Cubic critical, narrow range

Percentage Accurate: 54.9% → 92.0%

Time: 16.0s

Alternatives: 11

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0235:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0235)
   (/
    (- (sqrt (* (pow b 2.0) (+ 1.0 (/ (* a (* c -3.0)) (pow b 2.0))))) b)
    (* a 3.0))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
      (*
       a
       (+
        (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
        (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0235) {
		tmp = (sqrt((pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / pow(b, 2.0))))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 7.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.0235d0) then
        tmp = (sqrt(((b ** 2.0d0) * (1.0d0 + ((a * (c * (-3.0d0))) / (b ** 2.0d0))))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-1.0546875d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0235) {
		tmp = (Math.sqrt((Math.pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / Math.pow(b, 2.0))))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-1.0546875 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0)))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.0235:
		tmp = (math.sqrt((math.pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / math.pow(b, 2.0))))) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-1.0546875 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0)))))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0235)
		tmp = Float64(Float64(sqrt(Float64((b ^ 2.0) * Float64(1.0 + Float64(Float64(a * Float64(c * -3.0)) / (b ^ 2.0))))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.0235)
		tmp = (sqrt(((b ^ 2.0) * (1.0 + ((a * (c * -3.0)) / (b ^ 2.0))))) - b) / (a * 3.0);
	else
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-0.5625 * ((c ^ 3.0) / (b ^ 5.0))) + (-1.0546875 * ((a * (c ^ 4.0)) / (b ^ 7.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0235], N[(N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0235

   1. Initial program 89.5%

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

      1. /-rgt-identity 89.5%

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

      2. metadata-eval 89.5%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in b around inf 89.7%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

   if 0.0235 < b

   1. Initial program 52.5%

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

      1. /-rgt-identity 52.5%

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

      2. metadata-eval 52.5%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in a around 0 92.4%

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

   6. Taylor expanded in c around 0 92.4%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0235:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.026:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.026)
   (/
    (- (sqrt (* (pow b 2.0) (+ 1.0 (/ (* a (* c -3.0)) (pow b 2.0))))) b)
    (* a 3.0))
   (/
    (+
     (* c -0.5)
     (*
      a
      (+
       (* -0.5625 (/ (* a (pow c 3.0)) (pow b 4.0)))
       (* -0.375 (/ (pow c 2.0) (pow b 2.0))))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.026) {
		tmp = (sqrt((pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / pow(b, 2.0))))) - b) / (a * 3.0);
	} else {
		tmp = ((c * -0.5) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 4.0))) + (-0.375 * (pow(c, 2.0) / pow(b, 2.0)))))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.026d0) then
        tmp = (sqrt(((b ** 2.0d0) * (1.0d0 + ((a * (c * (-3.0d0))) / (b ** 2.0d0))))) - b) / (a * 3.0d0)
    else
        tmp = ((c * (-0.5d0)) + (a * (((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 4.0d0))) + ((-0.375d0) * ((c ** 2.0d0) / (b ** 2.0d0)))))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.026) {
		tmp = (Math.sqrt((Math.pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / Math.pow(b, 2.0))))) - b) / (a * 3.0);
	} else {
		tmp = ((c * -0.5) + (a * ((-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) + (-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 2.0)))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.026:
		tmp = (math.sqrt((math.pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / math.pow(b, 2.0))))) - b) / (a * 3.0)
	else:
		tmp = ((c * -0.5) + (a * ((-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 4.0))) + (-0.375 * (math.pow(c, 2.0) / math.pow(b, 2.0)))))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.026)
		tmp = Float64(Float64(sqrt(Float64((b ^ 2.0) * Float64(1.0 + Float64(Float64(a * Float64(c * -3.0)) / (b ^ 2.0))))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 2.0)))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.026)
		tmp = (sqrt(((b ^ 2.0) * (1.0 + ((a * (c * -3.0)) / (b ^ 2.0))))) - b) / (a * 3.0);
	else
		tmp = ((c * -0.5) + (a * ((-0.5625 * ((a * (c ^ 3.0)) / (b ^ 4.0))) + (-0.375 * ((c ^ 2.0) / (b ^ 2.0)))))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.026], N[(N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0259999999999999988

