quadp (p42, positive)

Percentage Accurate: 52.0% → 85.9%

Time: 16.6s

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: 52.0% 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: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+118)
   (- (/ c b) (/ b a))
   (if (<= b 1.1e-41)
     (/ (- (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 <= -2.5e+118) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.1e-41) {
		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 <= (-2.5d+118)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.1d-41) 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 <= -2.5e+118) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.1e-41) {
		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 <= -2.5e+118:
		tmp = (c / b) - (b / a)
	elif b <= 1.1e-41:
		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 <= -2.5e+118)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.1e-41)
		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 <= -2.5e+118)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.1e-41)
		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, -2.5e+118], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-41], 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 < -2.49999999999999986e118

   1. Initial program 57.3%

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

      1. *-commutative 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in b around -inf 96.7%

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

      1. +-commutative 96.7%

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

      2. mul-1-neg 96.7%

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

      3. unsub-neg 96.7%

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

   7. Simplified 96.7%

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

   if -2.49999999999999986e118 < b < 1.1e-41

   1. Initial program 84.7%

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

   if 1.1e-41 < b

   1. Initial program 12.4%

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

      1. *-commutative 12.4%

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

   3. Simplified 12.4%

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

   5. Taylor expanded in b around inf 90.1%

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

      1. associate-*r/ 90.1%

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

      2. neg-mul-1 90.1%

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

   7. Simplified 90.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-41}:\\ \;\;\;\;\left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-15)
   (- (/ c b) (/ b a))
   (if (<= b 2.9e-41)
     (* (- b (sqrt (* (* c a) -4.0))) (/ -0.5 a))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-15) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.9e-41) {
		tmp = (b - sqrt(((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 <= (-1.6d-15)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.9d-41) then
        tmp = (b - sqrt(((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 <= -1.6e-15) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.9e-41) {
		tmp = (b - Math.sqrt(((c * a) * -4.0))) * (-0.5 / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-15:
		tmp = (c / b) - (b / a)
	elif b <= 2.9e-41:
		tmp = (b - math.sqrt(((c * a) * -4.0))) * (-0.5 / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-15)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.9e-41)
		tmp = Float64(Float64(b - sqrt(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 <= -1.6e-15)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.9e-41)
		tmp = (b - sqrt(((c * a) * -4.0))) * (-0.5 / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-15], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e-41], N[(N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6e-15

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around -inf 93.5%

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

      1. +-commutative 93.5%

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

      2. mul-1-neg 93.5%

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

      3. unsub-neg 93.5%

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

   7. Simplified 93.5%

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

   if -1.6e-15 < b < 2.89999999999999977e-41

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in b around 0 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*r* 74.1%

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

   7. Simplified 74.1%

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

      1. frac-2neg 74.1%

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

      2. div-inv 74.0%

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

      3. distribute-neg-in 74.0%

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

      4. add-sqr-sqrt 44.3%

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

      5. sqrt-unprod 74.1%

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

      6. sqr-neg 74.1%

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

      7. sqrt-prod 29.8%

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

      8. add-sqr-sqrt 72.1%

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

      9. sub-neg 72.1%

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

      10. add-sqr-sqrt 42.3%

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

      11. sqrt-unprod 72.0%

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

      12. sqr-neg 72.0%

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

      13. sqrt-prod 29.7%

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

      14. add-sqr-sqrt 74.0%

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

      15. *-commutative 74.0%

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

      16. *-commutative 74.0%

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

      17. associate-*l* 74.0%

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

      18. distribute-rgt-neg-in 74.0%

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

      19. metadata-eval 74.0%

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

      20. metadata-eval 74.0%

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

      21. div-inv 74.0%

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

   9. Applied egg-rr 74.0%

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

       1. *-commutative 74.0%

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

   11. Simplified 74.0%

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

   if 2.89999999999999977e-41 < b

   1. Initial program 12.4%

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

      1. *-commutative 12.4%

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

   3. Simplified 12.4%

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

   5. Taylor expanded in b around inf 90.1%

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

      1. associate-*r/ 90.1%

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

      2. neg-mul-1 90.1%

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

   7. Simplified 90.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-41}:\\ \;\;\;\;\left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e-12)
   (- (/ c b) (/ b a))
   (if (<= b 1.2e-41)
     (/ (- b (sqrt (* (* c a) -4.0))) (* a -2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-12) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.2e-41) {
		tmp = (b - sqrt(((c * a) * -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 <= (-3.5d-12)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.2d-41) then
        tmp = (b - sqrt(((c * a) * (-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 <= -3.5e-12) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.2e-41) {
		tmp = (b - Math.sqrt(((c * a) * -4.0))) / (a * -2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.5e-12:
		tmp = (c / b) - (b / a)
	elif b <= 1.2e-41:
		tmp = (b - math.sqrt(((c * a) * -4.0))) / (a * -2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e-12)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.2e-41)
		tmp = Float64(Float64(b - sqrt(Float64(Float64(c * a) * -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 <= -3.5e-12)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.2e-41)
		tmp = (b - sqrt(((c * a) * -4.0))) / (a * -2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e-12], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.2e-41], N[(N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e-12

