Quadratic roots, full range

Percentage Accurate: 52.4% → 85.3%

Time: 11.7s

Alternatives: 7

Speedup: 19.1×

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: 52.4% 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: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+113}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+113)
   (- (/ c b) (/ b a))
   (if (<= b 9e-12)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+113) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9e-12) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - 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 <= (-4d+113)) then
        tmp = (c / b) - (b / a)
    else if (b <= 9d-12) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - 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 <= -4e+113) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9e-12) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e+113:
		tmp = (c / b) - (b / a)
	elif b <= 9e-12:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+113)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 9e-12)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e+113)
		tmp = (c / b) - (b / a);
	elseif (b <= 9e-12)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+113], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9e-12], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $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 < -4e113

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in b around -inf 90.7%

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

      1. +-commutative 90.7%

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

      2. mul-1-neg 90.7%

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

      3. unsub-neg 90.7%

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

   6. Simplified 90.7%

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

   if -4e113 < b < 8.99999999999999962e-12

   1. Initial program 86.9%

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

   if 8.99999999999999962e-12 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in b around inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. distribute-neg-frac 90.9%

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

   6. Simplified 90.9%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+113}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ t_1 := \frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b) (/ b a)))
        (t_1 (* (/ 0.5 a) (+ b (sqrt (* c (* a -4.0)))))))
   (if (<= b -1.4e-17)
     t_0
     (if (<= b -5e-66)
       t_1
       (if (<= b -4.5e-133) t_0 (if (<= b 1.55e-11) t_1 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	double tmp;
	if (b <= -1.4e-17) {
		tmp = t_0;
	} else if (b <= -5e-66) {
		tmp = t_1;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 1.55e-11) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c / b) - (b / a)
    t_1 = (0.5d0 / a) * (b + sqrt((c * (a * (-4.0d0)))))
    if (b <= (-1.4d-17)) then
        tmp = t_0
    else if (b <= (-5d-66)) then
        tmp = t_1
    else if (b <= (-4.5d-133)) then
        tmp = t_0
    else if (b <= 1.55d-11) then
        tmp = t_1
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = (0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	double tmp;
	if (b <= -1.4e-17) {
		tmp = t_0;
	} else if (b <= -5e-66) {
		tmp = t_1;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 1.55e-11) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c / b) - (b / a)
	t_1 = (0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	tmp = 0
	if b <= -1.4e-17:
		tmp = t_0
	elif b <= -5e-66:
		tmp = t_1
	elif b <= -4.5e-133:
		tmp = t_0
	elif b <= 1.55e-11:
		tmp = t_1
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) - Float64(b / a))
	t_1 = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))))
	tmp = 0.0
	if (b <= -1.4e-17)
		tmp = t_0;
	elseif (b <= -5e-66)
		tmp = t_1;
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 1.55e-11)
		tmp = t_1;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c / b) - (b / a);
	t_1 = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	tmp = 0.0;
	if (b <= -1.4e-17)
		tmp = t_0;
	elseif (b <= -5e-66)
		tmp = t_1;
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 1.55e-11)
		tmp = t_1;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.4e-17], t$95$0, If[LessEqual[b, -5e-66], t$95$1, If[LessEqual[b, -4.5e-133], t$95$0, If[LessEqual[b, 1.55e-11], t$95$1, N[((-c) / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.3999999999999999e-17 or -4.99999999999999962e-66 < b < -4.50000000000000009e-133

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in b around -inf 86.1%

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

      1. +-commutative 86.1%

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

      2. mul-1-neg 86.1%

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

      3. unsub-neg 86.1%

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

   6. Simplified 86.1%

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

   if -1.3999999999999999e-17 < b < -4.99999999999999962e-66 or -4.50000000000000009e-133 < b < 1.55000000000000014e-11