   1. Initial program 89.5%

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

      1. /-rgt-identity 89.5%

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

      2. metadata-eval 89.5%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in b around inf 89.7%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

   if 0.0259999999999999988 < b

   1. Initial program 52.5%

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

      1. /-rgt-identity 52.5%

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

      2. metadata-eval 52.5%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in b around inf 90.0%

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

   6. Taylor expanded in a around 0 90.0%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot c + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.026:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.028)
   (/
    (- (sqrt (* (pow b 2.0) (+ 1.0 (/ (* a (* c -3.0)) (pow b 2.0))))) b)
    (* a 3.0))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
      (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.028) {
		tmp = (sqrt((pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / pow(b, 2.0))))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.028d0) then
        tmp = (sqrt(((b ** 2.0d0) * (1.0d0 + ((a * (c * (-3.0d0))) / (b ** 2.0d0))))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + ((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.028) {
		tmp = (Math.sqrt((Math.pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / Math.pow(b, 2.0))))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0)))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.028:
		tmp = (math.sqrt((math.pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / math.pow(b, 2.0))))) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0)))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.028)
		tmp = Float64(Float64(sqrt(Float64((b ^ 2.0) * Float64(1.0 + Float64(Float64(a * Float64(c * -3.0)) / (b ^ 2.0))))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.028)
		tmp = (sqrt(((b ^ 2.0) * (1.0 + ((a * (c * -3.0)) / (b ^ 2.0))))) - b) / (a * 3.0);
	else
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (-0.5625 * ((a * (c ^ 3.0)) / (b ^ 5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.028], N[(N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0280000000000000006

   1. Initial program 89.5%

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

      1. /-rgt-identity 89.5%

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

      2. metadata-eval 89.5%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in b around inf 89.7%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

   if 0.0280000000000000006 < b

   1. Initial program 52.5%

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

      1. /-rgt-identity 52.5%

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

      2. metadata-eval 52.5%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in a around 0 90.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0295:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{4}} + -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0295)
   (/
    (- (sqrt (* (pow b 2.0) (+ 1.0 (/ (* a (* c -3.0)) (pow b 2.0))))) b)
    (* a 3.0))
   (/
    (*
     c
     (-
      (*
       c
       (+
        (* -0.5625 (/ (* c (pow a 2.0)) (pow b 4.0)))
        (* -0.375 (/ a (pow b 2.0)))))
      0.5))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0295) {
		tmp = (sqrt((pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / pow(b, 2.0))))) - b) / (a * 3.0);
	} else {
		tmp = (c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 4.0))) + (-0.375 * (a / pow(b, 2.0))))) - 0.5)) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.0295d0) then
        tmp = (sqrt(((b ** 2.0d0) * (1.0d0 + ((a * (c * (-3.0d0))) / (b ** 2.0d0))))) - b) / (a * 3.0d0)
    else
        tmp = (c * ((c * (((-0.5625d0) * ((c * (a ** 2.0d0)) / (b ** 4.0d0))) + ((-0.375d0) * (a / (b ** 2.0d0))))) - 0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0295) {
		tmp = (Math.sqrt((Math.pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / Math.pow(b, 2.0))))) - b) / (a * 3.0);
	} else {
		tmp = (c * ((c * ((-0.5625 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 4.0))) + (-0.375 * (a / Math.pow(b, 2.0))))) - 0.5)) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.0295:
		tmp = (math.sqrt((math.pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / math.pow(b, 2.0))))) - b) / (a * 3.0)
	else:
		tmp = (c * ((c * ((-0.5625 * ((c * math.pow(a, 2.0)) / math.pow(b, 4.0))) + (-0.375 * (a / math.pow(b, 2.0))))) - 0.5)) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0295)
		tmp = Float64(Float64(sqrt(Float64((b ^ 2.0) * Float64(1.0 + Float64(Float64(a * Float64(c * -3.0)) / (b ^ 2.0))))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 4.0))) + Float64(-0.375 * Float64(a / (b ^ 2.0))))) - 0.5)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.0295)
		tmp = (sqrt(((b ^ 2.0) * (1.0 + ((a * (c * -3.0)) / (b ^ 2.0))))) - b) / (a * 3.0);
	else
		tmp = (c * ((c * ((-0.5625 * ((c * (a ^ 2.0)) / (b ^ 4.0))) + (-0.375 * (a / (b ^ 2.0))))) - 0.5)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0295], N[(N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.029499999999999998

