Quadratic roots, full range

Percentage Accurate: 52.9% → 85.5%

Time: 11.2s

Alternatives: 7

Speedup: 11.6×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \frac{c}{b}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;-1 \cdot \left(t\_0 + \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ c b))))
   (if (<= b -5e+153)
     (* -1.0 (+ t_0 (/ b a)))
     (if (<= b 5e-112)
       (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
       t_0))))

C

double code(double a, double b, double c) {
	double t_0 = -1.0 * (c / b);
	double tmp;
	if (b <= -5e+153) {
		tmp = -1.0 * (t_0 + (b / a));
	} else if (b <= 5e-112) {
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) * (c / b)
    if (b <= (-5d+153)) then
        tmp = (-1.0d0) * (t_0 + (b / a))
    else if (b <= 5d-112) then
        tmp = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -1.0 * (c / b);
	double tmp;
	if (b <= -5e+153) {
		tmp = -1.0 * (t_0 + (b / a));
	} else if (b <= 5e-112) {
		tmp = (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -1.0 * (c / b)
	tmp = 0
	if b <= -5e+153:
		tmp = -1.0 * (t_0 + (b / a))
	elif b <= 5e-112:
		tmp = (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-1.0 * Float64(c / b))
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(-1.0 * Float64(t_0 + Float64(b / a)));
	elseif (b <= 5e-112)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -1.0 * (c / b);
	tmp = 0.0;
	if (b <= -5e+153)
		tmp = -1.0 * (t_0 + (b / a));
	elseif (b <= 5e-112)
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5e+153], N[(-1.0 * N[(t$95$0 + N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-112], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000018e153

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in b around -inf 98.5%

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

   6. Taylor expanded in c around 0 99.7%

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

   if -5.00000000000000018e153 < b < 5.00000000000000044e-112

   1. Initial program 82.9%

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

   if 5.00000000000000044e-112 < b

   1. Initial program 17.9%

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

      1. *-commutative 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \frac{c}{b}\\ \mathbf{if}\;b \leq -4.3 \cdot 10^{-44}:\\ \;\;\;\;-1 \cdot \left(t\_0 + \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-116}:\\ \;\;\;\;\left(b + \sqrt{\left(-4 \cdot a\right) \cdot c}\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ c b))))
   (if (<= b -4.3e-44)
     (* -1.0 (+ t_0 (/ b a)))
     (if (<= b 1.35e-116) (* (+ b (sqrt (* (* -4.0 a) c))) (/ 0.5 a)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = -1.0 * (c / b);
	double tmp;
	if (b <= -4.3e-44) {
		tmp = -1.0 * (t_0 + (b / a));
	} else if (b <= 1.35e-116) {
		tmp = (b + sqrt(((-4.0 * a) * c))) * (0.5 / a);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) * (c / b)
    if (b <= (-4.3d-44)) then
        tmp = (-1.0d0) * (t_0 + (b / a))
    else if (b <= 1.35d-116) then
        tmp = (b + sqrt((((-4.0d0) * a) * c))) * (0.5d0 / a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -1.0 * (c / b);
	double tmp;
	if (b <= -4.3e-44) {
		tmp = -1.0 * (t_0 + (b / a));
	} else if (b <= 1.35e-116) {
		tmp = (b + Math.sqrt(((-4.0 * a) * c))) * (0.5 / a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -1.0 * (c / b)
	tmp = 0
	if b <= -4.3e-44:
		tmp = -1.0 * (t_0 + (b / a))
	elif b <= 1.35e-116:
		tmp = (b + math.sqrt(((-4.0 * a) * c))) * (0.5 / a)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-1.0 * Float64(c / b))
	tmp = 0.0
	if (b <= -4.3e-44)
		tmp = Float64(-1.0 * Float64(t_0 + Float64(b / a)));
	elseif (b <= 1.35e-116)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(-4.0 * a) * c))) * Float64(0.5 / a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -1.0 * (c / b);
	tmp = 0.0;
	if (b <= -4.3e-44)
		tmp = -1.0 * (t_0 + (b / a));
	elseif (b <= 1.35e-116)
		tmp = (b + sqrt(((-4.0 * a) * c))) * (0.5 / a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.3e-44], N[(-1.0 * N[(t$95$0 + N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-116], N[(N[(b + N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.30000000000000013e-44

   1. Initial program 67.9%

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

      1. *-commutative 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in b around -inf 89.1%

