quadp (p42, positive)

Percentage Accurate: 52.0% → 87.8%

Time: 12.7s

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: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-42}:\\ \;\;\;\;\left(a \cdot \frac{-2}{a}\right) \cdot \frac{c}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+154)
   (/ (- b) a)
   (if (<= b -5.2e-250)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b 6.5e-42)
       (* (* a (/ -2.0 a)) (/ c (+ b (hypot b (sqrt (* a (* c -4.0)))))))
       (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = -b / a;
	} else if (b <= -5.2e-250) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 6.5e-42) {
		tmp = (a * (-2.0 / a)) * (c / (b + hypot(b, sqrt((a * (c * -4.0))))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = -b / a;
	} else if (b <= -5.2e-250) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 6.5e-42) {
		tmp = (a * (-2.0 / a)) * (c / (b + Math.hypot(b, Math.sqrt((a * (c * -4.0))))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+154:
		tmp = -b / a
	elif b <= -5.2e-250:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	elif b <= 6.5e-42:
		tmp = (a * (-2.0 / a)) * (c / (b + math.hypot(b, math.sqrt((a * (c * -4.0))))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -5.2e-250)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= 6.5e-42)
		tmp = Float64(Float64(a * Float64(-2.0 / a)) * Float64(c / Float64(b + hypot(b, sqrt(Float64(a * Float64(c * -4.0)))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e+154)
		tmp = -b / a;
	elseif (b <= -5.2e-250)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	elseif (b <= 6.5e-42)
		tmp = (a * (-2.0 / a)) * (c / (b + hypot(b, sqrt((a * (c * -4.0))))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -5.2e-250], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-42], N[(N[(a * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision] * N[(c / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.5000000000000002e154

   1. Initial program 31.1%

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

   2. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -8.5000000000000002e154 < b < -5.20000000000000016e-250

   1. Initial program 86.4%

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

   if -5.20000000000000016e-250 < b < 6.4999999999999998e-42

   1. Initial program 71.8%

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

      1. +-commutative 71.8%

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

      2. flip-+ 71.5%

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

      3. add-sqr-sqrt 71.5%

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

      4. fma-neg 71.5%

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

      5. distribute-lft-neg-in 71.5%

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

      6. associate-*r* 71.5%

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

      7. metadata-eval 71.5%

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

      8. sqr-neg 71.5%

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

      9. add-sqr-sqrt 22.2%

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

      10. sqrt-unprod 67.6%

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

      11. sqr-neg 67.6%

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

      12. sqrt-prod 45.7%

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

      13. add-sqr-sqrt 67.9%

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

      14. unsub-neg 67.9%

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

      15. +-commutative 67.9%

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

   3. Applied egg-rr 71.5%

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

      1. fma-def 71.5%

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

      2. +-commutative 71.5%

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

      3. associate-*r* 71.5%

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

      4. *-commutative 71.5%

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

      5. fma-udef 71.5%

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

      6. *-commutative 71.5%

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

      7. *-commutative 71.5%

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

   5. Simplified 71.5%

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

   6. Taylor expanded in a around 0 75.9%

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

      1. div-inv 75.9%

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

      2. associate-/l* 75.8%

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

      3. *-commutative 75.8%

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

      4. *-commutative 75.8%

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

      5. associate-*r* 75.8%

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

      6. *-commutative 75.8%

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

   8. Applied egg-rr 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-/r/ 75.8%

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

      3. associate-*l/ 75.9%

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

      4. times-frac 62.3%

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

      5. *-lft-identity 62.3%

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

      6. times-frac 75.9%

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

      7. *-commutative 75.9%

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

      8. associate-/r* 75.9%

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

      9. metadata-eval 75.9%

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

      10. *-commutative 75.9%

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

      11. associate-*l* 75.9%

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

   10. Simplified 75.9%

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

       1. frac-times 62.3%

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

   12. Applied egg-rr 62.3%

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

       1. associate-*r* 62.3%

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

       2. times-frac 85.0%

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

       3. associate-*l/ 84.7%

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

   14. Simplified 84.7%

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

   if 6.4999999999999998e-42 < b

   1. Initial program 11.4%

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

   2. Taylor expanded in b around inf 89.8%

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

      1. associate-*r/ 89.8%

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

      2. neg-mul-1 89.8%

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

   4. Simplified 89.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-42}:\\ \;\;\;\;\left(a \cdot \frac{-2}{a}\right) \cdot \frac{c}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+154)
   (/ (- b) a)
   (if (<= b 4.7e-99)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = -b / a;
	} else if (b <= 4.7e-99) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.5d+154)) then
        tmp = -b / a
    else if (b <= 4.7d-99) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = -b / a;
	} else if (b <= 4.7e-99) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+154:
		tmp = -b / a
	elif b <= 4.7e-99:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.7e-99)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e+154)
		tmp = -b / a;
	elseif (b <= 4.7e-99)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.7e-99], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5000000000000002e154

   1. Initial program 31.1%

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

   2. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -8.5000000000000002e154 < b < 4.69999999999999989e-99

