Cubic critical

Percentage Accurate: 51.9% → 85.4%

Time: 12.1s

Alternatives: 12

Speedup: 16.4×

Specification

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: 51.9% 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: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+154)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 4.8e-99)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 4.8e-99) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} 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) :: tmp
    if (b <= (-8.5d+154)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 4.8d-99) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 4.8e-99) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+154:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 4.8e-99:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+154)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 4.8e-99)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e+154)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 4.8e-99)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+154], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.8e-99], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5000000000000002e154

   1. Initial program 31.0%

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

      1. sqr-neg 31.0%

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

      2. sqr-neg 31.0%

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

      3. associate-*l* 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in b around -inf 99.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   7. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if -8.5000000000000002e154 < b < 4.8000000000000001e-99

   1. Initial program 82.7%

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

   if 4.8000000000000001e-99 < b

   1. Initial program 15.3%

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

      1. sqr-neg 15.3%

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

      2. sqr-neg 15.3%

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

      3. associate-*l* 15.3%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in b around inf 86.5%

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

      1. associate-*r/ 86.6%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+124}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+124)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 4.8e-99)
     (* (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) (/ 0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+124) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 4.8e-99) {
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) * (0.3333333333333333 / a);
	} 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) :: tmp
    if (b <= (-2d+124)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 4.8d-99) then
        tmp = (sqrt(((b * b) - (a * (3.0d0 * c)))) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+124) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 4.8e-99) {
		tmp = (Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+124:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 4.8e-99:
		tmp = (math.sqrt(((b * b) - (a * (3.0 * c)))) - b) * (0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+124)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 4.8e-99)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+124)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 4.8e-99)
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) * (0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+124], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.8e-99], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9999999999999999e124

   1. Initial program 40.0%

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

      1. sqr-neg 40.0%

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

      2. sqr-neg 40.0%

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

      3. associate-*l* 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   6. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if -1.9999999999999999e124 < b < 4.8000000000000001e-99

   1. Initial program 81.8%

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

      1. neg-sub0 81.8%

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

      2. sqr-neg 81.8%

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

      3. associate-+l- 81.8%

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

      4. sub0-neg 81.8%

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

      5. neg-mul-1 81.8%

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

   3. Simplified 81.7%

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

      1. associate-*r* 81.7%

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

      2. metadata-eval 81.7%

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

      3. distribute-rgt-neg-in 81.7%

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

      4. *-commutative 81.7%

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

      5. fma-neg 81.7%

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

      6. associate-*r* 81.7%

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

      7. *-commutative 81.7%

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

      8. associate-*l* 81.7%

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

   5. Applied egg-rr 81.7%

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

      1. *-commutative 81.7%

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

   7. Simplified 81.7%

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

      1. div-inv 81.6%

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

      2. clear-num 81.7%

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

   9. Applied egg-rr 81.7%

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

   if 4.8000000000000001e-99 < b

   1. Initial program 15.3%

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

      1. sqr-neg 15.3%

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

      2. sqr-neg 15.3%

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

      3. associate-*l* 15.3%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in b around inf 86.5%

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

      1. associate-*r/ 86.6%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+124}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+154)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 4.8e-99)
     (* (/ (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) a) 0.3333333333333333)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 4.8e-99) {
		tmp = ((sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333;
	} 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) :: tmp
    if (b <= (-8.5d+154)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 4.8d-99) then
        tmp = ((sqrt(((b * b) - (a * (3.0d0 * c)))) - b) / a) * 0.3333333333333333d0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 4.8e-99) {
		tmp = ((Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+154:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 4.8e-99:
		tmp = ((math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+154)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 4.8e-99)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e+154)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 4.8e-99)
		tmp = ((sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+154], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.8e-99], N[(N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5000000000000002e154

   1. Initial program 31.0%

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

      1. sqr-neg 31.0%

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

      2. sqr-neg 31.0%

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

      3. associate-*l* 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in b around -inf 99.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   7. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if -8.5000000000000002e154 < b < 4.8000000000000001e-99

