quadp (p42, positive)

Percentage Accurate: 51.4% → 85.6%

Time: 20.4s

Alternatives: 6

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: 51.4% 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: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+153)
   (- (/ b a))
   (if (<= b 3.5e-117)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+153) {
		tmp = -(b / a);
	} else if (b <= 3.5e-117) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - 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 <= (-2.3d+153)) then
        tmp = -(b / a)
    else if (b <= 3.5d-117) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - 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 <= -2.3e+153) {
		tmp = -(b / a);
	} else if (b <= 3.5e-117) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e+153:
		tmp = -(b / a)
	elif b <= 3.5e-117:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e+153)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 3.5e-117)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - 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 <= -2.3e+153)
		tmp = -(b / a);
	elseif (b <= 3.5e-117)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e+153], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 3.5e-117], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $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 < -2.3000000000000001e153

   1. Initial program 38.6%

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

      1. *-commutative 38.6%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in b around -inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. mul-1-neg 98.1%

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

   7. Simplified 98.1%

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

   if -2.3000000000000001e153 < b < 3.4999999999999998e-117

   1. Initial program 84.8%

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

   if 3.4999999999999998e-117 < b

   1. Initial program 14.7%

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

      1. *-commutative 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. neg-mul-1 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-110}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-110)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 3.5e-117)
     (* (/ (- b (sqrt (* a (* c -4.0)))) a) -0.5)
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-110) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 3.5e-117) {
		tmp = ((b - sqrt((a * (c * -4.0)))) / a) * -0.5;
	} 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 <= (-9.5d-110)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 3.5d-117) then
        tmp = ((b - sqrt((a * (c * (-4.0d0))))) / a) * (-0.5d0)
    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 <= -9.5e-110) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 3.5e-117) {
		tmp = ((b - Math.sqrt((a * (c * -4.0)))) / a) * -0.5;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-110:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 3.5e-117:
		tmp = ((b - math.sqrt((a * (c * -4.0)))) / a) * -0.5
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-110)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 3.5e-117)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(a * Float64(c * -4.0)))) / a) * -0.5);
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e-110)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 3.5e-117)
		tmp = ((b - sqrt((a * (c * -4.0)))) / a) * -0.5;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-110], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.5e-117], N[(N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.50000000000000004e-110

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in b around -inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. *-commutative 87.1%

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

      3. distribute-rgt-neg-in 87.1%

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

      4. +-commutative 87.1%

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

      5. mul-1-neg 87.1%

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

      6. unsub-neg 87.1%

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

   7. Simplified 87.1%

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

   if -9.50000000000000004e-110 < b < 3.4999999999999998e-117

   1. Initial program 75.9%

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

      1. *-commutative 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in b around 0 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*r* 74.9%

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

   7. Simplified 74.9%

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

      1. frac-2neg 74.9%

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

      2. div-inv 74.7%

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

      3. distribute-neg-in 74.7%

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

      4. add-sqr-sqrt 37.2%

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

      5. sqrt-unprod 74.5%

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

      6. sqr-neg 74.5%

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

      7. sqrt-unprod 37.7%

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

      8. add-sqr-sqrt 74.5%

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

      9. sub-neg 74.5%

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

      10. add-sqr-sqrt 36.8%

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

      11. sqrt-unprod 74.3%

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

      12. sqr-neg 74.3%

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

      13. sqrt-unprod 37.5%

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

      14. add-sqr-sqrt 74.7%

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

      15. distribute-rgt-neg-in 74.7%

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

      16. metadata-eval 74.7%

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

   9. Applied egg-rr 74.7%

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

       1. associate-*r/ 74.9%

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

       2. times-frac 74.9%

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

       3. metadata-eval 74.9%

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

   11. Simplified 74.9%

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

   if 3.4999999999999998e-117 < b

   1. Initial program 14.7%

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

      1. *-commutative 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. neg-mul-1 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-110}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+67}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2.8e+67) (- (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e+67) {
		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 <= 2.8d+67) 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 <= 2.8e+67) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.8e+67:
		tmp = -(b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.8e+67)
		tmp = Float64(-Float64(b / a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.8e+67)
		tmp = -(b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.8e+67], (-N[(b / a), $MachinePrecision]), N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999998e67

   1. Initial program 60.4%

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

      1. *-commutative 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in b around -inf 46.9%