   1. Initial program 82.8%

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

   if 4.69999999999999989e-99 < b

   1. Initial program 15.2%

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

   2. Taylor expanded in b around inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

   4. Simplified 86.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-89}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.18 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{c \cdot \frac{a}{-0.25}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-89)
   (- (/ c b) (/ b a))
   (if (<= b 1.18e-99)
     (/ (- (sqrt (* c (/ a -0.25))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-89) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.18e-99) {
		tmp = (sqrt((c * (a / -0.25))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-89)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.18d-99) then
        tmp = (sqrt((c * (a / (-0.25d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-89) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.18e-99) {
		tmp = (Math.sqrt((c * (a / -0.25))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-89:
		tmp = (c / b) - (b / a)
	elif b <= 1.18e-99:
		tmp = (math.sqrt((c * (a / -0.25))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-89)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.18e-99)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a / -0.25))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-89)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.18e-99)
		tmp = (sqrt((c * (a / -0.25))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-89], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.18e-99], N[(N[(N[Sqrt[N[(c * N[(a / -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000004e-89

   1. Initial program 64.7%

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

   2. Taylor expanded in b around -inf 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

   4. Simplified 83.5%

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

   if -1.00000000000000004e-89 < b < 1.1800000000000001e-99

   1. Initial program 77.2%

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

      1. add-sqr-sqrt 76.9%

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

      2. pow2 76.9%

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

      3. pow1/2 76.9%

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

      4. sqrt-pow1 76.9%

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

      5. fma-neg 76.9%

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

      6. distribute-lft-neg-in 76.9%

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

      7. associate-*r* 76.9%

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

      8. metadata-eval 76.9%

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

      9. metadata-eval 76.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{2 \cdot a} \]

   3. Applied egg-rr 76.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{0.25}\right)}^{2}}}{2 \cdot a} \]

   4. Taylor expanded in c around -inf 46.4%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2} - b}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. unpow2 46.4%

         \[\leadsto \frac{\color{blue}{e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} - b}{2 \cdot a} \]

      2. exp-prod 45.3%

         \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} \cdot e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} - b}{2 \cdot a} \]

      3. exp-prod 44.6%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} - b}{2 \cdot a} \]

      4. pow-sqr 44.6%

         \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)\right)}} - b}{2 \cdot a} \]

      5. mul-1-neg 44.6%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(4 \cdot a\right) + \color{blue}{\left(-\log \left(\frac{-1}{c}\right)\right)}\right)\right)} - b}{2 \cdot a} \]

      6. unsub-neg 44.6%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \color{blue}{\left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)}\right)} - b}{2 \cdot a} \]

      7. *-commutative 44.6%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \color{blue}{\left(a \cdot 4\right)} - \log \left(\frac{-1}{c}\right)\right)\right)} - b}{2 \cdot a} \]

   6. Simplified 44.6%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(a \cdot 4\right) - \log \left(\frac{-1}{c}\right)\right)\right)} - b}}{2 \cdot a} \]

   7. Taylor expanded in a around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. +-commutative 46.3%

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

      3. log-prod 46.4%

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

      4. log-div 68.4%

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

      5. exp-to-pow 72.6%

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

      6. unpow1/2 72.6%

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

      7. associate-/r/ 72.6%

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

      8. *-commutative 72.6%

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

      9. associate-/l* 72.6%

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

      10. metadata-eval 72.6%

          \[\leadsto \frac{\sqrt{c \cdot \frac{a}{\color{blue}{-0.25}}} - b}{2 \cdot a} \]

   9. Simplified 72.6%

      \[\leadsto \frac{\color{blue}{\sqrt{c \cdot \frac{a}{-0.25}} - b}}{2 \cdot a} \]

   if 1.1800000000000001e-99 < b

   1. Initial program 15.2%

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

   2. Taylor expanded in b around inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

   4. Simplified 86.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-89}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.18 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{c \cdot \frac{a}{-0.25}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \frac{a}{-0.25}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-87)
   (- (/ c b) (/ b a))
   (if (<= b 4.2e-99) (* 0.5 (/ (sqrt (* c (/ a -0.25))) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.2e-99) {
		tmp = 0.5 * (sqrt((c * (a / -0.25))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.4d-87)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4.2d-99) then
        tmp = 0.5d0 * (sqrt((c * (a / (-0.25d0)))) / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.2e-99) {
		tmp = 0.5 * (Math.sqrt((c * (a / -0.25))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-87:
		tmp = (c / b) - (b / a)
	elif b <= 4.2e-99:
		tmp = 0.5 * (math.sqrt((c * (a / -0.25))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-87)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.2e-99)
		tmp = Float64(0.5 * Float64(sqrt(Float64(c * Float64(a / -0.25))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-87)
		tmp = (c / b) - (b / a);
	elseif (b <= 4.2e-99)
		tmp = 0.5 * (sqrt((c * (a / -0.25))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-87], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.2e-99], N[(0.5 * N[(N[Sqrt[N[(c * N[(a / -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4e-87

