quadp (p42, positive)

Percentage Accurate: 52.4% → 84.8%

Time: 12.0s

Alternatives: 9

Speedup: 12.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * 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) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * 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) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+152)
   (- (/ c b) (/ b a))
   (if (<= b 6.6)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+152) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = 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+152)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.6d0) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+152) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+152:
		tmp = (c / b) - (b / a)
	elif b <= 6.6:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+152)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+152)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.6)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+152], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e152

   1. Initial program 41.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 41.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 41.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 94.7%

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

      1. mul-1-neg 94.7%

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

      2. distribute-rgt-neg-in 94.7%

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

      3. +-commutative 94.7%

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

      4. mul-1-neg 94.7%

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

      5. unsub-neg 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in a around 0 89.1%

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

   9. Taylor expanded in a around inf 94.8%

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

       1. +-commutative 94.8%

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

       2. neg-mul-1 94.8%

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

       3. unsub-neg 94.8%

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

   11. Simplified 94.8%

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

   if -1e152 < b < 6.5999999999999996

   1. Initial program 83.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   if 6.5999999999999996 < b

   1. Initial program 8.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 8.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 91.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.8:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.8e+123)
   (- (/ c b) (/ b a))
   (if (<= b 6.8)
     (* (- b (sqrt (+ (* b b) (* (* c a) -4.0)))) (/ -0.5 a))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.8e+123) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.8) {
		tmp = (b - sqrt(((b * b) + ((c * a) * -4.0)))) * (-0.5 / a);
	} else {
		tmp = 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 <= (-4.8d+123)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.8d0) then
        tmp = (b - sqrt(((b * b) + ((c * a) * (-4.0d0))))) * ((-0.5d0) / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.8e+123) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.8) {
		tmp = (b - Math.sqrt(((b * b) + ((c * a) * -4.0)))) * (-0.5 / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.8e+123:
		tmp = (c / b) - (b / a)
	elif b <= 6.8:
		tmp = (b - math.sqrt(((b * b) + ((c * a) * -4.0)))) * (-0.5 / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.8e+123)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.8)
		tmp = Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -4.0)))) * Float64(-0.5 / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.8e+123)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.8)
		tmp = (b - sqrt(((b * b) + ((c * a) * -4.0)))) * (-0.5 / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.8e+123], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8], N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.79999999999999978e123

   1. Initial program 58.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 58.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 96.1%

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

      1. mul-1-neg 96.1%

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

      2. distribute-rgt-neg-in 96.1%

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

      3. +-commutative 96.1%

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

      4. mul-1-neg 96.1%

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

      5. unsub-neg 96.1%

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

   7. Simplified 96.1%

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

   8. Taylor expanded in a around 0 92.3%

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

   9. Taylor expanded in a around inf 96.3%

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

       1. +-commutative 96.3%

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

       2. neg-mul-1 96.3%

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

       3. unsub-neg 96.3%

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

   11. Simplified 96.3%

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

   if -4.79999999999999978e123 < b < 6.79999999999999982

   1. Initial program 81.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 81.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 81.5%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{-a \cdot 2}} \]

      2. div-inv 81.3%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   6. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}} \]
   7. Step-by-step derivation

      1. fma-undefine 81.3%

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + {b}^{2}}}\right) \cdot \frac{1}{a \cdot -2} \]

   8. Applied egg-rr 81.3%

      \[\leadsto \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + {b}^{2}}}\right) \cdot \frac{1}{a \cdot -2} \]
   9. Step-by-step derivation

      1. unpow2 81.3%

         \[\leadsto \left(b - \sqrt{\left(a \cdot c\right) \cdot -4 + \color{blue}{b \cdot b}}\right) \cdot \frac{1}{a \cdot -2} \]

   10. Applied egg-rr 81.3%

       \[\leadsto \left(b - \sqrt{\left(a \cdot c\right) \cdot -4 + \color{blue}{b \cdot b}}\right) \cdot \frac{1}{a \cdot -2} \]

   11. Taylor expanded in a around 0 81.3%

       \[\leadsto \left(b - \sqrt{\left(a \cdot c\right) \cdot -4 + b \cdot b}\right) \cdot \color{blue}{\frac{-0.5}{a}} \]

   if 6.79999999999999982 < b

   1. Initial program 8.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 8.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 91.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.8:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6:\\ \;\;\;\;\frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-85)
   (- (/ c b) (/ b a))
   (if (<= b 6.6) (/ (- b (sqrt (* a (* c -4.0)))) (* a -2.0)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-85) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6) {
		tmp = (b - sqrt((a * (c * -4.0)))) / (a * -2.0);
	} else {
		tmp = 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 <= (-4.3d-85)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.6d0) then
        tmp = (b - sqrt((a * (c * (-4.0d0))))) / (a * (-2.0d0))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-85) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6) {
		tmp = (b - Math.sqrt((a * (c * -4.0)))) / (a * -2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.3e-85:
		tmp = (c / b) - (b / a)
	elif b <= 6.6:
		tmp = (b - math.sqrt((a * (c * -4.0)))) / (a * -2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.3e-85)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.6)
		tmp = Float64(Float64(b - sqrt(Float64(a * Float64(c * -4.0)))) / Float64(a * -2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.3e-85)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.6)
		tmp = (b - sqrt((a * (c * -4.0)))) / (a * -2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.3e-85], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6], N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.29999999999999999e-85