   1. Initial program 89.5%

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

      1. /-rgt-identity 89.5%

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

      2. metadata-eval 89.5%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in b around inf 89.7%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

   if 0.029499999999999998 < b

   1. Initial program 52.5%

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

      1. /-rgt-identity 52.5%

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

      2. metadata-eval 52.5%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in b around inf 90.0%

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

   6. Taylor expanded in c around 0 89.9%

      \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0295:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{4}} + -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.2)
   (/
    (- (sqrt (* (pow b 2.0) (+ 1.0 (/ (* a (* c -3.0)) (pow b 2.0))))) b)
    (* a 3.0))
   (/ 1.0 (/ (fma (* a (/ c b)) 1.5 (* b -2.0)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2) {
		tmp = (sqrt((pow(b, 2.0) * (1.0 + ((a * (c * -3.0)) / pow(b, 2.0))))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (fma((a * (c / b)), 1.5, (b * -2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(sqrt(Float64((b ^ 2.0) * Float64(1.0 + Float64(Float64(a * Float64(c * -3.0)) / (b ^ 2.0))))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(a * Float64(c / b)), 1.5, Float64(b * -2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2], N[(N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 1.5 + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 80.0%

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

      1. /-rgt-identity 80.0%

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

      2. metadata-eval 80.0%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in b around inf 80.1%

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

      1. associate-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. associate-*r* 80.2%

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

   7. Simplified 80.2%

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

   if 6.20000000000000018 < b

   1. Initial program 48.0%

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

      1. /-rgt-identity 48.0%

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

      2. metadata-eval 48.0%

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

   3. Simplified 48.2%

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

   5. Taylor expanded in b around inf 86.7%

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

      1. clear-num 86.7%

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

      2. inv-pow 86.7%

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

      3. *-commutative 86.7%

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

      4. +-commutative 86.7%

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

      5. fma-define 86.7%

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

      6. div-inv 86.7%

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

      7. pow-prod-down 86.7%

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

      8. pow-flip 86.7%

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

      9. metadata-eval 86.7%

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

   7. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-2}, -1.5 \cdot \left(a \cdot c\right)\right)}{b}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 86.7%

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

      2. associate-/r/ 86.7%

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

      3. fma-undefine 86.7%

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

      4. associate-*r* 86.7%

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

      5. fma-define 86.7%

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

      6. *-commutative 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in c around 0 87.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]
   11. Step-by-step derivation

       1. +-commutative 87.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} + -2 \cdot b}}{c}} \]

       2. *-commutative 87.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} \cdot 1.5} + -2 \cdot b}{c}} \]

       3. fma-define 87.4%

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

       4. associate-/l* 87.4%

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

       5. *-commutative 87.4%

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

   12. Simplified 87.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{a \cdot \left(c \cdot -3\right)}{{b}^{2}}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.2)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ 1.0 (/ (fma (* a (/ c b)) 1.5 (* b -2.0)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (fma((a * (c / b)), 1.5, (b * -2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(a * Float64(c / b)), 1.5, Float64(b * -2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 1.5 + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 80.0%

