The quadratic formula (r1)

Percentage Accurate: 52.2% → 85.7%

Time: 17.3s

Alternatives: 9

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e+71)
   (/ (- (* c (* (/ (* a -4.0) b) -0.25)) b) a)
   (if (<= b 2.9e-39)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e+71) {
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	} else if (b <= 2.9e-39) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.1d+71)) then
        tmp = ((c * (((a * (-4.0d0)) / b) * (-0.25d0))) - b) / a
    else if (b <= 2.9d-39) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e+71) {
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	} else if (b <= 2.9e-39) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.1e+71:
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a
	elif b <= 2.9e-39:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e+71)
		tmp = Float64(Float64(Float64(c * Float64(Float64(Float64(a * -4.0) / b) * -0.25)) - b) / a);
	elseif (b <= 2.9e-39)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.1e+71)
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	elseif (b <= 2.9e-39)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e+71], N[(N[(N[(c * N[(N[(N[(a * -4.0), $MachinePrecision] / b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.9e-39], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.09999999999999997e71

   1. Initial program 50.4%

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

      1. *-commutative 50.4%

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

   3. Simplified 50.4%

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

      1. prod-diff 50.1%

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

      2. *-commutative 50.1%

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

      3. fma-def 50.1%

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

      4. associate-+l+ 50.1%

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

      5. pow2 50.1%

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

      6. distribute-lft-neg-in 50.1%

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

      7. *-commutative 50.1%

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

      8. distribute-rgt-neg-in 50.1%

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

      9. metadata-eval 50.1%

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

      10. associate-*r* 50.1%

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

      11. *-commutative 50.1%

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

      12. *-commutative 50.1%

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

      13. fma-udef 50.1%

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

   5. Applied egg-rr 50.1%

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

      1. fma-def 50.1%

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

      2. fma-def 50.2%

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

      3. associate-*l* 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in b around -inf 89.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + -0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b}} \]
   9. Step-by-step derivation

      1. +-commutative 89.2%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 89.2%

         \[\leadsto -0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      3. unsub-neg 89.2%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} - \frac{b}{a}} \]

      4. *-commutative 89.2%

         \[\leadsto \color{blue}{\frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} \cdot -0.25} - \frac{b}{a} \]

      5. associate-*r* 89.2%

         \[\leadsto \frac{\color{blue}{\left(-8 \cdot a\right) \cdot c} + 4 \cdot \left(a \cdot c\right)}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      6. associate-*r* 89.2%

         \[\leadsto \frac{\left(-8 \cdot a\right) \cdot c + \color{blue}{\left(4 \cdot a\right) \cdot c}}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      7. distribute-rgt-in 89.2%

         \[\leadsto \frac{\color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right)}}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      8. times-frac 79.5%

         \[\leadsto \color{blue}{\left(\frac{c}{a} \cdot \frac{-8 \cdot a + 4 \cdot a}{b}\right)} \cdot -0.25 - \frac{b}{a} \]

      9. distribute-rgt-out 79.5%

         \[\leadsto \left(\frac{c}{a} \cdot \frac{\color{blue}{a \cdot \left(-8 + 4\right)}}{b}\right) \cdot -0.25 - \frac{b}{a} \]

      10. metadata-eval 79.5%

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

   10. Simplified 79.5%

       \[\leadsto \color{blue}{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}} \]
   11. Step-by-step derivation

       1. add-cube-cbrt 78.1%

          \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}} \cdot \sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}\right) \cdot \sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}} \]

       2. pow3 78.1%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}\right)}^{3}} \]

       3. associate-*l* 78.1%

          \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{c}{a} \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right)} - \frac{b}{a}}\right)}^{3} \]

   12. Applied egg-rr 78.1%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{c}{a} \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - \frac{b}{a}}\right)}^{3}} \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 79.5%

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

       2. sub-neg 79.5%

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

       3. *-commutative 79.5%

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

       4. *-un-lft-identity 79.5%

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

       5. times-frac 79.5%

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

       6. metadata-eval 79.5%

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

   14. Applied egg-rr 79.5%

       \[\leadsto \color{blue}{\frac{c}{a} \cdot \left(\left(-4 \cdot \frac{a}{b}\right) \cdot -0.25\right) + \left(-\frac{b}{a}\right)} \]
   15. Step-by-step derivation