   1. Initial program 64.7%

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

   2. Taylor expanded in b around -inf 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

   4. Simplified 83.5%

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

   if -2.4e-87 < b < 4.19999999999999968e-99

   1. Initial program 77.2%

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

      1. add-sqr-sqrt 76.9%

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

      2. pow2 76.9%

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

      3. pow1/2 76.9%

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

      4. sqrt-pow1 76.9%

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

      5. fma-neg 76.9%

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

      6. distribute-lft-neg-in 76.9%

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

      7. associate-*r* 76.9%

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

      8. metadata-eval 76.9%

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

      9. metadata-eval 76.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{2 \cdot a} \]

   3. Applied egg-rr 76.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{0.25}\right)}^{2}}}{2 \cdot a} \]

   4. Taylor expanded in c around -inf 46.4%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2} - b}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. unpow2 46.4%

         \[\leadsto \frac{\color{blue}{e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} - b}{2 \cdot a} \]

      2. exp-prod 45.3%

         \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} \cdot e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} - b}{2 \cdot a} \]

      3. exp-prod 44.6%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} - b}{2 \cdot a} \]

      4. pow-sqr 44.6%

         \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)\right)}} - b}{2 \cdot a} \]

      5. mul-1-neg 44.6%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(4 \cdot a\right) + \color{blue}{\left(-\log \left(\frac{-1}{c}\right)\right)}\right)\right)} - b}{2 \cdot a} \]

      6. unsub-neg 44.6%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \color{blue}{\left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)}\right)} - b}{2 \cdot a} \]

      7. *-commutative 44.6%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \color{blue}{\left(a \cdot 4\right)} - \log \left(\frac{-1}{c}\right)\right)\right)} - b}{2 \cdot a} \]

   6. Simplified 44.6%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(a \cdot 4\right) - \log \left(\frac{-1}{c}\right)\right)\right)} - b}}{2 \cdot a} \]

   7. Taylor expanded in b around 0 46.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{0.5 \cdot \left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)}}{a}} \]
   8. Step-by-step derivation

      1. log-prod 46.2%

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

      2. +-commutative 46.2%

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

      3. log-prod 46.3%

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

      4. log-div 67.7%

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

      5. *-commutative 67.7%

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

      6. exp-to-pow 71.8%

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

      7. unpow1/2 71.8%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\frac{a \cdot 4}{\frac{-1}{c}}}}}{a} \]

      8. associate-/r/ 71.8%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\frac{a \cdot 4}{-1} \cdot c}}}{a} \]

      9. *-commutative 71.8%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{c \cdot \frac{a \cdot 4}{-1}}}}{a} \]

      10. associate-/l* 71.8%

          \[\leadsto 0.5 \cdot \frac{\sqrt{c \cdot \color{blue}{\frac{a}{\frac{-1}{4}}}}}{a} \]

      11. metadata-eval 71.8%

          \[\leadsto 0.5 \cdot \frac{\sqrt{c \cdot \frac{a}{\color{blue}{-0.25}}}}{a} \]

   9. Simplified 71.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{c \cdot \frac{a}{-0.25}}}{a}} \]

   if 4.19999999999999968e-99 < b

   1. Initial program 15.2%

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

   2. Taylor expanded in b around inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

   4. Simplified 86.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \frac{a}{-0.25}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.0%

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

   2. Taylor expanded in b around -inf 64.1%

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

      1. +-commutative 64.1%

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

      2. mul-1-neg 64.1%

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

      3. unsub-neg 64.1%

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

   4. Simplified 64.1%

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

   if -9.999999999999969e-311 < b

   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 66.4%

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

      1. associate-*r/ 66.4%

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

      2. neg-mul-1 66.4%

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

   4. Simplified 66.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 44.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.65 \cdot 10^{-22}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 3.65e-22) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.65e-22) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.65d-22) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.65e-22) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.65e-22:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.65e-22)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.65e-22)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.65e-22], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.65000000000000014e-22

   1. Initial program 67.4%

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

   2. Taylor expanded in b around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. mul-1-neg 47.2%

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

   4. Simplified 47.2%

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

   if 3.65000000000000014e-22 < b

   1. Initial program 11.8%

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

   2. Taylor expanded in b around -inf 2.2%

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

   3. Taylor expanded in b around 0 28.6%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.65 \cdot 10^{-22}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7: 68.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.0%

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

   2. Taylor expanded in b around -inf 63.6%

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

      1. associate-*r/ 63.6%

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

      2. mul-1-neg 63.6%

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

   4. Simplified 63.6%

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

   if -9.999999999999969e-311 < b

   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 66.4%

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

      1. associate-*r/ 66.4%

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

      2. neg-mul-1 66.4%

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

   4. Simplified 66.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 11.2% 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 50.4%

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

2. Taylor expanded in b around -inf 31.2%

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

3. Taylor expanded in b around 0 11.0%

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

4. Final simplification 11.0%

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

Developer target: 70.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* 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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-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(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * 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 = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-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[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023271 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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

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