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

      1. /-rgt-identity 80.0%

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

      2. metadata-eval 80.0%

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

   3. Simplified 80.1%

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

   if 6.20000000000000018 < b

   1. Initial program 48.0%

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

      1. /-rgt-identity 48.0%

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

      2. metadata-eval 48.0%

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

   3. Simplified 48.2%

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

   5. Taylor expanded in b around inf 86.7%

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

      1. clear-num 86.7%

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

      2. inv-pow 86.7%

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

      3. *-commutative 86.7%

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

      4. +-commutative 86.7%

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

      5. fma-define 86.7%

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

      6. div-inv 86.7%

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

      7. pow-prod-down 86.7%

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

      8. pow-flip 86.7%

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

      9. metadata-eval 86.7%

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

   7. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-2}, -1.5 \cdot \left(a \cdot c\right)\right)}{b}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 86.7%

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

      2. associate-/r/ 86.7%

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

      3. fma-undefine 86.7%

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

      4. associate-*r* 86.7%

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

      5. fma-define 86.7%

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

      6. *-commutative 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in c around 0 87.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]
   11. Step-by-step derivation

       1. +-commutative 87.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} + -2 \cdot b}}{c}} \]

       2. *-commutative 87.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} \cdot 1.5} + -2 \cdot b}{c}} \]

       3. fma-define 87.4%

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

       4. associate-/l* 87.4%

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

       5. *-commutative 87.4%

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

   12. Simplified 87.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.2)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (/ 1.0 (/ (fma (* a (/ c b)) 1.5 (* b -2.0)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (fma((a * (c / b)), 1.5, (b * -2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(a * Float64(c / b)), 1.5, Float64(b * -2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2], 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], N[(1.0 / N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 1.5 + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 80.0%

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

   if 6.20000000000000018 < b

   1. Initial program 48.0%

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

      1. /-rgt-identity 48.0%

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

      2. metadata-eval 48.0%

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

   3. Simplified 48.2%

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

   5. Taylor expanded in b around inf 86.7%

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

      1. clear-num 86.7%

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

      2. inv-pow 86.7%

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

      3. *-commutative 86.7%

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

      4. +-commutative 86.7%

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

      5. fma-define 86.7%

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

      6. div-inv 86.7%

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

      7. pow-prod-down 86.7%

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

      8. pow-flip 86.7%

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

      9. metadata-eval 86.7%

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

   7. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-2}, -1.5 \cdot \left(a \cdot c\right)\right)}{b}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 86.7%

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

      2. associate-/r/ 86.7%

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

      3. fma-undefine 86.7%

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

      4. associate-*r* 86.7%

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

      5. fma-define 86.7%

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

      6. *-commutative 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in c around 0 87.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]
   11. Step-by-step derivation

       1. +-commutative 87.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} + -2 \cdot b}}{c}} \]

       2. *-commutative 87.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} \cdot 1.5} + -2 \cdot b}{c}} \]

       3. fma-define 87.4%

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

       4. associate-/l* 87.4%

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

       5. *-commutative 87.4%

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

   12. Simplified 87.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (fma (* a (/ c b)) 1.5 (* b -2.0)) c)))

C

double code(double a, double b, double c) {
	return 1.0 / (fma((a * (c / b)), 1.5, (b * -2.0)) / c);
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(fma(Float64(a * Float64(c / b)), 1.5, Float64(b * -2.0)) / c))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 1.5 + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

   1. /-rgt-identity 55.4%

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

   2. metadata-eval 55.4%

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

3. Simplified 55.6%

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

5. Taylor expanded in b around inf 80.9%

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

   1. clear-num 80.9%

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

   2. inv-pow 80.9%

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

   3. *-commutative 80.9%

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

   4. +-commutative 80.9%

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

   5. fma-define 80.9%

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

   6. div-inv 80.9%

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

   7. pow-prod-down 80.9%

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

   8. pow-flip 80.9%

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

   9. metadata-eval 80.9%

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

7. Applied egg-rr 80.9%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-2}, -1.5 \cdot \left(a \cdot c\right)\right)}{b}}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 80.9%