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around -inf 93.5%

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

      1. +-commutative 93.5%

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

      2. mul-1-neg 93.5%

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

      3. unsub-neg 93.5%

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

   7. Simplified 93.5%

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

   if -3.5e-12 < b < 1.20000000000000011e-41

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in b around 0 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*r* 74.1%

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

   7. Simplified 74.1%

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

      1. frac-2neg 74.1%

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

      2. distribute-frac-neg2 74.1%

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

      3. distribute-neg-in 74.1%

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

      4. add-sqr-sqrt 44.4%

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

      5. sqrt-unprod 74.2%

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

      6. sqr-neg 74.2%

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

      7. sqrt-prod 29.8%

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

      8. add-sqr-sqrt 72.2%

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

      9. sub-neg 72.2%

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

      10. add-sqr-sqrt 42.3%

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

      11. sqrt-unprod 72.0%

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

      12. sqr-neg 72.0%

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

      13. sqrt-prod 29.7%

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

      14. add-sqr-sqrt 74.1%

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

      15. *-commutative 74.1%

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

      16. *-commutative 74.1%

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

      17. associate-*l* 74.1%

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

   9. Applied egg-rr 74.1%

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

       1. distribute-neg-frac2 74.1%

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

       2. distribute-rgt-neg-in 74.1%

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

       3. metadata-eval 74.1%

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

   11. Simplified 74.1%

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

   if 1.20000000000000011e-41 < b

   1. Initial program 12.4%

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

      1. *-commutative 12.4%

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

   3. Simplified 12.4%

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

   5. Taylor expanded in b around inf 90.1%

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

      1. associate-*r/ 90.1%

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

      2. neg-mul-1 90.1%

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

   7. Simplified 90.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5e-310) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 76.1%

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

      1. *-commutative 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in b around -inf 65.8%

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

      1. associate-*r/ 65.8%

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

      2. mul-1-neg 65.8%

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

   7. Simplified 65.8%

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

   if -4.999999999999985e-310 < 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. Step-by-step derivation

      1. frac-2neg 30.9%

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

      2. div-inv 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. fma-define 30.9%

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

      2. *-commutative 30.9%

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

      3. fma-define 30.9%

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

      4. *-commutative 30.9%

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

      5. *-commutative 30.9%

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

      6. associate-/r* 30.9%

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

      7. metadata-eval 30.9%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{\color{blue}{-0.5}}{a} \]

   8. Simplified 30.9%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 30.9%

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}{a}} \]

      2. clear-num 30.9%

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

   10. Applied egg-rr 30.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}}} \]
   11. Step-by-step derivation

       1. clear-num 30.9%

          \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}{a}} \]

       2. add-sqr-sqrt 15.5%

          \[\leadsto \frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}} \]

       3. times-frac 15.5%

          \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}}} \]

       4. pow1/2 15.5%

          \[\leadsto \frac{b - \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.5}}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}} \]

       5. metadata-eval 15.5%

          \[\leadsto \frac{b - {\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}} \]

       6. pow-pow 9.6%

          \[\leadsto \frac{b - \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}} \]

       7. pow1/3 10.3%

          \[\leadsto \frac{b - \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}} \]

       8. times-frac 10.3%

          \[\leadsto \color{blue}{\frac{\left(b - \sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right) \cdot -0.5}{\sqrt{a} \cdot \sqrt{a}}} \]

       9. add-sqr-sqrt 23.7%

          \[\leadsto \frac{\left(b - \sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right) \cdot -0.5}{\color{blue}{a}} \]

       10. associate-*r/ 23.7%

           \[\leadsto \color{blue}{\left(b - \sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right) \cdot \frac{-0.5}{a}} \]

       11. *-commutative 23.7%

           \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right)} \]

       12. sub-neg 23.7%

           \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(b + \left(-\sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right)\right)} \]

