The quadratic formula (r2)

Percentage Accurate: 52.9% → 91.2%

Time: 16.7s

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: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-307}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+75}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ (- c) b)
   (if (<= b 8.5e-307)
     (* -0.5 (/ (* c 4.0) (- b (sqrt (fma a (* c -4.0) (* b b))))))
     (if (<= b 6.5e+75)
       (* -0.5 (/ (+ b (sqrt (- (* b b) (* 4.0 (* c a))))) a))
       (- (/ c b) (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+153) {
		tmp = -c / b;
	} else if (b <= 8.5e-307) {
		tmp = -0.5 * ((c * 4.0) / (b - sqrt(fma(a, (c * -4.0), (b * b)))));
	} else if (b <= 6.5e+75) {
		tmp = -0.5 * ((b + sqrt(((b * b) - (4.0 * (c * a))))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+153)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 8.5e-307)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b))))));
	elseif (b <= 6.5e+75)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+153], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 8.5e-307], N[(-0.5 * N[(N[(c * 4.0), $MachinePrecision] / N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+75], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1e153

   1. Initial program 1.7%

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

   2. Taylor expanded in b around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. neg-mul-1 97.7%

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

   4. Simplified 97.7%

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

   if -1e153 < b < 8.4999999999999995e-307

   1. Initial program 42.3%

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

      1. sub-neg 42.3%

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

      2. distribute-neg-out 42.3%

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

      3. neg-mul-1 42.3%

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

      4. times-frac 42.3%

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

      5. metadata-eval 42.3%

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

      6. remove-double-neg 42.3%

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

      7. neg-sub0 42.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(0 - \left(-b\right)\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \]

      8. associate-+l- 42.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0 - \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{a} \]

   3. Simplified 42.3%

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

      1. fma-udef 42.3%

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

      2. *-commutative 42.3%

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

      3. metadata-eval 42.3%

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

      4. cancel-sign-sub-inv 42.3%

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

      5. prod-diff 42.1%

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

      6. *-commutative 42.1%

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

      7. fma-neg 42.1%

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

      8. associate-+l- 42.1%

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

      9. add-sqr-sqrt 4.7%

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

      10. sqrt-unprod 6.2%

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

      11. *-commutative 6.2%

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

      12. *-commutative 6.2%

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

      13. swap-sqr 6.2%

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

      14. metadata-eval 6.2%

          \[\leadsto -0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(\sqrt{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{16}} - \mathsf{fma}\left(-a \cdot c, 4, \left(a \cdot c\right) \cdot 4\right)\right)}}{a} \]

      15. metadata-eval 6.2%

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

      16. swap-sqr 6.2%

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

      17. sqrt-unprod 3.8%

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

      18. add-sqr-sqrt 5.1%

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

      19. associate-*l* 5.1%

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

   5. Applied egg-rr 42.1%

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

      1. associate--r+ 42.1%

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

      2. +-inverses 42.3%

         \[\leadsto -0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)}}{a} \]

      3. neg-sub0 42.3%

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

      4. associate-*r* 42.3%

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

      5. distribute-rgt-neg-in 42.3%

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

      6. metadata-eval 42.3%

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

   7. Simplified 42.3%

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

      1. flip-+ 42.1%

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

      2. add-sqr-sqrt 42.1%

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

      3. associate-*l* 42.1%

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

      4. associate-*l* 42.1%

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

   9. Applied egg-rr 42.1%

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

       1. associate-+l- 69.1%

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

       2. +-inverses 69.1%

          \[\leadsto -0.5 \cdot \frac{\frac{\color{blue}{0} + a \cdot \left(c \cdot 4\right)}{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}}{a} \]

       3. +-commutative 69.1%

          \[\leadsto -0.5 \cdot \frac{\frac{\color{blue}{a \cdot \left(c \cdot 4\right) + 0}}{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}}{a} \]

       4. sub-neg 69.1%

          \[\leadsto -0.5 \cdot \frac{\frac{a \cdot \left(c \cdot 4\right) + 0}{b - \sqrt{\color{blue}{b \cdot b + \left(-a \cdot \left(c \cdot 4\right)\right)}}}}{a} \]