       1. sub-neg 79.5%

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

       2. associate-*l/ 97.7%

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

       3. div-sub 97.8%

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

       4. associate-*r/ 97.8%

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

       5. *-commutative 97.8%

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

   16. Simplified 97.8%

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

   if -1.09999999999999997e71 < b < 2.89999999999999988e-39

   1. Initial program 80.6%

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

   if 2.89999999999999988e-39 < b

   1. Initial program 13.5%

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

      1. *-commutative 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in b around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. distribute-neg-frac 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-48)
   (/ (- (* c (* (/ (* a -4.0) b) -0.25)) b) a)
   (if (<= b 4.2e-39)
     (/ 1.0 (/ a (/ (- (sqrt (* -4.0 (* c a))) b) 2.0)))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-48) {
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	} else if (b <= 4.2e-39) {
		tmp = 1.0 / (a / ((sqrt((-4.0 * (c * a))) - b) / 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 <= (-5d-48)) then
        tmp = ((c * (((a * (-4.0d0)) / b) * (-0.25d0))) - b) / a
    else if (b <= 4.2d-39) then
        tmp = 1.0d0 / (a / ((sqrt(((-4.0d0) * (c * a))) - b) / 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 <= -5e-48) {
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	} else if (b <= 4.2e-39) {
		tmp = 1.0 / (a / ((Math.sqrt((-4.0 * (c * a))) - b) / 2.0));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-48:
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a
	elif b <= 4.2e-39:
		tmp = 1.0 / (a / ((math.sqrt((-4.0 * (c * a))) - b) / 2.0))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-48)
		tmp = Float64(Float64(Float64(c * Float64(Float64(Float64(a * -4.0) / b) * -0.25)) - b) / a);
	elseif (b <= 4.2e-39)
		tmp = Float64(1.0 / Float64(a / Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - b) / 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 <= -5e-48)
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	elseif (b <= 4.2e-39)
		tmp = 1.0 / (a / ((sqrt((-4.0 * (c * a))) - b) / 2.0));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-48], N[(N[(N[(c * N[(N[(N[(a * -4.0), $MachinePrecision] / b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.2e-39], N[(1.0 / N[(a / N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999999e-48

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

   3. Simplified 65.0%

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

      1. prod-diff 64.8%

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

      2. *-commutative 64.8%

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

      3. fma-def 64.8%

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

      4. associate-+l+ 64.8%

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

      5. pow2 64.8%

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

      6. distribute-lft-neg-in 64.8%

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

      7. *-commutative 64.8%

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

      8. distribute-rgt-neg-in 64.8%

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

      9. metadata-eval 64.8%

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

      10. associate-*r* 64.8%

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

      11. *-commutative 64.8%

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

      12. *-commutative 64.8%

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

      13. fma-udef 64.8%

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

   5. Applied egg-rr 64.8%

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

      1. fma-def 64.8%

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

      2. fma-def 64.8%

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

      3. associate-*l* 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in b around -inf 82.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + -0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b}} \]
   9. Step-by-step derivation

      1. +-commutative 82.9%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 82.9%

         \[\leadsto -0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      3. unsub-neg 82.9%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} - \frac{b}{a}} \]

      4. *-commutative 82.9%

         \[\leadsto \color{blue}{\frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} \cdot -0.25} - \frac{b}{a} \]

      5. associate-*r* 82.9%

         \[\leadsto \frac{\color{blue}{\left(-8 \cdot a\right) \cdot c} + 4 \cdot \left(a \cdot c\right)}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      6. associate-*r* 82.9%

         \[\leadsto \frac{\left(-8 \cdot a\right) \cdot c + \color{blue}{\left(4 \cdot a\right) \cdot c}}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      7. distribute-rgt-in 82.9%

         \[\leadsto \frac{\color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right)}}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      8. times-frac 76.3%

         \[\leadsto \color{blue}{\left(\frac{c}{a} \cdot \frac{-8 \cdot a + 4 \cdot a}{b}\right)} \cdot -0.25 - \frac{b}{a} \]

      9. distribute-rgt-out 76.3%

         \[\leadsto \left(\frac{c}{a} \cdot \frac{\color{blue}{a \cdot \left(-8 + 4\right)}}{b}\right) \cdot -0.25 - \frac{b}{a} \]

      10. metadata-eval 76.3%

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

   10. Simplified 76.3%

       \[\leadsto \color{blue}{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}} \]
   11. Step-by-step derivation

       1. add-cube-cbrt 74.9%

          \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}} \cdot \sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}\right) \cdot \sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}} \]

