quadm (p42, negative)

Percentage Accurate: 51.9% → 84.7%

Time: 13.0s

Alternatives: 8

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -1.65 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}}{-2}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b)))
   (if (<= b -1.65e-17)
     t_0
     (if (<= b -1.6e-65)
       (/ (/ (+ b (sqrt (* c (* a -4.0)))) a) -2.0)
       (if (<= b -4.5e-133)
         t_0
         (if (<= b 4.4e+111)
           (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
           (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double tmp;
	if (b <= -1.65e-17) {
		tmp = t_0;
	} else if (b <= -1.6e-65) {
		tmp = ((b + sqrt((c * (a * -4.0)))) / a) / -2.0;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 4.4e+111) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 = -c / b
    if (b <= (-1.65d-17)) then
        tmp = t_0
    else if (b <= (-1.6d-65)) then
        tmp = ((b + sqrt((c * (a * (-4.0d0))))) / a) / (-2.0d0)
    else if (b <= (-4.5d-133)) then
        tmp = t_0
    else if (b <= 4.4d+111) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -c / b;
	double tmp;
	if (b <= -1.65e-17) {
		tmp = t_0;
	} else if (b <= -1.6e-65) {
		tmp = ((b + Math.sqrt((c * (a * -4.0)))) / a) / -2.0;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 4.4e+111) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -c / b
	tmp = 0
	if b <= -1.65e-17:
		tmp = t_0
	elif b <= -1.6e-65:
		tmp = ((b + math.sqrt((c * (a * -4.0)))) / a) / -2.0
	elif b <= -4.5e-133:
		tmp = t_0
	elif b <= 4.4e+111:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	tmp = 0.0
	if (b <= -1.65e-17)
		tmp = t_0;
	elseif (b <= -1.6e-65)
		tmp = Float64(Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / a) / -2.0);
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 4.4e+111)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -c / b;
	tmp = 0.0;
	if (b <= -1.65e-17)
		tmp = t_0;
	elseif (b <= -1.6e-65)
		tmp = ((b + sqrt((c * (a * -4.0)))) / a) / -2.0;
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 4.4e+111)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -1.65e-17], t$95$0, If[LessEqual[b, -1.6e-65], N[(N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -2.0), $MachinePrecision], If[LessEqual[b, -4.5e-133], t$95$0, If[LessEqual[b, 4.4e+111], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.65e-17 or -1.6e-65 < b < -4.50000000000000009e-133

   1. Initial program 14.6%

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

   2. Taylor expanded in b around -inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. neg-mul-1 86.0%

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

   4. Simplified 86.0%

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

   if -1.65e-17 < b < -1.6e-65

   1. Initial program 99.7%

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

      1. add-cube-cbrt 98.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}} \]

      2. pow3 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in b around 0 98.9%

      \[\leadsto {\left(\sqrt[3]{\frac{\frac{b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{-2}}{a}}\right)}^{3} \]
   5. Step-by-step derivation

      1. *-commutative 98.9%

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

   6. Simplified 98.9%

      \[\leadsto {\left(\sqrt[3]{\frac{\frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}{-2}}{a}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-/r* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l* 99.7%