       5. +-commutative 69.1%

          \[\leadsto -0.5 \cdot \frac{\frac{a \cdot \left(c \cdot 4\right) + 0}{b - \sqrt{\color{blue}{\left(-a \cdot \left(c \cdot 4\right)\right) + b \cdot b}}}}{a} \]

       6. distribute-rgt-neg-in 69.1%

          \[\leadsto -0.5 \cdot \frac{\frac{a \cdot \left(c \cdot 4\right) + 0}{b - \sqrt{\color{blue}{a \cdot \left(-c \cdot 4\right)} + b \cdot b}}}{a} \]

       7. fma-def 69.1%

          \[\leadsto -0.5 \cdot \frac{\frac{a \cdot \left(c \cdot 4\right) + 0}{b - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c \cdot 4, b \cdot b\right)}}}}{a} \]

       8. distribute-rgt-neg-in 69.1%

          \[\leadsto -0.5 \cdot \frac{\frac{a \cdot \left(c \cdot 4\right) + 0}{b - \sqrt{\mathsf{fma}\left(a, \color{blue}{c \cdot \left(-4\right)}, b \cdot b\right)}}}{a} \]

       9. metadata-eval 69.1%

          \[\leadsto -0.5 \cdot \frac{\frac{a \cdot \left(c \cdot 4\right) + 0}{b - \sqrt{\mathsf{fma}\left(a, c \cdot \color{blue}{-4}, b \cdot b\right)}}}{a} \]

   11. Simplified 69.1%

       \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{a \cdot \left(c \cdot 4\right) + 0}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 54.9%

          \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{a \cdot \left(c \cdot 4\right) + 0}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)\right)} \]

       2. expm1-udef 16.5%

          \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{a \cdot \left(c \cdot 4\right) + 0}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)} - 1\right)} \]

       3. +-rgt-identity 16.5%

          \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{a \cdot \left(c \cdot 4\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)} - 1\right) \]

       4. associate-/l/ 16.2%

          \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{a \cdot \left(c \cdot 4\right)}{a \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}}\right)} - 1\right) \]

   13. Applied egg-rr 16.2%

       \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 4\right)}{a \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 54.3%

          \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 4\right)}{a \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\right)\right)} \]

       2. expm1-log1p 68.5%

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

       3. times-frac 87.1%

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

       4. *-inverses 87.1%

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

   15. Simplified 87.1%

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

   if 8.4999999999999995e-307 < b < 6.4999999999999998e75

   1. Initial program 85.2%

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

      1. sub-neg 85.2%

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

      2. distribute-neg-out 85.2%

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

      3. neg-mul-1 85.2%

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

      4. times-frac 86.7%

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

      5. metadata-eval 86.7%

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

      6. remove-double-neg 86.7%

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

      7. neg-sub0 86.7%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(0 - \left(-b\right)\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \]

      8. associate-+l- 86.7%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0 - \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{a} \]

   3. Simplified 86.7%

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

      1. fma-udef 86.7%

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

      2. *-commutative 86.7%

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

      3. metadata-eval 86.7%

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

      4. cancel-sign-sub-inv 86.7%

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

      5. prod-diff 86.3%

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

      6. *-commutative 86.3%

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

      7. fma-neg 86.3%

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

      8. associate-+l- 86.3%

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

      9. add-sqr-sqrt 29.6%

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

      10. sqrt-unprod 35.9%

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

      11. *-commutative 35.9%

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

      12. *-commutative 35.9%

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

      13. swap-sqr 35.9%

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

      14. metadata-eval 35.9%

          \[\leadsto -0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(\sqrt{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{16}} - \mathsf{fma}\left(-a \cdot c, 4, \left(a \cdot c\right) \cdot 4\right)\right)}}{a} \]

      15. metadata-eval 35.9%

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

      16. swap-sqr 35.9%

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

      17. sqrt-unprod 21.9%

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

      18. add-sqr-sqrt 35.8%

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

      19. associate-*l* 35.8%

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

   5. Applied egg-rr 86.3%

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

      1. associate--r+ 86.3%

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

      2. +-inverses 86.7%

         \[\leadsto -0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)}}{a} \]