   1. Initial program 74.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 74.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. distribute-rgt-neg-in 82.9%

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

      3. +-commutative 82.9%

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

      4. mul-1-neg 82.9%

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

      5. unsub-neg 82.9%

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

   7. Simplified 82.9%

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

   8. Taylor expanded in a around 0 81.1%

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

   9. Taylor expanded in a around inf 83.2%

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

       1. +-commutative 83.2%

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

       2. neg-mul-1 83.2%

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

       3. unsub-neg 83.2%

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

   11. Simplified 83.2%

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

   if -4.29999999999999999e-85 < b < 6.5999999999999996

   1. Initial program 75.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 75.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 75.4%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{-a \cdot 2}} \]

      2. div-inv 75.3%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   6. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}} \]

   7. Taylor expanded in a around inf 70.0%

      \[\leadsto \left(b - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{1}{a \cdot -2} \]
   8. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \cdot \frac{1}{a \cdot -2} \]

      2. associate-*l* 70.0%

         \[\leadsto \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \cdot \frac{1}{a \cdot -2} \]

   9. Simplified 70.0%

      \[\leadsto \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \cdot \frac{1}{a \cdot -2} \]
   10. Step-by-step derivation

       1. un-div-inv 70.1%

          \[\leadsto \color{blue}{\frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]

   11. Applied egg-rr 70.1%

       \[\leadsto \color{blue}{\frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]

   if 6.5999999999999996 < b

   1. Initial program 8.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 8.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 91.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6:\\ \;\;\;\;\frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-87)
   (- (/ c b) (/ b a))
   (if (<= b 6.6) (* (/ -0.5 a) (- b (sqrt (* (* c a) -4.0)))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6) {
		tmp = (-0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	} else {
		tmp = 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 <= (-5d-87)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.6d0) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((c * a) * (-4.0d0))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((c * a) * -4.0)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-87:
		tmp = (c / b) - (b / a)
	elif b <= 6.6:
		tmp = (-0.5 / a) * (b - math.sqrt(((c * a) * -4.0)))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-87)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.6)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(c * a) * -4.0))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-87)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.6)
		tmp = (-0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-87], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000042e-87

   1. Initial program 74.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 74.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. distribute-rgt-neg-in 82.9%

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

      3. +-commutative 82.9%

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

      4. mul-1-neg 82.9%

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

      5. unsub-neg 82.9%

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

   7. Simplified 82.9%

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

   8. Taylor expanded in a around 0 81.1%

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

   9. Taylor expanded in a around inf 83.2%

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

       1. +-commutative 83.2%

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

       2. neg-mul-1 83.2%

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

       3. unsub-neg 83.2%

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

   11. Simplified 83.2%

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

   if -5.00000000000000042e-87 < b < 6.5999999999999996

   1. Initial program 75.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 75.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 75.4%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{-a \cdot 2}} \]

      2. div-inv 75.3%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   6. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -2}} \]

   7. Taylor expanded in a around inf 70.0%

      \[\leadsto \left(b - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{1}{a \cdot -2} \]
   8. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \cdot \frac{1}{a \cdot -2} \]

      2. associate-*l* 70.0%

         \[\leadsto \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \cdot \frac{1}{a \cdot -2} \]

   9. Simplified 70.0%

      \[\leadsto \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \cdot \frac{1}{a \cdot -2} \]
   10. Step-by-step derivation

       1. *-commutative 70.0%

          \[\leadsto \color{blue}{\frac{1}{a \cdot -2} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]

       2. sub-neg 70.0%

          \[\leadsto \frac{1}{a \cdot -2} \cdot \color{blue}{\left(b + \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)} \]

       3. distribute-lft-in 70.0%

          \[\leadsto \color{blue}{\frac{1}{a \cdot -2} \cdot b + \frac{1}{a \cdot -2} \cdot \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]

       4. associate-/r* 70.0%

          \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-2}} \cdot b + \frac{1}{a \cdot -2} \cdot \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right) \]

       5. div-inv 70.0%

          \[\leadsto \color{blue}{\left(\frac{1}{a} \cdot \frac{1}{-2}\right)} \cdot b + \frac{1}{a \cdot -2} \cdot \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right) \]

       6. metadata-eval 70.0%

          \[\leadsto \left(\frac{1}{a} \cdot \color{blue}{-0.5}\right) \cdot b + \frac{1}{a \cdot -2} \cdot \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right) \]

       7. associate-/r* 70.0%

          \[\leadsto \left(\frac{1}{a} \cdot -0.5\right) \cdot b + \color{blue}{\frac{\frac{1}{a}}{-2}} \cdot \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right) \]

       8. div-inv 70.0%

          \[\leadsto \left(\frac{1}{a} \cdot -0.5\right) \cdot b + \color{blue}{\left(\frac{1}{a} \cdot \frac{1}{-2}\right)} \cdot \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right) \]