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

   8. Applied egg-rr 99.7%

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

   if -4.50000000000000009e-133 < b < 4.39999999999999997e111

   1. Initial program 88.8%

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

   if 4.39999999999999997e111 < b

   1. Initial program 40.7%

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

   2. Taylor expanded in b around inf 97.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-17}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}}{-2}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b)) (t_1 (* (+ b (sqrt (* c (* a -4.0)))) (/ -0.5 a))))
   (if (<= b -2.4e-17)
     t_0
     (if (<= b -6.5e-66)
       t_1
       (if (<= b -4.5e-133)
         t_0
         (if (<= b 4.3e-12) t_1 (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = (b + sqrt((c * (a * -4.0)))) * (-0.5 / a);
	double tmp;
	if (b <= -2.4e-17) {
		tmp = t_0;
	} else if (b <= -6.5e-66) {
		tmp = t_1;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 4.3e-12) {
		tmp = t_1;
	} else {
		tmp = (c / b) - (b / a);
	}
	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
    t_1 = (b + sqrt((c * (a * (-4.0d0))))) * ((-0.5d0) / a)
    if (b <= (-2.4d-17)) then
        tmp = t_0
    else if (b <= (-6.5d-66)) then
        tmp = t_1
    else if (b <= (-4.5d-133)) then
        tmp = t_0
    else if (b <= 4.3d-12) then
        tmp = t_1
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = (b + Math.sqrt((c * (a * -4.0)))) * (-0.5 / a);
	double tmp;
	if (b <= -2.4e-17) {
		tmp = t_0;
	} else if (b <= -6.5e-66) {
		tmp = t_1;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 4.3e-12) {
		tmp = t_1;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -c / b
	t_1 = (b + math.sqrt((c * (a * -4.0)))) * (-0.5 / a)
	tmp = 0
	if b <= -2.4e-17:
		tmp = t_0
	elif b <= -6.5e-66:
		tmp = t_1
	elif b <= -4.5e-133:
		tmp = t_0
	elif b <= 4.3e-12:
		tmp = t_1
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	t_1 = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) * Float64(-0.5 / a))
	tmp = 0.0
	if (b <= -2.4e-17)
		tmp = t_0;
	elseif (b <= -6.5e-66)
		tmp = t_1;
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 4.3e-12)
		tmp = t_1;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -c / b;
	t_1 = (b + sqrt((c * (a * -4.0)))) * (-0.5 / a);
	tmp = 0.0;
	if (b <= -2.4e-17)
		tmp = t_0;
	elseif (b <= -6.5e-66)
		tmp = t_1;
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 4.3e-12)
		tmp = t_1;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.4e-17], t$95$0, If[LessEqual[b, -6.5e-66], t$95$1, If[LessEqual[b, -4.5e-133], t$95$0, If[LessEqual[b, 4.3e-12], t$95$1, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.39999999999999986e-17 or -6.50000000000000024e-66 < b < -4.50000000000000009e-133

   1. Initial program 14.6%

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

   2. Taylor expanded in b around -inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. neg-mul-1 86.0%

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

   4. Simplified 86.0%

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

   if -2.39999999999999986e-17 < b < -6.50000000000000024e-66 or -4.50000000000000009e-133 < b < 4.29999999999999985e-12

   1. Initial program 87.4%

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

      1. add-cube-cbrt 85.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}} \]

      2. pow3 85.9%

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

   3. Applied egg-rr 85.9%

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

   4. Taylor expanded in b around 0 78.2%

      \[\leadsto {\left(\sqrt[3]{\frac{\frac{b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{-2}}{a}}\right)}^{3} \]
   5. Step-by-step derivation

      1. *-commutative 78.2%

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

   6. Simplified 78.2%

      \[\leadsto {\left(\sqrt[3]{\frac{\frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}{-2}}{a}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 79.4%

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

      2. expm1-log1p-u 60.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{-2}}{a}\right)\right)} \]

      3. expm1-udef 26.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{-2}}{a}\right)} - 1} \]

      4. associate-/l/ 26.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{a \cdot -2}}\right)} - 1 \]

      5. *-commutative 26.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}{a \cdot -2}\right)} - 1 \]

      6. associate-*l* 26.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{a \cdot -2}\right)} - 1 \]

   8. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 60.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\right)\right)} \]