      3. neg-sub0 86.7%

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

      4. associate-*r* 86.7%

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

      5. distribute-rgt-neg-in 86.7%

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

      6. metadata-eval 86.7%

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

   7. Simplified 86.7%

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

   if 6.4999999999999998e75 < b

   1. Initial program 56.9%

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

   2. Taylor expanded in b around inf 96.3%

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

      1. +-commutative 96.3%

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

      2. mul-1-neg 96.3%

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

      3. unsub-neg 96.3%

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

   4. Simplified 96.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-307}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+75}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-78}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{+75}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e-78)
   (/ (- c) b)
   (if (<= b 7.8e+75)
     (* -0.5 (/ (+ b (sqrt (- (* b b) (* 4.0 (* c a))))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e-78) {
		tmp = -c / b;
	} else if (b <= 7.8e+75) {
		tmp = -0.5 * ((b + sqrt(((b * b) - (4.0 * (c * a))))) / 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 <= (-1.02d-78)) then
        tmp = -c / b
    else if (b <= 7.8d+75) then
        tmp = (-0.5d0) * ((b + sqrt(((b * b) - (4.0d0 * (c * a))))) / 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 <= -1.02e-78) {
		tmp = -c / b;
	} else if (b <= 7.8e+75) {
		tmp = -0.5 * ((b + Math.sqrt(((b * b) - (4.0 * (c * a))))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.02e-78:
		tmp = -c / b
	elif b <= 7.8e+75:
		tmp = -0.5 * ((b + math.sqrt(((b * b) - (4.0 * (c * a))))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.02e-78)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 7.8e+75)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / 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 <= -1.02e-78)
		tmp = -c / b;
	elseif (b <= 7.8e+75)
		tmp = -0.5 * ((b + sqrt(((b * b) - (4.0 * (c * a))))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e-78], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 7.8e+75], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $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 < -1.02e-78

   1. Initial program 11.2%

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

   2. Taylor expanded in b around -inf 89.9%

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

      1. associate-*r/ 89.9%

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

      2. neg-mul-1 89.9%

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

   4. Simplified 89.9%

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

   if -1.02e-78 < b < 7.80000000000000075e75

   1. Initial program 78.1%

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

      1. sub-neg 78.1%

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

      2. distribute-neg-out 78.1%

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

      3. neg-mul-1 78.1%

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

      4. times-frac 79.0%

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

      5. metadata-eval 79.0%

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

      6. remove-double-neg 79.0%

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

      7. neg-sub0 79.0%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(0 - \left(-b\right)\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \]

      8. associate-+l- 79.0%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0 - \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{a} \]

   3. Simplified 79.0%

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

      1. fma-udef 79.0%

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

      2. *-commutative 79.0%

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

      3. metadata-eval 79.0%

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

      4. cancel-sign-sub-inv 79.0%

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

      5. prod-diff 78.7%

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

      6. *-commutative 78.7%

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

      7. fma-neg 78.7%

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

      8. associate-+l- 78.7%

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

      9. add-sqr-sqrt 19.5%

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

      10. sqrt-unprod 23.5%

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

      11. *-commutative 23.5%

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

      12. *-commutative 23.5%

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

      13. swap-sqr 23.5%

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

      14. metadata-eval 23.5%

          \[\leadsto -0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(\sqrt{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{16}} - \mathsf{fma}\left(-a \cdot c, 4, \left(a \cdot c\right) \cdot 4\right)\right)}}{a} \]

      15. metadata-eval 23.5%

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

      16. swap-sqr 23.5%

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

      17. sqrt-unprod 14.5%

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

      18. add-sqr-sqrt 23.4%

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

      19. associate-*l* 23.4%

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

   5. Applied egg-rr 78.7%

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

      1. associate--r+ 78.7%

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

      2. +-inverses 79.0%

         \[\leadsto -0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)}}{a} \]

