quadm (p42, negative)

Percentage Accurate: 52.3% → 85.5%

Time: 13.4s

Alternatives: 7

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: 52.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+86}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-49)
   (/ (- c) b)
   (if (<= b 1.15e+86)
     (* -0.5 (/ (+ b (sqrt (fma b b (* a (* c -4.0))))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-49) {
		tmp = -c / b;
	} else if (b <= 1.15e+86) {
		tmp = -0.5 * ((b + sqrt(fma(b, b, (a * (c * -4.0))))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-49)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.15e+86)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(fma(b, b, Float64(a * Float64(c * -4.0))))) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-49], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.15e+86], N[(-0.5 * N[(N[(b + N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999985e-49

   1. Initial program 16.1%

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

   2. Taylor expanded in b around -inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. neg-mul-1 88.9%

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

   4. Simplified 88.9%

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

   if -1.79999999999999985e-49 < b < 1.14999999999999995e86

   1. Initial program 83.0%

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

      1. /-rgt-identity 83.0%

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

      2. metadata-eval 83.0%

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

      3. associate-/l* 82.9%

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

      4. associate-/r/ 82.9%

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

      5. *-commutative 82.9%

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

      6. metadata-eval 82.9%

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

      7. metadata-eval 82.9%

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

      8. associate-*l/ 82.9%

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

      9. associate-/r/ 82.9%

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

      10. times-frac 83.0%

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

      11. *-commutative 83.0%

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

      12. times-frac 83.0%

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

      13. metadata-eval 83.0%

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

      14. associate-/r/ 83.0%

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

      15. *-commutative 83.0%

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

      16. div-sub 83.0%

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

   3. Simplified 83.0%

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

   if 1.14999999999999995e86 < b

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. unsub-neg 97.0%

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

   4. Simplified 97.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+86}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (+ b (sqrt (* c (* a -4.0)))) a))))
   (if (<= b -4.5e-103)
     (/ (- c) b)
     (if (<= b 3.8e-63)
       t_0
       (if (<= b 5.6e-34)
         (/ (+ (* 2.0 (/ (* c a) b)) (* b -2.0)) (* a 2.0))
         (if (<= b 1.8e+33) t_0 (/ (- b) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = -0.5 * ((b + sqrt((c * (a * -4.0)))) / a);
	double tmp;
	if (b <= -4.5e-103) {
		tmp = -c / b;
	} else if (b <= 3.8e-63) {
		tmp = t_0;
	} else if (b <= 5.6e-34) {
		tmp = ((2.0 * ((c * a) / b)) + (b * -2.0)) / (a * 2.0);
	} else if (b <= 1.8e+33) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * ((b + sqrt((c * (a * (-4.0d0))))) / a)
    if (b <= (-4.5d-103)) then
        tmp = -c / b
    else if (b <= 3.8d-63) then
        tmp = t_0
    else if (b <= 5.6d-34) then
        tmp = ((2.0d0 * ((c * a) / b)) + (b * (-2.0d0))) / (a * 2.0d0)
    else if (b <= 1.8d+33) then
        tmp = t_0
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -0.5 * ((b + Math.sqrt((c * (a * -4.0)))) / a);
	double tmp;
	if (b <= -4.5e-103) {
		tmp = -c / b;
	} else if (b <= 3.8e-63) {
		tmp = t_0;
	} else if (b <= 5.6e-34) {
		tmp = ((2.0 * ((c * a) / b)) + (b * -2.0)) / (a * 2.0);
	} else if (b <= 1.8e+33) {
		tmp = t_0;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -0.5 * ((b + math.sqrt((c * (a * -4.0)))) / a)
	tmp = 0
	if b <= -4.5e-103:
		tmp = -c / b
	elif b <= 3.8e-63:
		tmp = t_0
	elif b <= 5.6e-34:
		tmp = ((2.0 * ((c * a) / b)) + (b * -2.0)) / (a * 2.0)
	elif b <= 1.8e+33:
		tmp = t_0
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / a))
	tmp = 0.0
	if (b <= -4.5e-103)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 3.8e-63)
		tmp = t_0;
	elseif (b <= 5.6e-34)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(c * a) / b)) + Float64(b * -2.0)) / Float64(a * 2.0));
	elseif (b <= 1.8e+33)
		tmp = t_0;
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -0.5 * ((b + sqrt((c * (a * -4.0)))) / a);
	tmp = 0.0;
	if (b <= -4.5e-103)
		tmp = -c / b;
	elseif (b <= 3.8e-63)
		tmp = t_0;
	elseif (b <= 5.6e-34)
		tmp = ((2.0 * ((c * a) / b)) + (b * -2.0)) / (a * 2.0);
	elseif (b <= 1.8e+33)
		tmp = t_0;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.5 * N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.5e-103], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 3.8e-63], t$95$0, If[LessEqual[b, 5.6e-34], N[(N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e+33], t$95$0, N[((-b) / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.5e-103