      2. expm1-log1p 79.4%

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

      3. associate-/l/ 79.4%

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

      4. metadata-eval 79.4%

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

      5. associate-/l* 79.4%

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

      6. /-rgt-identity 79.4%

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

      7. associate-*r/ 79.3%

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

   10. Simplified 79.3%

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

   if 4.29999999999999985e-12 < b

   1. Initial program 60.8%

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

   2. Taylor expanded in b around inf 91.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-66}:\\ \;\;\;\;\left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-12}:\\ \;\;\;\;\left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := \frac{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}}{-2}\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.85 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b)) (t_1 (/ (/ (+ b (sqrt (* c (* a -4.0)))) a) -2.0)))
   (if (<= b -1.05e-16)
     t_0
     (if (<= b -1.85e-66)
       t_1
       (if (<= b -4.5e-133)
         t_0
         (if (<= b 5.3e-12) t_1 (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = ((b + sqrt((c * (a * -4.0)))) / a) / -2.0;
	double tmp;
	if (b <= -1.05e-16) {
		tmp = t_0;
	} else if (b <= -1.85e-66) {
		tmp = t_1;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 5.3e-12) {
		tmp = t_1;
	} else {
		tmp = (c / b) - (b / a);
	}
	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
    t_1 = ((b + sqrt((c * (a * (-4.0d0))))) / a) / (-2.0d0)
    if (b <= (-1.05d-16)) then
        tmp = t_0
    else if (b <= (-1.85d-66)) then
        tmp = t_1
    else if (b <= (-4.5d-133)) then
        tmp = t_0
    else if (b <= 5.3d-12) then
        tmp = t_1
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = ((b + Math.sqrt((c * (a * -4.0)))) / a) / -2.0;
	double tmp;
	if (b <= -1.05e-16) {
		tmp = t_0;
	} else if (b <= -1.85e-66) {
		tmp = t_1;
	} else if (b <= -4.5e-133) {
		tmp = t_0;
	} else if (b <= 5.3e-12) {
		tmp = t_1;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -c / b
	t_1 = ((b + math.sqrt((c * (a * -4.0)))) / a) / -2.0
	tmp = 0
	if b <= -1.05e-16:
		tmp = t_0
	elif b <= -1.85e-66:
		tmp = t_1
	elif b <= -4.5e-133:
		tmp = t_0
	elif b <= 5.3e-12:
		tmp = t_1
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	t_1 = Float64(Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / a) / -2.0)
	tmp = 0.0
	if (b <= -1.05e-16)
		tmp = t_0;
	elseif (b <= -1.85e-66)
		tmp = t_1;
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 5.3e-12)
		tmp = t_1;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -c / b;
	t_1 = ((b + sqrt((c * (a * -4.0)))) / a) / -2.0;
	tmp = 0.0;
	if (b <= -1.05e-16)
		tmp = t_0;
	elseif (b <= -1.85e-66)
		tmp = t_1;
	elseif (b <= -4.5e-133)
		tmp = t_0;
	elseif (b <= 5.3e-12)
		tmp = t_1;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -2.0), $MachinePrecision]}, If[LessEqual[b, -1.05e-16], t$95$0, If[LessEqual[b, -1.85e-66], t$95$1, If[LessEqual[b, -4.5e-133], t$95$0, If[LessEqual[b, 5.3e-12], t$95$1, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0500000000000001e-16 or -1.8500000000000001e-66 < b < -4.50000000000000009e-133

   1. Initial program 14.6%

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

   2. Taylor expanded in b around -inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. neg-mul-1 86.0%

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

   4. Simplified 86.0%

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

   if -1.0500000000000001e-16 < b < -1.8500000000000001e-66 or -4.50000000000000009e-133 < b < 5.29999999999999963e-12

   1. Initial program 87.4%

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

      1. add-cube-cbrt 85.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}} \]

      2. pow3 85.9%

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

   3. Applied egg-rr 85.9%

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

   4. Taylor expanded in b around 0 78.2%

      \[\leadsto {\left(\sqrt[3]{\frac{\frac{b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{-2}}{a}}\right)}^{3} \]
   5. Step-by-step derivation

      1. *-commutative 78.2%

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

   6. Simplified 78.2%

      \[\leadsto {\left(\sqrt[3]{\frac{\frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}{-2}}{a}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 79.4%

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

      2. associate-/l/ 79.4%

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

      3. associate-/r* 79.4%

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

      4. *-commutative 79.4%

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

      5. associate-*l* 79.4%

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

   8. Applied egg-rr 79.4%

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

   if 5.29999999999999963e-12 < b

   1. Initial program 60.8%

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

   2. Taylor expanded in b around inf 91.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.85 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}}{-2}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 67.6% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 30.3%