      3. neg-sub0 79.0%

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

      4. associate-*r* 79.0%

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

      5. distribute-rgt-neg-in 79.0%

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

      6. metadata-eval 79.0%

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

   7. Simplified 79.0%

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

   if 7.80000000000000075e75 < b

   1. Initial program 56.9%

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

   2. Taylor expanded in b around inf 96.3%

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

      1. +-commutative 96.3%

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

      2. mul-1-neg 96.3%

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

      3. unsub-neg 96.3%

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

   4. Simplified 96.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-78}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{+75}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-71)
   (/ (- c) b)
   (if (<= b 5.2e-70)
     (* -0.5 (/ (+ b (sqrt (* a (* c -4.0)))) a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-71) {
		tmp = -c / b;
	} else if (b <= 5.2e-70) {
		tmp = -0.5 * ((b + sqrt((a * (c * -4.0)))) / a);
	} 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 <= (-3.8d-71)) then
        tmp = -c / b
    else if (b <= 5.2d-70) then
        tmp = (-0.5d0) * ((b + sqrt((a * (c * (-4.0d0))))) / a)
    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 <= -3.8e-71) {
		tmp = -c / b;
	} else if (b <= 5.2e-70) {
		tmp = -0.5 * ((b + Math.sqrt((a * (c * -4.0)))) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-71:
		tmp = -c / b
	elif b <= 5.2e-70:
		tmp = -0.5 * ((b + math.sqrt((a * (c * -4.0)))) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-71)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 5.2e-70)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(a * Float64(c * -4.0)))) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-71)
		tmp = -c / b;
	elseif (b <= 5.2e-70)
		tmp = -0.5 * ((b + sqrt((a * (c * -4.0)))) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-71], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 5.2e-70], N[(-0.5 * N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.79999999999999992e-71

   1. Initial program 11.2%

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

   2. Taylor expanded in b around -inf 89.9%

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

      1. associate-*r/ 89.9%

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

      2. neg-mul-1 89.9%

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

   4. Simplified 89.9%

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

   if -3.79999999999999992e-71 < b < 5.20000000000000004e-70

   1. Initial program 74.3%

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

      1. sub-neg 74.3%

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

      2. distribute-neg-out 74.3%

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

      3. neg-mul-1 74.3%

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

      4. times-frac 74.3%

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

      5. metadata-eval 74.3%

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

      6. remove-double-neg 74.3%

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

      7. neg-sub0 74.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(0 - \left(-b\right)\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \]

      8. associate-+l- 74.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0 - \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{a} \]

   3. Simplified 74.3%

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

   4. Taylor expanded in b around 0 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-*r* 69.5%

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

   6. Simplified 69.5%

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

   if 5.20000000000000004e-70 < b

   1. Initial program 65.5%

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

   2. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. mul-1-neg 87.4%

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

   4. Simplified 87.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4: 67.4% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 27.3%

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

   2. Taylor expanded in b around -inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. neg-mul-1 71.4%

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

   4. Simplified 71.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 70.3%

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

   2. Taylor expanded in b around inf 69.7%

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

      1. +-commutative 69.7%

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

      2. mul-1-neg 69.7%

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

      3. unsub-neg 69.7%

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

   4. Simplified 69.7%

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

4. Final simplification 70.5%

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

Alternative 5: 67.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-266}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-266) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-266) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.8d-266)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-266) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e-266:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-266)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e-266)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-266], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.8e-266

   1. Initial program 25.6%

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

   2. Taylor expanded in b around -inf 73.8%

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

      1. associate-*r/ 73.8%

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

      2. neg-mul-1 73.8%

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

   4. Simplified 73.8%

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

   if -2.8e-266 < b

   1. Initial program 70.4%

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

   2. Taylor expanded in b around inf 67.8%

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

      1. associate-*r/ 67.8%

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

      2. mul-1-neg 67.8%

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

   4. Simplified 67.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-266}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6: 34.6% 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 51.0%

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

2. Taylor expanded in b around inf 39.6%

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

   1. associate-*r/ 39.6%

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

   2. mul-1-neg 39.6%

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

4. Simplified 39.6%

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

5. Final simplification 39.6%

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

Alternative 7: 2.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

   1. add-cube-cbrt 50.2%

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

   2. pow3 50.3%

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

3. Applied egg-rr 1.0%

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

4. Taylor expanded in b around -inf 2.3%

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

5. Final simplification 2.3%

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

Developer target: 71.4% 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 2023275 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))