   1. Initial program 19.3%

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

   2. Taylor expanded in b around -inf 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 -4.5e-103 < b < 3.80000000000000017e-63 or 5.59999999999999994e-34 < b < 1.8000000000000001e33

   1. Initial program 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in a around inf 73.6%

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

      1. *-commutative 73.6%

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

      2. *-commutative 73.6%

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

      3. *-commutative 73.6%

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

      4. associate-*l* 73.6%

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

   6. Simplified 73.6%

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

   if 3.80000000000000017e-63 < b < 5.59999999999999994e-34

   1. Initial program 100.0%

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

   2. Taylor expanded in b around inf 93.5%

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

   if 1.8000000000000001e33 < b

   1. Initial program 63.8%

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

   2. Taylor expanded in b around inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. mul-1-neg 94.1%

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

   4. Simplified 94.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+33}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{if}\;b \leq -8 \cdot 10^{-103}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{0.5}{\frac{c}{b}}\right)}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (+ b (sqrt (* c (* a -4.0)))) a))))
   (if (<= b -8e-103)
     (* -0.5 (/ 1.0 (fma -0.5 (/ a b) (/ 0.5 (/ c b)))))
     (if (<= b 7e-63)
       t_0
       (if (<= b 1.8e-33)
         (/ (+ (* 2.0 (/ (* c a) b)) (* b -2.0)) (* a 2.0))
         (if (<= b 1.8e+33) t_0 (/ (- b) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = -0.5 * ((b + sqrt((c * (a * -4.0)))) / a);
	double tmp;
	if (b <= -8e-103) {
		tmp = -0.5 * (1.0 / fma(-0.5, (a / b), (0.5 / (c / b))));
	} else if (b <= 7e-63) {
		tmp = t_0;
	} else if (b <= 1.8e-33) {
		tmp = ((2.0 * ((c * a) / b)) + (b * -2.0)) / (a * 2.0);
	} else if (b <= 1.8e+33) {
		tmp = t_0;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / a))
	tmp = 0.0
	if (b <= -8e-103)
		tmp = Float64(-0.5 * Float64(1.0 / fma(-0.5, Float64(a / b), Float64(0.5 / Float64(c / b)))));
	elseif (b <= 7e-63)
		tmp = t_0;
	elseif (b <= 1.8e-33)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(c * a) / b)) + Float64(b * -2.0)) / Float64(a * 2.0));
	elseif (b <= 1.8e+33)
		tmp = t_0;
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.5 * N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8e-103], N[(-0.5 * N[(1.0 / N[(-0.5 * N[(a / b), $MachinePrecision] + N[(0.5 / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-63], t$95$0, If[LessEqual[b, 1.8e-33], N[(N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e+33], t$95$0, N[((-b) / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -7.99999999999999966e-103

   1. Initial program 19.3%

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

      1. /-rgt-identity 19.3%

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

      2. metadata-eval 19.3%

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

      3. associate-/l* 19.3%

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

      4. associate-/r/ 19.3%

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

      5. *-commutative 19.3%

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

      6. metadata-eval 19.3%

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

      7. metadata-eval 19.3%

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

      8. associate-*l/ 19.3%

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

      9. associate-/r/ 19.3%

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

      10. times-frac 19.3%

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

      11. *-commutative 19.3%

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

      12. times-frac 19.3%

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

      13. metadata-eval 19.3%

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

      14. associate-/r/ 19.3%

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

      15. *-commutative 19.3%

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

      16. div-sub 18.6%

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

   3. Simplified 19.3%

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

      1. fma-udef 19.3%

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

   5. Applied egg-rr 19.3%

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

      1. clear-num 19.3%

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

      2. inv-pow 19.3%

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

   7. Applied egg-rr 25.1%

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

      1. unpow-1 25.1%

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

   9. Simplified 25.1%

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

   10. Taylor expanded in b around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. fma-def 0.0%

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

       3. associate-*r/ 0.0%

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

       4. *-commutative 0.0%

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

       5. times-frac 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 85.9%

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

       8. metadata-eval 85.9%

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

   12. Simplified 85.9%

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

       1. expm1-log1p-u 81.7%

          \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\right)\right)} \]

       2. expm1-udef 31.2%

          \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\right)} - 1\right)} \]

       3. associate-*r/ 31.2%

          \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{\frac{0.5 \cdot b}{c}}\right)}\right)} - 1\right) \]

   14. Applied egg-rr 31.2%

       \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{0.5 \cdot b}{c}\right)}\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 81.7%

          \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{0.5 \cdot b}{c}\right)}\right)\right)} \]