       13. distribute-lft-in 23.4%

           \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot b + \frac{-0.5}{a} \cdot \left(-\sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right)} \]

   12. Applied egg-rr 31.5%

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

   13. Taylor expanded in a around 0 19.3%

       \[\leadsto \color{blue}{\frac{-0.5 \cdot b + 0.5 \cdot b}{a}} \]
   14. Step-by-step derivation

       1. distribute-rgt-out 19.3%

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

       2. metadata-eval 19.3%

          \[\leadsto \frac{b \cdot \color{blue}{0}}{a} \]

       3. associate-*l/ 11.3%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot 0} \]

       4. mul0-rgt 19.3%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 19.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8e-232) (/ (- b) a) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8e-232) {
		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 <= 1.8d-232) 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 <= 1.8e-232) {
		tmp = -b / a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.8e-232:
		tmp = -b / a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8e-232)
		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 <= 1.8e-232)
		tmp = -b / a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8e-232], N[((-b) / a), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000008e-232

   1. Initial program 75.2%

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

      1. *-commutative 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in b around -inf 62.8%

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

      1. associate-*r/ 62.8%

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

      2. mul-1-neg 62.8%

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

   7. Simplified 62.8%

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

   if 1.80000000000000008e-232 < b

   1. Initial program 29.2%

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

      1. *-commutative 29.2%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in b around inf 72.0%

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

      1. associate-*r/ 72.0%

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

      2. neg-mul-1 72.0%

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

   7. Simplified 72.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 11.4% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 55.3%

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

   1. *-commutative 55.3%

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

3. Simplified 55.3%

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

   1. frac-2neg 55.3%

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

   2. div-inv 55.2%

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

6. Applied egg-rr 55.3%

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

   1. fma-define 55.2%

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

   2. *-commutative 55.2%

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

   3. fma-define 55.3%

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

   4. *-commutative 55.3%

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

   5. *-commutative 55.3%

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

   6. associate-/r* 55.3%

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

   7. metadata-eval 55.3%

      \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{\color{blue}{-0.5}}{a} \]

8. Simplified 55.3%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 55.3%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}{a}} \]

   2. clear-num 55.2%

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

10. Applied egg-rr 55.2%

    \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}}} \]
11. Step-by-step derivation

    1. clear-num 55.3%

       \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}{a}} \]

    2. add-sqr-sqrt 26.1%

       \[\leadsto \frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}} \]

    3. times-frac 26.1%

       \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}}} \]

    4. pow1/2 26.1%

       \[\leadsto \frac{b - \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.5}}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}} \]

    5. metadata-eval 26.1%

       \[\leadsto \frac{b - {\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}} \]

    6. pow-pow 20.8%

       \[\leadsto \frac{b - \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}} \]

    7. pow1/3 21.8%

       \[\leadsto \frac{b - \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}}}{\sqrt{a}} \cdot \frac{-0.5}{\sqrt{a}} \]

    8. times-frac 21.8%

       \[\leadsto \color{blue}{\frac{\left(b - \sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right) \cdot -0.5}{\sqrt{a} \cdot \sqrt{a}}} \]

    9. add-sqr-sqrt 46.2%

       \[\leadsto \frac{\left(b - \sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right) \cdot -0.5}{\color{blue}{a}} \]

    10. associate-*r/ 46.2%

        \[\leadsto \color{blue}{\left(b - \sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right) \cdot \frac{-0.5}{a}} \]

    11. *-commutative 46.2%

        \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right)} \]

    12. sub-neg 46.2%

        \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(b + \left(-\sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right)\right)} \]

    13. distribute-lft-in 46.1%

        \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot b + \frac{-0.5}{a} \cdot \left(-\sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{1.5}}\right)} \]

12. Applied egg-rr 52.0%

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

13. Taylor expanded in a around 0 10.3%

    \[\leadsto \color{blue}{\frac{-0.5 \cdot b + 0.5 \cdot b}{a}} \]
14. Step-by-step derivation

    1. distribute-rgt-out 10.3%

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

    2. metadata-eval 10.3%

       \[\leadsto \frac{b \cdot \color{blue}{0}}{a} \]

    3. associate-*l/ 6.5%

       \[\leadsto \color{blue}{\frac{b}{a} \cdot 0} \]

    4. mul0-rgt 10.3%

       \[\leadsto \color{blue}{0} \]

15. Simplified 10.3%

    \[\leadsto \color{blue}{0} \]

16. Final simplification 10.3%

    \[\leadsto 0 \]
17. 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 2024059 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (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)))