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

   6. Taylor expanded in c around 0 89.8%

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

   if -4.30000000000000013e-44 < b < 1.35e-116

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in b around 0 70.5%

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

      1. div-inv 70.4%

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

      2. add-sqr-sqrt 45.6%

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

      3. sqrt-unprod 69.2%

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

      4. sqr-neg 69.2%

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

      5. sqrt-prod 25.1%

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

      6. add-sqr-sqrt 68.3%

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

      7. associate-*r* 68.4%

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

      8. metadata-eval 68.4%

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

      9. div-inv 68.4%

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

      10. clear-num 68.4%

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

   7. Applied egg-rr 68.4%

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

   if 1.35e-116 < b

   1. Initial program 17.9%

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

      1. *-commutative 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \frac{c}{b}\\ \mathbf{if}\;b \leq -7.2 \cdot 10^{-42}:\\ \;\;\;\;-1 \cdot \left(t\_0 + \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ c b))))
   (if (<= b -7.2e-42)
     (* -1.0 (+ t_0 (/ b a)))
     (if (<= b 6.2e-112) (/ (- (sqrt (* (* -4.0 a) c)) b) (* a 2.0)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = -1.0 * (c / b);
	double tmp;
	if (b <= -7.2e-42) {
		tmp = -1.0 * (t_0 + (b / a));
	} else if (b <= 6.2e-112) {
		tmp = (sqrt(((-4.0 * a) * c)) - b) / (a * 2.0);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) * (c / b)
    if (b <= (-7.2d-42)) then
        tmp = (-1.0d0) * (t_0 + (b / a))
    else if (b <= 6.2d-112) then
        tmp = (sqrt((((-4.0d0) * a) * c)) - b) / (a * 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -1.0 * (c / b);
	double tmp;
	if (b <= -7.2e-42) {
		tmp = -1.0 * (t_0 + (b / a));
	} else if (b <= 6.2e-112) {
		tmp = (Math.sqrt(((-4.0 * a) * c)) - b) / (a * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -1.0 * (c / b)
	tmp = 0
	if b <= -7.2e-42:
		tmp = -1.0 * (t_0 + (b / a))
	elif b <= 6.2e-112:
		tmp = (math.sqrt(((-4.0 * a) * c)) - b) / (a * 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-1.0 * Float64(c / b))
	tmp = 0.0
	if (b <= -7.2e-42)
		tmp = Float64(-1.0 * Float64(t_0 + Float64(b / a)));
	elseif (b <= 6.2e-112)
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a * 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -1.0 * (c / b);
	tmp = 0.0;
	if (b <= -7.2e-42)
		tmp = -1.0 * (t_0 + (b / a));
	elseif (b <= 6.2e-112)
		tmp = (sqrt(((-4.0 * a) * c)) - b) / (a * 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -7.2e-42], N[(-1.0 * N[(t$95$0 + N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-112], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.2000000000000004e-42

   1. Initial program 67.9%

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

      1. *-commutative 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in b around -inf 89.1%

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

   6. Taylor expanded in c around 0 89.8%

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

   if -7.2000000000000004e-42 < b < 6.1999999999999995e-112

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in b around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. unsub-neg 70.5%

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

      3. associate-*r* 70.5%

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

   7. Applied egg-rr 70.5%

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

   if 6.1999999999999995e-112 < b

   1. Initial program 17.9%

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

      1. *-commutative 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 66.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \frac{c}{b}\\ \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1 \cdot \left(t\_0 + \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ c b))))
   (if (<= b -5e-311) (* -1.0 (+ t_0 (/ b a))) t_0)))

C

double code(double a, double b, double c) {
	double t_0 = -1.0 * (c / b);
	double tmp;
	if (b <= -5e-311) {
		tmp = -1.0 * (t_0 + (b / a));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) * (c / b)
    if (b <= (-5d-311)) then
        tmp = (-1.0d0) * (t_0 + (b / a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -1.0 * (c / b);
	double tmp;
	if (b <= -5e-311) {
		tmp = -1.0 * (t_0 + (b / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -1.0 * (c / b)
	tmp = 0
	if b <= -5e-311:
		tmp = -1.0 * (t_0 + (b / a))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-1.0 * Float64(c / b))
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(-1.0 * Float64(t_0 + Float64(b / a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -1.0 * (c / b);
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = -1.0 * (t_0 + (b / a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5e-311], N[(-1.0 * N[(t$95$0 + N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in b around -inf 64.7%

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

   6. Taylor expanded in c around 0 68.8%

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

   if -5.00000000000023e-311 < b

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in b around inf 73.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (* -1.0 (/ b a)) (* -1.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-311)) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (-1.0d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = -1.0 * (b / a)
	else:
		tmp = -1.0 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = -1.0 * (b / a);
	else
		tmp = -1.0 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in b around -inf 68.1%

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

   if -5.00000000000023e-311 < b

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in b around inf 73.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 34.7% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \frac{b}{a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. *-commutative 49.4%

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

3. Simplified 49.4%

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

5. Taylor expanded in b around -inf 34.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
6. Add Preprocessing

Alternative 7: 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 49.4%

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

   1. *-commutative 49.4%

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

3. Simplified 49.4%

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

   1. *-un-lft-identity 49.4%

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

   2. *-un-lft-identity 49.4%

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

   3. prod-diff 49.4%

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

   4. *-commutative 49.4%

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

   5. *-un-lft-identity 49.4%

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

   6. fma-define 49.4%

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

   7. *-un-lft-identity 49.4%

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

   8. +-commutative 49.4%

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

   9. add-sqr-sqrt 35.5%

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

   10. sqrt-unprod 46.4%

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

   11. sqr-neg 46.4%

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

   12. sqrt-prod 11.2%

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

   13. add-sqr-sqrt 31.5%

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

   14. pow2 31.5%

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

   15. add-sqr-sqrt 21.7%

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

   16. sqrt-unprod 31.5%

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

   17. sqr-neg 31.5%

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

   18. sqrt-prod 11.2%

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

   19. add-sqr-sqrt 31.2%

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

   20. *-commutative 31.2%

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

   21. *-un-lft-identity 31.2%

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

6. Applied egg-rr 31.2%

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

7. Taylor expanded in b around -inf 2.4%

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

Reproduce

herbie shell --seed 2024050 -o generate:simplify
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))