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

   3. Simplified 81.6%

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

      1. add-sqr-sqrt 34.3%

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

      2. pow2 34.3%

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

   5. Applied egg-rr 33.0%

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

   6. Taylor expanded in b around 0 32.9%

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

      1. unpow2 32.9%

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

      2. add-sqr-sqrt 77.7%

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

      3. *-commutative 77.7%

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

      4. *-commutative 77.7%

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

      5. *-commutative 77.7%

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

      6. associate-*r* 77.7%

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

   8. Applied egg-rr 77.7%

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

   if 1.55000000000000014e-11 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in b around inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. distribute-neg-frac 90.9%

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

   6. Simplified 90.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-66}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ t_1 := \frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b) (/ b a)))
        (t_1 (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))))
   (if (<= b -6.6e-15)
     t_0
     (if (<= b -2.5e-66)
       t_1
       (if (<= b -4.5e-133) t_0 (if (<= b 3e-12) t_1 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -6.6e-15) {
		tmp = t_0;
	} else if (b <= -2.5e-66) {
		tmp = t_1;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 3e-12) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c / b) - (b / a)
    t_1 = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    if (b <= (-6.6d-15)) then
        tmp = t_0
    else if (b <= (-2.5d-66)) then
        tmp = t_1
    else if (b <= (-4.5d-133)) then
        tmp = t_0
    else if (b <= 3d-12) then
        tmp = t_1
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -6.6e-15) {
		tmp = t_0;
	} else if (b <= -2.5e-66) {
		tmp = t_1;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 3e-12) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c / b) - (b / a)
	t_1 = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	tmp = 0
	if b <= -6.6e-15:
		tmp = t_0
	elif b <= -2.5e-66:
		tmp = t_1
	elif b <= -4.5e-133:
		tmp = t_0
	elif b <= 3e-12:
		tmp = t_1
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) - Float64(b / a))
	t_1 = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -6.6e-15)
		tmp = t_0;
	elseif (b <= -2.5e-66)
		tmp = t_1;
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 3e-12)
		tmp = t_1;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c / b) - (b / a);
	t_1 = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	tmp = 0.0;
	if (b <= -6.6e-15)
		tmp = t_0;
	elseif (b <= -2.5e-66)
		tmp = t_1;
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 3e-12)
		tmp = t_1;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.6e-15], t$95$0, If[LessEqual[b, -2.5e-66], t$95$1, If[LessEqual[b, -4.5e-133], t$95$0, If[LessEqual[b, 3e-12], t$95$1, N[((-c) / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.6e-15 or -2.49999999999999981e-66 < b < -4.50000000000000009e-133

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in b around -inf 86.1%

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

      1. +-commutative 86.1%

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

      2. mul-1-neg 86.1%

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

      3. unsub-neg 86.1%

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

   6. Simplified 86.1%

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

   if -6.6e-15 < b < -2.49999999999999981e-66 or -4.50000000000000009e-133 < b < 3.0000000000000001e-12

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in b around 0 78.8%

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

      1. *-commutative 78.8%

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

      2. *-commutative 78.8%

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

      3. associate-*r* 78.8%

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

   6. Simplified 78.8%

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

   if 3.0000000000000001e-12 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in b around inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. distribute-neg-frac 90.9%

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

   6. Simplified 90.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 67.5% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in b around -inf 69.3%

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

      1. +-commutative 69.3%

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

      2. mul-1-neg 69.3%

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

      3. unsub-neg 69.3%

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

   6. Simplified 69.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 39.8%

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

      1. *-commutative 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in b around inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. distribute-neg-frac 60.9%

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

   6. Simplified 60.9%

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

4. Final simplification 65.6%

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

Alternative 5: 43.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.38 \cdot 10^{-11}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.38e-11) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.38e-11) {
		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 <= 1.38d-11) 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 <= 1.38e-11) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.38e-11:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.38e-11)
		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 <= 1.38e-11)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.38e-11], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.38e-11

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in b around -inf 52.8%

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

      1. associate-*r/ 52.8%

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

      2. mul-1-neg 52.8%

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

   6. Simplified 52.8%

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

   if 1.38e-11 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in b around inf 68.6%