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

      1. associate-*r/ 46.9%

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

      2. mul-1-neg 46.9%

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

   7. Simplified 46.9%

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

   if 2.7999999999999998e67 < b

   1. Initial program 8.3%

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

      1. *-commutative 8.3%

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

   3. Simplified 8.3%

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

   5. Taylor expanded in b around inf 80.0%

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

      1. associate-/l* 90.2%

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

   7. Simplified 90.2%

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

      1. *-un-lft-identity 90.2%

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

      2. times-frac 90.0%

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

      3. associate-*r* 90.0%

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

      4. metadata-eval 90.0%

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

      5. distribute-lft-neg-in 90.0%

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

      6. *-commutative 90.0%

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

      7. frac-2neg 90.0%

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

      8. distribute-frac-neg 90.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 37.1%

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

      11. sqr-neg 37.1%

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

      12. sqrt-unprod 36.5%

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

      13. add-sqr-sqrt 36.5%

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

      14. distribute-rgt-neg-in 36.5%

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

      15. *-commutative 36.5%

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

      16. distribute-lft-neg-in 36.5%

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

      17. metadata-eval 36.5%

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

      18. associate-*r* 36.5%

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

      19. distribute-lft-neg-in 36.5%

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

      20. metadata-eval 36.5%

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

   9. Applied egg-rr 36.5%

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

       1. associate-*r* 36.5%

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

       2. *-commutative 36.5%

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

       3. clear-num 36.5%

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

       4. un-div-inv 36.5%

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

   11. Applied egg-rr 36.5%

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

   12. Taylor expanded in a around 0 36.4%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+67}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-285}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.4e-285) (- (/ b a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-285) {
		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 <= 4.4d-285) 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 <= 4.4e-285) {
		tmp = -(b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.4e-285:
		tmp = -(b / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4e-285)
		tmp = Float64(-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 <= 4.4e-285)
		tmp = -(b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4e-285], (-N[(b / a), $MachinePrecision]), N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.3999999999999998e-285

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in b around -inf 70.7%

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

      1. associate-*r/ 70.7%

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

      2. mul-1-neg 70.7%

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

   7. Simplified 70.7%

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

   if 4.3999999999999998e-285 < b

   1. Initial program 24.7%

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

      1. *-commutative 24.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in b around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. neg-mul-1 72.3%

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

   7. Simplified 72.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-285}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 46.6%

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

   1. *-commutative 46.6%

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

   2. +-commutative 46.6%

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

   3. unsub-neg 46.6%

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

   4. fma-neg 46.6%

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

   5. *-commutative 46.6%

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

   6. associate-*r* 46.6%

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

   7. distribute-lft-neg-in 46.6%

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

   8. *-commutative 46.6%

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

   9. distribute-rgt-neg-in 46.6%

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

   10. associate-*r* 46.2%

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

   11. metadata-eval 46.2%

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

3. Simplified 46.2%

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

   1. *-un-lft-identity 46.2%

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

   2. *-un-lft-identity 46.2%

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

   3. prod-diff 46.2%

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

6. Applied egg-rr 22.3%

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

   1. associate-+l+ 22.3%

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

   2. fma-undefine 22.3%

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

   3. *-rgt-identity 22.3%

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

8. Simplified 22.3%

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

9. Taylor expanded in b around -inf 2.4%

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

10. Final simplification 2.4%

    \[\leadsto \frac{b}{a} \]
11. Add Preprocessing

Alternative 6: 10.4% 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 46.6%

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

   1. *-commutative 46.6%

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

3. Simplified 46.6%

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

5. Taylor expanded in b around inf 30.1%

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

   1. associate-/l* 34.8%

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

7. Simplified 34.8%

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

   1. *-un-lft-identity 34.8%

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

   2. times-frac 34.7%

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

   3. associate-*r* 34.7%

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

   4. metadata-eval 34.7%

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

   5. distribute-lft-neg-in 34.7%

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

   6. *-commutative 34.7%

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

   7. frac-2neg 34.7%

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

   8. distribute-frac-neg 34.7%

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

   9. add-sqr-sqrt 1.1%

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

   10. sqrt-unprod 11.4%

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

   11. sqr-neg 11.4%

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

   12. sqrt-unprod 10.2%

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

   13. add-sqr-sqrt 11.9%

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

   14. distribute-rgt-neg-in 11.9%

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

   15. *-commutative 11.9%

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

   16. distribute-lft-neg-in 11.9%

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

   17. metadata-eval 11.9%

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

   18. associate-*r* 11.9%

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

   19. distribute-lft-neg-in 11.9%

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

   20. metadata-eval 11.9%

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

9. Applied egg-rr 11.9%

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

    1. associate-*r* 11.9%

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

    2. *-commutative 11.9%

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

    3. clear-num 11.9%

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

    4. un-div-inv 11.9%

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

11. Applied egg-rr 11.9%

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

12. Taylor expanded in a around 0 11.8%

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

13. Final simplification 11.8%

    \[\leadsto \frac{c}{b} \]
14. Add Preprocessing

Developer target: 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 2024055 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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