   1. Initial program 82.7%

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

      1. neg-sub0 82.7%

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

      2. sqr-neg 82.7%

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

      3. associate-+l- 82.7%

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

      4. sub0-neg 82.7%

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

      5. neg-mul-1 82.7%

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

   3. Simplified 82.5%

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

      1. associate-*r* 82.5%

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

      2. metadata-eval 82.5%

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

      3. distribute-rgt-neg-in 82.5%

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

      4. *-commutative 82.5%

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

      5. fma-neg 82.5%

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

      6. associate-*r* 82.5%

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

      7. *-commutative 82.5%

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

      8. associate-*l* 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. *-commutative 82.5%

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

   7. Simplified 82.5%

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

      1. associate-/r/ 82.6%

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

   9. Applied egg-rr 82.6%

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

   if 4.8000000000000001e-99 < b

   1. Initial program 15.3%

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

      1. sqr-neg 15.3%

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

      2. sqr-neg 15.3%

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

      3. associate-*l* 15.3%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in b around inf 86.5%

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

      1. associate-*r/ 86.6%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+154)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.9e-99)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-99) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} 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) :: tmp
    if (b <= (-8.5d+154)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.9d-99) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-99) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+154:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.9e-99:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+154)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.9e-99)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e+154)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.9e-99)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+154], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.9e-99], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5000000000000002e154

   1. Initial program 31.0%

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

      1. sqr-neg 31.0%

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

      2. sqr-neg 31.0%

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

      3. associate-*l* 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in b around -inf 99.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   7. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if -8.5000000000000002e154 < b < 1.8999999999999998e-99

   1. Initial program 82.7%

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

      1. sqr-neg 82.7%

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

      2. sqr-neg 82.7%

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

      3. associate-*l* 82.6%

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

   3. Simplified 82.6%

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

   if 1.8999999999999998e-99 < b

   1. Initial program 15.3%

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

      1. sqr-neg 15.3%

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

      2. sqr-neg 15.3%

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

      3. associate-*l* 15.3%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in b around inf 86.5%

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

      1. associate-*r/ 86.6%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-87)
   (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
   (if (<= b 4.5e-99)
     (* (/ 0.3333333333333333 a) (- (sqrt (* c (* a -3.0))) b))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-87) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 4.5e-99) {
		tmp = (0.3333333333333333 / a) * (sqrt((c * (a * -3.0))) - b);
	} 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) :: tmp
    if (b <= (-1.35d-87)) then
        tmp = ((((a * 1.5d0) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 4.5d-99) then
        tmp = (0.3333333333333333d0 / a) * (sqrt((c * (a * (-3.0d0)))) - b)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-87) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 4.5e-99) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt((c * (a * -3.0))) - b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e-87:
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 4.5e-99:
		tmp = (0.3333333333333333 / a) * (math.sqrt((c * (a * -3.0))) - b)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-87)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 4.5e-99)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(c * Float64(a * -3.0))) - b));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.35e-87)
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 4.5e-99)
		tmp = (0.3333333333333333 / a) * (sqrt((c * (a * -3.0))) - b);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-87], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-99], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.34999999999999992e-87

   1. Initial program 64.6%

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

      1. sqr-neg 64.6%

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

      2. sqr-neg 64.6%

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

      3. associate-*l* 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in b around -inf 76.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-commutative 76.9%

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

      2. mul-1-neg 76.9%

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

      3. unsub-neg 76.9%

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

      4. associate-/l* 83.2%

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

      5. associate-*r/ 83.2%

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

   6. Simplified 83.2%

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

   if -1.34999999999999992e-87 < b < 4.5000000000000003e-99

   1. Initial program 77.1%

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

      1. neg-sub0 77.1%

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

      2. sqr-neg 77.1%

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

      3. associate-+l- 77.1%

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

      4. sub0-neg 77.1%

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

      5. neg-mul-1 77.1%

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

   3. Simplified 77.0%

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

      1. associate-*r* 77.0%

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

      2. metadata-eval 77.0%

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

      3. distribute-rgt-neg-in 77.0%

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

      4. *-commutative 77.0%

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

      5. fma-neg 77.0%

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

      6. associate-*r* 77.0%

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

      7. *-commutative 77.0%

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

      8. associate-*l* 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

      1. expm1-log1p-u 57.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{\frac{a}{0.3333333333333333}}\right)\right)} \]