       9. metadata-eval 70.0%

          \[\leadsto \left(\frac{1}{a} \cdot -0.5\right) \cdot b + \left(\frac{1}{a} \cdot \color{blue}{-0.5}\right) \cdot \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right) \]

   11. Applied egg-rr 70.0%

       \[\leadsto \color{blue}{\left(\frac{1}{a} \cdot -0.5\right) \cdot b + \left(\frac{1}{a} \cdot -0.5\right) \cdot \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]
   12. Step-by-step derivation

       1. distribute-lft-out 70.0%

          \[\leadsto \color{blue}{\left(\frac{1}{a} \cdot -0.5\right) \cdot \left(b + \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)} \]

       2. associate-*l/ 70.0%

          \[\leadsto \color{blue}{\frac{1 \cdot -0.5}{a}} \cdot \left(b + \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \]

       3. metadata-eval 70.0%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \left(-\sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \]

       4. sub-neg 70.0%

          \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]

       5. associate-*r* 70.0%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

       6. *-commutative 70.0%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \]

   13. Simplified 70.0%

       \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)} \]

   if 6.5999999999999996 < b

   1. Initial program 8.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 8.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 91.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 78.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 78.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 69.5%

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

      1. mul-1-neg 69.5%

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

      2. distribute-rgt-neg-in 69.5%

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

      3. +-commutative 69.5%

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

      4. mul-1-neg 69.5%

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

      5. unsub-neg 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in a around 0 68.2%

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

   9. Taylor expanded in a around inf 69.8%

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

       1. +-commutative 69.8%

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

       2. neg-mul-1 69.8%

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

       3. unsub-neg 69.8%

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

   11. Simplified 69.8%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 30.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 30.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 65.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 65.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 65.6%

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

   7. Simplified 65.6%

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

4. Final simplification 67.7%

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

Alternative 6: 66.9% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 78.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 78.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 69.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 69.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. mul-1-neg 69.3%

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

   7. Simplified 69.3%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 30.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 30.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 65.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 65.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 65.6%

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

   7. Simplified 65.6%

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

4. Final simplification 67.5%

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

Alternative 7: 42.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 4.3e+17) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.3e+17) {
		tmp = b / -a;
	} else {
		tmp = 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 <= 4.3d+17) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.3e+17) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.3e+17:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.3e+17)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.3e+17)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.3e+17], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.3e17

   1. Initial program 74.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 74.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 50.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 50.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. mul-1-neg 50.8%

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

   7. Simplified 50.8%

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

   if 4.3e17 < b

   1. Initial program 8.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 8.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 2.3%

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

      1. mul-1-neg 2.3%

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

      2. distribute-rgt-neg-in 2.3%

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

      3. +-commutative 2.3%

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

      4. mul-1-neg 2.3%

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

      5. unsub-neg 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in b around 0 30.6%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.1% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 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 = c / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 55.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

3. Simplified 55.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around -inf 36.2%

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

   1. mul-1-neg 36.2%

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

   2. distribute-rgt-neg-in 36.2%

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

   3. +-commutative 36.2%

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

   4. mul-1-neg 36.2%

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

   5. unsub-neg 36.2%

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

7. Simplified 36.2%

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

8. Taylor expanded in b around 0 11.2%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
9. Add Preprocessing

Alternative 9: 2.5% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return b / 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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 55.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

3. Simplified 55.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 54.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}} \]

   2. inv-pow 54.9%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{-1}} \]

   3. add-sqr-sqrt 39.7%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{-1} \]

   4. sqrt-unprod 53.2%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{-1} \]

   5. sqr-neg 53.2%

      \[\leadsto {\left(\frac{a \cdot 2}{\sqrt{\color{blue}{b \cdot b}} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{-1} \]

   6. sqrt-prod 13.6%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{-1} \]

   7. add-sqr-sqrt 32.4%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{b} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{-1} \]

   8. sub-neg 32.4%

      \[\leadsto {\left(\frac{a \cdot 2}{b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}}\right)}^{-1} \]

   9. +-commutative 32.4%

      \[\leadsto {\left(\frac{a \cdot 2}{b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}}\right)}^{-1} \]

   10. *-commutative 32.4%

       \[\leadsto {\left(\frac{a \cdot 2}{b + \sqrt{\left(-\color{blue}{\left(a \cdot c\right) \cdot 4}\right) + b \cdot b}}\right)}^{-1} \]

   11. distribute-rgt-neg-in 32.4%

       \[\leadsto {\left(\frac{a \cdot 2}{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)} + b \cdot b}}\right)}^{-1} \]

   12. fma-define 32.4%

       \[\leadsto {\left(\frac{a \cdot 2}{b + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}\right)}^{-1} \]

   13. metadata-eval 32.4%

       \[\leadsto {\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}\right)}^{-1} \]

   14. pow2 32.4%

       \[\leadsto {\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, \color{blue}{{b}^{2}}\right)}}\right)}^{-1} \]

6. Applied egg-rr 32.4%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 32.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   2. *-commutative 32.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}} \]

8. Simplified 32.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

9. Taylor expanded in a around 0 2.6%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
10. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024123 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))