       2. expm1-log1p 85.9%

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

       3. associate-/l* 86.1%

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

   16. Simplified 86.1%

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

   if -7.99999999999999966e-103 < b < 7.00000000000000006e-63 or 1.80000000000000017e-33 < b < 1.8000000000000001e33

   1. Initial program 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in a around inf 73.6%

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

      1. *-commutative 73.6%

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

      2. *-commutative 73.6%

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

      3. *-commutative 73.6%

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

      4. associate-*l* 73.6%

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

   6. Simplified 73.6%

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

   if 7.00000000000000006e-63 < b < 1.80000000000000017e-33

   1. Initial program 100.0%

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

   2. Taylor expanded in b around inf 93.5%

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

   if 1.8000000000000001e33 < b

   1. Initial program 63.8%

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

   2. Taylor expanded in b around inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. mul-1-neg 94.1%

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

   4. Simplified 94.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-103}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{0.5}{\frac{c}{b}}\right)}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-63}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+33}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-51}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+85}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e-51)
   (/ (- c) b)
   (if (<= b 5e+85)
     (* -0.5 (/ (+ b (sqrt (+ (* a (* c -4.0)) (* b b)))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e-51) {
		tmp = -c / b;
	} else if (b <= 5e+85) {
		tmp = -0.5 * ((b + sqrt(((a * (c * -4.0)) + (b * b)))) / a);
	} 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 <= (-7d-51)) then
        tmp = -c / b
    else if (b <= 5d+85) then
        tmp = (-0.5d0) * ((b + sqrt(((a * (c * (-4.0d0))) + (b * b)))) / a)
    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 <= -7e-51) {
		tmp = -c / b;
	} else if (b <= 5e+85) {
		tmp = -0.5 * ((b + Math.sqrt(((a * (c * -4.0)) + (b * b)))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7e-51:
		tmp = -c / b
	elif b <= 5e+85:
		tmp = -0.5 * ((b + math.sqrt(((a * (c * -4.0)) + (b * b)))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e-51)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 5e+85)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(a * Float64(c * -4.0)) + Float64(b * b)))) / a));
	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 <= -7e-51)
		tmp = -c / b;
	elseif (b <= 5e+85)
		tmp = -0.5 * ((b + sqrt(((a * (c * -4.0)) + (b * b)))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e-51], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 5e+85], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.9999999999999995e-51

   1. Initial program 16.1%

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

   2. Taylor expanded in b around -inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. neg-mul-1 88.9%

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

   4. Simplified 88.9%

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

   if -6.9999999999999995e-51 < b < 5.0000000000000001e85

   1. Initial program 83.0%

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

      1. /-rgt-identity 83.0%

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

      2. metadata-eval 83.0%

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

      3. associate-/l* 82.9%

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

      4. associate-/r/ 82.9%

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

      5. *-commutative 82.9%

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

      6. metadata-eval 82.9%

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

      7. metadata-eval 82.9%

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

      8. associate-*l/ 82.9%

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

      9. associate-/r/ 82.9%

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

      10. times-frac 83.0%

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

      11. *-commutative 83.0%

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

      12. times-frac 83.0%

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

      13. metadata-eval 83.0%

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

      14. associate-/r/ 83.0%

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

      15. *-commutative 83.0%

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

      16. div-sub 83.0%

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

   3. Simplified 83.0%

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

      1. fma-udef 83.0%

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

   5. Applied egg-rr 83.0%

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

   if 5.0000000000000001e85 < b

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. unsub-neg 97.0%

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

   4. Simplified 97.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-51}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+85}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 68.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-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 <= -2e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-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 < -1.999999999999994e-310

   1. Initial program 33.6%

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

   2. Taylor expanded in b around -inf 66.7%

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

      1. associate-*r/ 66.7%

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

      2. neg-mul-1 66.7%

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

   4. Simplified 66.7%

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

   if -1.999999999999994e-310 < b

   1. Initial program 74.0%

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

   2. Taylor expanded in b around inf 66.4%

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

      1. mul-1-neg 66.4%

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

      2. unsub-neg 66.4%

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

   4. Simplified 66.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 68.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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 <= (-2d-310)) 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 <= -2e-310) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		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 <= -2e-310)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 33.6%

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

   2. Taylor expanded in b around -inf 66.7%

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

      1. associate-*r/ 66.7%

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

      2. neg-mul-1 66.7%

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

   4. Simplified 66.7%

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

   if -1.999999999999994e-310 < b

   1. Initial program 74.0%

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

   2. Taylor expanded in b around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. mul-1-neg 66.2%

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

   4. Simplified 66.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7: 35.9% accurate, 29.0× 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(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 55.1%

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

2. Taylor expanded in b around inf 36.5%

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

   1. associate-*r/ 36.5%

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

   2. mul-1-neg 36.5%

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

4. Simplified 36.5%

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

5. Final simplification 36.5%

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

Developer target: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * 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 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023258 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

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