}}}{b}\right)\right)}{3 \cdot a} \]

   13. metadata-eval 93.1%

       \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{{\left(\frac{a}{b} \cdot c\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{a}{b} \cdot c\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}}{b}\right)\right)}{3 \cdot a} \]

   14. pow-sqr 93.1%

       \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\color{blue}{{\left(\frac{a}{b} \cdot c\right)}^{\left(2 \cdot \frac{2}{2}\right)}}}{b}\right)\right)}{3 \cdot a} \]

   15. *-commutative 93.1%

       \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{{\color{blue}{\left(c \cdot \frac{a}{b}\right)}}^{\left(2 \cdot \frac{2}{2}\right)}}{b}\right)\right)}{3 \cdot a} \]

   16. metadata-eval 93.1%

       \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{{\left(c \cdot \frac{a}{b}\right)}^{\left(2 \cdot \color{blue}{1}\right)}}{b}\right)\right)}{3 \cdot a} \]

   17. metadata-eval 93.1%

       \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{{\left(c \cdot \frac{a}{b}\right)}^{\color{blue}{2}}}{b}\right)\right)}{3 \cdot a} \]

6. Applied egg-rr 93.1%

   \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \color{blue}{\frac{{\left(c \cdot \frac{a}{b}\right)}^{2}}{b}}\right)\right)}{3 \cdot a} \]
7. Step-by-step derivation

   1. unpow2 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\color{blue}{\left(c \cdot \frac{a}{b}\right) \cdot \left(c \cdot \frac{a}{b}\right)}}{b}\right)\right)}{3 \cdot a} \]

   2. associate-*r/ 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\left(c \cdot \frac{a}{b}\right) \cdot \color{blue}{\frac{c \cdot a}{b}}}{b}\right)\right)}{3 \cdot a} \]

   3. *-commutative 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\left(c \cdot \frac{a}{b}\right) \cdot \frac{\color{blue}{a \cdot c}}{b}}{b}\right)\right)}{3 \cdot a} \]

   4. associate-*r/ 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\color{blue}{\frac{\left(c \cdot \frac{a}{b}\right) \cdot \left(a \cdot c\right)}{b}}}{b}\right)\right)}{3 \cdot a} \]

   5. associate-*r/ 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\frac{\color{blue}{\frac{c \cdot a}{b}} \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)}{3 \cdot a} \]

   6. *-commutative 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\frac{\frac{\color{blue}{a \cdot c}}{b} \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)}{3 \cdot a} \]

   7. *-un-lft-identity 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\frac{\frac{a \cdot c}{\color{blue}{1 \cdot b}} \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)}{3 \cdot a} \]

   8. times-frac 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\frac{\color{blue}{\left(\frac{a}{1} \cdot \frac{c}{b}\right)} \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)}{3 \cdot a} \]

   9. /-rgt-identity 93.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\frac{\left(\color{blue}{a} \cdot \frac{c}{b}\right) \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)}{3 \cdot a} \]

8. Applied egg-rr 93.1%

   \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, \frac{a}{b} \cdot c, -1.125 \cdot \frac{\color{blue}{\frac{\left(a \cdot \frac{c}{b}\right) \cdot \left(a \cdot c\right)}{b}}}{b}\right)\right)}{3 \cdot a} \]

9. Final simplification 93.1%

   \[\leadsto \frac{\mathsf{fma}\left(-1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-1.5, c \cdot \frac{a}{b}, -1.125 \cdot \frac{\frac{\left(a \cdot c\right) \cdot \left(a \cdot \frac{c}{b}\right)}{b}}{b}\right)\right)}{a \cdot 3} \]

Alternative 8: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

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

Derivation

1. Initial program 32.1%

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

2. Taylor expanded in b around inf 90.4%

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

3. Final simplification 90.4%

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

Alternative 9: 81.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

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) / (b / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.1%

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

2. Taylor expanded in b around inf 94.4%

   \[\leadsto \frac{\color{blue}{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + \left(-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{{b}^{7}}\right)\right)}}{3 \cdot a} \]

3. Taylor expanded in a around 0 80.7%

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

   1. associate-*r/ 80.7%

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

   2. associate-/l* 80.5%

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

5. Simplified 80.5%

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

6. Final simplification 80.5%

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

Alternative 10: 81.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.1%

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

2. Taylor expanded in b around inf 80.7%

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

   1. associate-*r/ 80.7%

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

   2. *-commutative 80.7%

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

4. Simplified 80.7%

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

5. Final simplification 80.7%

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

Reproduce

herbie shell --seed 2023336 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))