      2. expm1-udef 20.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{\frac{a}{0.3333333333333333}}\right)} - 1} \]

      3. div-inv 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right) \cdot \frac{1}{\frac{a}{0.3333333333333333}}}\right)} - 1 \]

      4. clear-num 20.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right) \cdot \color{blue}{\frac{0.3333333333333333}{a}}\right)} - 1 \]

   9. Applied egg-rr 20.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 57.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\right)\right)} \]

       2. expm1-log1p 77.0%

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

       3. associate-*r* 77.1%

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

       4. *-commutative 77.1%

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

       5. cancel-sign-sub-inv 77.1%

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

       6. distribute-lft-neg-in 77.1%

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

       7. +-commutative 77.1%

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

       8. distribute-lft-neg-in 77.1%

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

       9. metadata-eval 77.1%

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

       10. fma-def 77.0%

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

   11. Simplified 77.0%

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

   12. Taylor expanded in a around inf 72.5%

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

       1. *-commutative 72.5%

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

       2. *-commutative 72.5%

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

       3. associate-*l* 72.5%

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

   14. Simplified 72.5%

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

   if 4.5000000000000003e-99 < b

   1. Initial program 15.3%

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

      1. sqr-neg 15.3%

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

      2. sqr-neg 15.3%

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

      3. associate-*l* 15.3%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in b around inf 86.5%

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

      1. associate-*r/ 86.6%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.76 \cdot 10^{-99}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-87)
   (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
   (if (<= b 1.76e-99)
     (* 0.3333333333333333 (/ (- (sqrt (* (* a c) -3.0)) b) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-87) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 1.76e-99) {
		tmp = 0.3333333333333333 * ((sqrt(((a * c) * -3.0)) - b) / a);
	} 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) :: tmp
    if (b <= (-2.6d-87)) then
        tmp = ((((a * 1.5d0) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 1.76d-99) then
        tmp = 0.3333333333333333d0 * ((sqrt(((a * c) * (-3.0d0))) - b) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-87) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 1.76e-99) {
		tmp = 0.3333333333333333 * ((Math.sqrt(((a * c) * -3.0)) - b) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e-87:
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 1.76e-99:
		tmp = 0.3333333333333333 * ((math.sqrt(((a * c) * -3.0)) - b) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e-87)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 1.76e-99)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.6e-87)
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 1.76e-99)
		tmp = 0.3333333333333333 * ((sqrt(((a * c) * -3.0)) - b) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e-87], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.76e-99], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.60000000000000002e-87

   1. Initial program 64.6%

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

      1. sqr-neg 64.6%

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

      2. sqr-neg 64.6%

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

      3. associate-*l* 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in b around -inf 76.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-commutative 76.9%

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

      2. mul-1-neg 76.9%

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

      3. unsub-neg 76.9%

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

      4. associate-/l* 83.2%

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

      5. associate-*r/ 83.2%

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

   6. Simplified 83.2%

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

   if -2.60000000000000002e-87 < b < 1.75999999999999991e-99

   1. Initial program 77.1%

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

      1. neg-sub0 77.1%

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

      2. sqr-neg 77.1%

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

      3. associate-+l- 77.1%

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

      4. sub0-neg 77.1%

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

      5. neg-mul-1 77.1%

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

   3. Simplified 77.0%

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

      1. associate-*r* 77.0%

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

      2. metadata-eval 77.0%

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

      3. distribute-rgt-neg-in 77.0%

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

      4. *-commutative 77.0%

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

      5. fma-neg 77.0%

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

      6. associate-*r* 77.0%

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

      7. *-commutative 77.0%

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

      8. associate-*l* 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

      1. associate-/r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

   10. Taylor expanded in b around 0 72.5%

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

   if 1.75999999999999991e-99 < b

   1. Initial program 15.3%

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

      1. sqr-neg 15.3%

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

      2. sqr-neg 15.3%

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

      3. associate-*l* 15.3%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in b around inf 86.5%