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

   2. associate-/r/ 80.9%

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

   3. fma-undefine 80.9%

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

   4. associate-*r* 80.9%

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

   5. fma-define 80.9%

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

   6. *-commutative 80.9%

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

9. Simplified 80.9%

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

10. Taylor expanded in c around 0 81.8%

    \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]
11. Step-by-step derivation

    1. +-commutative 81.8%

       \[\leadsto \frac{1}{\frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} + -2 \cdot b}}{c}} \]

    2. *-commutative 81.8%

       \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} \cdot 1.5} + -2 \cdot b}{c}} \]

    3. fma-define 81.8%

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

    4. associate-/l* 81.8%

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

    5. *-commutative 81.8%

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

12. Simplified 81.8%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, b \cdot -2\right)}{c}}} \]
13. Add Preprocessing

Alternative 9: 82.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{a \cdot c}{b}}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (+ (* b -2.0) (* 1.5 (/ (* a c) b))) c)))

C

double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + (1.5 * ((a * c) / 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 = 1.0d0 / (((b * (-2.0d0)) + (1.5d0 * ((a * c) / b))) / c)
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + (1.5 * ((a * c) / b))) / c);
}

Python

def code(a, b, c):
	return 1.0 / (((b * -2.0) + (1.5 * ((a * c) / b))) / c)

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(1.5 * Float64(Float64(a * c) / b))) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (((b * -2.0) + (1.5 * ((a * c) / b))) / c);
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

   1. /-rgt-identity 55.4%

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

   2. metadata-eval 55.4%

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

3. Simplified 55.6%

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

5. Taylor expanded in b around inf 80.9%

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

   1. clear-num 80.9%

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

   2. inv-pow 80.9%

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

   3. *-commutative 80.9%

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

   4. +-commutative 80.9%

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

   5. fma-define 80.9%

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

   6. div-inv 80.9%

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

   7. pow-prod-down 80.9%

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

   8. pow-flip 80.9%

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

   9. metadata-eval 80.9%

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

7. Applied egg-rr 80.9%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-2}, -1.5 \cdot \left(a \cdot c\right)\right)}{b}}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 80.9%

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

   2. associate-/r/ 80.9%

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

   3. fma-undefine 80.9%

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

   4. associate-*r* 80.9%

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

   5. fma-define 80.9%

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

   6. *-commutative 80.9%

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

9. Simplified 80.9%

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

10. Taylor expanded in c around 0 81.8%

    \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]

11. Final simplification 81.8%

    \[\leadsto \frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{a \cdot c}{b}}{c}} \]
12. Add Preprocessing

Alternative 10: 82.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

   1. /-rgt-identity 55.4%

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

   2. metadata-eval 55.4%

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

3. Simplified 55.6%

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

5. Taylor expanded in b around inf 80.9%

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

   1. clear-num 80.9%

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

   2. inv-pow 80.9%

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

   3. *-commutative 80.9%

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

   4. +-commutative 80.9%

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

   5. fma-define 80.9%

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

   6. div-inv 80.9%

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

   7. pow-prod-down 80.9%

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

   8. pow-flip 80.9%

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

   9. metadata-eval 80.9%

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

7. Applied egg-rr 80.9%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-2}, -1.5 \cdot \left(a \cdot c\right)\right)}{b}}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 80.9%

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

   2. associate-/r/ 80.9%

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

   3. fma-undefine 80.9%

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

   4. associate-*r* 80.9%

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

   5. fma-define 80.9%

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

   6. *-commutative 80.9%

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

9. Simplified 80.9%

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

10. Taylor expanded in a around 0 81.7%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
11. Add Preprocessing

Alternative 11: 64.8% 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 55.4%

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

   1. /-rgt-identity 55.4%

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

   2. metadata-eval 55.4%

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

3. Simplified 55.6%

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

5. Taylor expanded in b around inf 63.8%

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

Reproduce

herbie shell --seed 2024136 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))