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

      1. associate-*r/ 68.6%

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

      2. associate-*r* 68.6%

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

      3. *-commutative 68.6%

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

   6. Simplified 68.6%

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

      1. div-inv 68.5%

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

      2. *-commutative 68.5%

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

      3. times-frac 69.2%

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

      4. associate-*l* 69.2%

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

   8. Applied egg-rr 69.2%

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

      1. associate-/l* 69.3%

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

      2. *-commutative 69.3%

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

      3. associate-/l/ 68.5%

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

   10. Simplified 68.5%

       \[\leadsto \color{blue}{\frac{a}{\frac{2}{c \cdot -2}} \cdot \frac{1}{a \cdot b}} \]
   11. Step-by-step derivation

       1. frac-2neg 68.5%

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

       2. frac-2neg 68.5%

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

       3. metadata-eval 68.5%

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

       4. frac-times 71.6%

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

       5. frac-2neg 71.6%

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

       6. metadata-eval 71.6%

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

       7. distribute-neg-frac 71.6%

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

       8. metadata-eval 71.6%

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

       9. distribute-rgt-neg-in 71.6%

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

       10. metadata-eval 71.6%

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

       11. distribute-rgt-neg-in 71.6%

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

       12. add-sqr-sqrt 0.0%

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

       13. sqrt-unprod 26.2%

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

       14. sqr-neg 26.2%

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

       15. sqrt-unprod 26.0%

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

       16. add-sqr-sqrt 26.0%

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

       17. *-commutative 26.0%

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

   12. Applied egg-rr 26.0%

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

       1. *-commutative 26.0%

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

       2. mul-1-neg 26.0%

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

       3. *-commutative 26.0%

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

       4. associate-*l* 25.8%

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

   14. Simplified 25.8%

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

   15. Taylor expanded in a around 0 25.6%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.38 \cdot 10^{-11}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6: 67.4% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in b around -inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. mul-1-neg 69.2%

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

   6. Simplified 69.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 39.8%

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

      1. *-commutative 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in b around inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. distribute-neg-frac 60.9%

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

   6. Simplified 60.9%

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

4. Final simplification 65.5%

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

Alternative 7: 10.9% 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 60.8%

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

   1. *-commutative 60.8%

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

3. Simplified 60.8%

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

4. Taylor expanded in b around inf 21.3%

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

   1. associate-*r/ 21.3%

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

   2. associate-*r* 21.3%

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

   3. *-commutative 21.3%

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

6. Simplified 21.3%

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

   1. div-inv 21.2%

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

   2. *-commutative 21.2%

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

   3. times-frac 21.3%

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

   4. associate-*l* 21.3%

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

8. Applied egg-rr 21.3%

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

   1. associate-/l* 21.3%

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

   2. *-commutative 21.3%

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

   3. associate-/l/ 21.1%

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

10. Simplified 21.1%

    \[\leadsto \color{blue}{\frac{a}{\frac{2}{c \cdot -2}} \cdot \frac{1}{a \cdot b}} \]
11. Step-by-step derivation

    1. frac-2neg 21.1%

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

    2. frac-2neg 21.1%

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

    3. metadata-eval 21.1%

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

    4. frac-times 23.5%

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

    5. frac-2neg 23.5%

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

    6. metadata-eval 23.5%

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

    7. distribute-neg-frac 23.5%

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

    8. metadata-eval 23.5%

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

    9. distribute-rgt-neg-in 23.5%

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

    10. metadata-eval 23.5%

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

    11. distribute-rgt-neg-in 23.5%

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

    12. add-sqr-sqrt 1.7%

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

    13. sqrt-unprod 8.6%

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

    14. sqr-neg 8.6%

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

    15. sqrt-unprod 7.0%

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

    16. add-sqr-sqrt 9.5%

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

    17. *-commutative 9.5%

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

12. Applied egg-rr 9.5%

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

    1. *-commutative 9.5%

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

    2. mul-1-neg 9.5%

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

    3. *-commutative 9.5%

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

    4. associate-*l* 10.8%

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

14. Simplified 10.8%

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

15. Taylor expanded in a around 0 8.8%

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

16. Final simplification 8.8%

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

Reproduce

herbie shell --seed 2023305 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))