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

      1. associate-*r/ 86.6%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.76 \cdot 10^{-99}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 68.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} 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) :: tmp
    if (b <= (-1d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.0%

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

      1. sqr-neg 70.0%

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

      2. sqr-neg 70.0%

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

      3. associate-*l* 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in b around -inf 63.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -9.999999999999969e-311 < b

   1. Initial program 30.2%

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

      1. sqr-neg 30.2%

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

      2. sqr-neg 30.2%

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

      3. associate-*l* 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in b around inf 66.4%

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

      1. associate-*r/ 66.4%

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

   6. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8: 68.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (* b -2.0) (* a 3.0)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (b * -2.0) / (a * 3.0);
	} 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) :: tmp
    if (b <= (-1d-310)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.0%

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

      1. sqr-neg 70.0%

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

      2. sqr-neg 70.0%

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

      3. associate-*l* 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in b around -inf 63.5%

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

      1. *-commutative 63.5%

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   6. Simplified 63.5%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if -9.999999999999969e-311 < b

   1. Initial program 30.2%

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

      1. sqr-neg 30.2%

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

      2. sqr-neg 30.2%

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

      3. associate-*l* 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in b around inf 66.4%

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

      1. associate-*r/ 66.4%

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

   6. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9: 68.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.0%

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

      1. neg-sub0 70.0%

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

      2. sqr-neg 70.0%

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

      3. associate-+l- 70.0%

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

      4. sub0-neg 70.0%

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

      5. neg-mul-1 70.0%

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

   3. Simplified 69.8%

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

      1. associate-*r* 69.8%

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

      2. metadata-eval 69.8%

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

      3. distribute-rgt-neg-in 69.8%

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

      4. *-commutative 69.8%

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

      5. fma-neg 69.8%

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

      6. associate-*r* 69.8%

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

      7. *-commutative 69.8%

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

      8. associate-*l* 69.8%

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

   5. Applied egg-rr 69.8%

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

      1. *-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in b around -inf 63.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 63.3%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 63.3%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

   10. Simplified 63.3%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 63.4%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   12. Applied egg-rr 63.4%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   if -9.999999999999969e-311 < b

   1. Initial program 30.2%

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

      1. sqr-neg 30.2%

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

      2. sqr-neg 30.2%

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

      3. associate-*l* 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in b around inf 66.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 68.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (* -0.6666666666666666 (/ b a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.0%

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

      1. sqr-neg 70.0%

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

      2. sqr-neg 70.0%

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

      3. associate-*l* 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in b around -inf 63.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. *-commutative 63.4%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   6. Simplified 63.4%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   if -9.999999999999969e-311 < b

   1. Initial program 30.2%

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

      1. sqr-neg 30.2%

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

      2. sqr-neg 30.2%

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

      3. associate-*l* 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in b around inf 66.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 68.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (* -0.6666666666666666 (/ b a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} 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) :: tmp
    if (b <= (-1d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.0%

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

      1. sqr-neg 70.0%

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

      2. sqr-neg 70.0%

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

      3. associate-*l* 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in b around -inf 63.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. *-commutative 63.4%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   6. Simplified 63.4%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   if -9.999999999999969e-311 < b

   1. Initial program 30.2%

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

      1. sqr-neg 30.2%

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

      2. sqr-neg 30.2%

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

      3. associate-*l* 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in b around inf 66.4%

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

      1. associate-*r/ 66.4%

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

   6. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 12: 34.9% 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 50.4%

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

   1. sqr-neg 50.4%

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

   2. sqr-neg 50.4%

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

   3. associate-*l* 50.4%

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

3. Simplified 50.4%

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

4. Taylor expanded in b around inf 33.8%

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

5. Final simplification 33.8%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))