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

   2. Taylor expanded in b around -inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. neg-mul-1 69.2%

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

   4. Simplified 69.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 72.6%

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

   2. Taylor expanded in b around inf 61.2%

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

4. Final simplification 65.7%

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

Alternative 5: 43.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -48:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -48.0) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -48.0) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	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 <= (-48.0d0)) then
        tmp = c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -48.0) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -48.0:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -48.0)
		tmp = Float64(c / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -48.0)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -48.0], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -48

   1. Initial program 13.2%

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

      1. div-inv 13.2%

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

      2. *-commutative 13.2%

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

      3. associate-/r* 13.2%

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

      4. metadata-eval 13.2%

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

      5. add-sqr-sqrt 10.9%

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

      6. sqrt-unprod 12.5%

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

      7. sqr-neg 12.5%

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

      8. sqrt-prod 0.0%

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

      9. add-sqr-sqrt 6.8%

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

      10. fma-neg 6.8%

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

      11. distribute-lft-neg-in 6.8%

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

      12. *-commutative 6.8%

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

      13. associate-*r* 6.8%

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

      14. metadata-eval 6.8%

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

   3. Applied egg-rr 6.8%

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

   4. Taylor expanded in a around 0 17.8%

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

   if -48 < b

   1. Initial program 70.4%

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

   2. Taylor expanded in b around inf 43.9%

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

      1. associate-*r/ 43.9%

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

      2. mul-1-neg 43.9%

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

   4. Simplified 43.9%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -48:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6: 67.3% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-268}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-268) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-268) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	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 <= (-2.05d-268)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-268) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-268:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-268)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.05e-268)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-268], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.0499999999999999e-268

   1. Initial program 28.3%

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

   2. Taylor expanded in b around -inf 71.1%

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

      1. associate-*r/ 71.1%

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

      2. neg-mul-1 71.1%

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

   4. Simplified 71.1%

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

   if -2.0499999999999999e-268 < b

   1. Initial program 73.6%

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

   2. Taylor expanded in b around inf 59.0%

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

      1. associate-*r/ 59.0%

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

      2. mul-1-neg 59.0%

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

   4. Simplified 59.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-268}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 49.0%

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

   1. div-inv 48.9%

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

   2. *-commutative 48.9%

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

   3. associate-/r* 48.9%

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

   4. metadata-eval 48.9%

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

   5. add-sqr-sqrt 16.0%

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

   6. sqrt-unprod 31.9%

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

   7. sqr-neg 31.9%

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

   8. sqrt-prod 19.4%

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

   9. add-sqr-sqrt 33.0%

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

   10. fma-neg 33.0%

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

   11. distribute-lft-neg-in 33.0%

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

   12. *-commutative 33.0%

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

   13. associate-*r* 33.0%

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

   14. metadata-eval 33.0%

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

3. Applied egg-rr 33.0%

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

4. Taylor expanded in b around -inf 2.7%

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

5. Final simplification 2.7%

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

Alternative 8: 11.1% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.0%

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

   1. div-inv 48.9%

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

   2. *-commutative 48.9%

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

   3. associate-/r* 48.9%

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

   4. metadata-eval 48.9%

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

   5. add-sqr-sqrt 16.0%

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

   6. sqrt-unprod 31.9%

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

   7. sqr-neg 31.9%

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

   8. sqrt-prod 19.4%

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

   9. add-sqr-sqrt 33.0%

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

   10. fma-neg 33.0%

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

   11. distribute-lft-neg-in 33.0%

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

   12. *-commutative 33.0%

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

   13. associate-*r* 33.0%

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

   14. metadata-eval 33.0%

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

3. Applied egg-rr 33.0%

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

4. Taylor expanded in a around 0 8.7%

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

5. Final simplification 8.7%

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

Developer target: 99.7% accurate, 0.1× speedup

Math

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

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) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	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(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	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 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	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[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2023305 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ c (- (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))) (/ (+ (/ 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)))))) (- a)))

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