       2. pow3 75.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}\right)}^{3}} \]

       3. associate-*l* 75.0%

          \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{c}{a} \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right)} - \frac{b}{a}}\right)}^{3} \]

   12. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{c}{a} \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - \frac{b}{a}}\right)}^{3}} \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 76.3%

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

       2. sub-neg 76.3%

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

       3. *-commutative 76.3%

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

       4. *-un-lft-identity 76.3%

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

       5. times-frac 76.3%

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

       6. metadata-eval 76.3%

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

   14. Applied egg-rr 76.3%

       \[\leadsto \color{blue}{\frac{c}{a} \cdot \left(\left(-4 \cdot \frac{a}{b}\right) \cdot -0.25\right) + \left(-\frac{b}{a}\right)} \]
   15. Step-by-step derivation

       1. sub-neg 76.3%

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

       2. associate-*l/ 88.7%

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

       3. div-sub 88.8%

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

       4. associate-*r/ 88.8%

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

       5. *-commutative 88.8%

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

   16. Simplified 88.8%

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

   if -4.9999999999999999e-48 < b < 4.19999999999999987e-39

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

   3. Simplified 73.5%

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

      1. add-sqr-sqrt 73.1%

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

      2. pow2 73.1%

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

      3. pow1/2 73.1%

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

      4. sqrt-pow1 73.2%

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

      5. sub-neg 73.2%

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

      6. +-commutative 73.2%

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

      7. distribute-lft-neg-in 73.2%

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

      8. *-commutative 73.2%

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

      9. distribute-rgt-neg-in 73.2%

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

      10. metadata-eval 73.2%

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

      11. associate-*r* 71.8%

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

      12. *-commutative 71.8%

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

      13. fma-udef 71.8%

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

      14. pow2 71.8%

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

      15. metadata-eval 71.8%

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

   5. Applied egg-rr 71.8%

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

   6. Taylor expanded in c around -inf 48.1%

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

      1. clear-num 48.1%

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

      2. inv-pow 48.1%

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

   8. Applied egg-rr 71.8%

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

      1. unpow-1 71.8%

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

      2. associate-/l* 71.8%

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

      3. unpow2 71.8%

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

      4. pow-sqr 72.2%

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

      5. metadata-eval 72.2%

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

      6. unpow1/2 72.2%

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

      7. associate-*l* 70.8%

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

      8. associate-/r/ 70.8%

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

      9. metadata-eval 70.8%

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

      10. associate-*r* 70.8%

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

      11. metadata-eval 70.8%

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

      12. *-commutative 70.8%

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

      13. associate-*r* 72.1%

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

      14. *-commutative 72.1%

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

   10. Simplified 72.1%

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

   if 4.19999999999999987e-39 < b

   1. Initial program 13.5%

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

      1. *-commutative 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in b around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. distribute-neg-frac 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-48)
   (/ (- (* c (* (/ (* a -4.0) b) -0.25)) b) a)
   (if (<= b 4.2e-39)
     (/ (- (sqrt (* -4.0 (* c a))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-48) {
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	} else if (b <= 4.2e-39) {
		tmp = (sqrt((-4.0 * (c * a))) - 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 <= (-1.6d-48)) then
        tmp = ((c * (((a * (-4.0d0)) / b) * (-0.25d0))) - b) / a
    else if (b <= 4.2d-39) then
        tmp = (sqrt(((-4.0d0) * (c * a))) - 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 <= -1.6e-48) {
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	} else if (b <= 4.2e-39) {
		tmp = (Math.sqrt((-4.0 * (c * a))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-48:
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a
	elif b <= 4.2e-39:
		tmp = (math.sqrt((-4.0 * (c * a))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-48)
		tmp = Float64(Float64(Float64(c * Float64(Float64(Float64(a * -4.0) / b) * -0.25)) - b) / a);
	elseif (b <= 4.2e-39)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - 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 <= -1.6e-48)
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	elseif (b <= 4.2e-39)
		tmp = (sqrt((-4.0 * (c * a))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-48], N[(N[(N[(c * N[(N[(N[(a * -4.0), $MachinePrecision] / b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.2e-39], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $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 < -1.5999999999999999e-48

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

   3. Simplified 65.0%

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

      1. prod-diff 64.8%

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

      2. *-commutative 64.8%

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

      3. fma-def 64.8%

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

      4. associate-+l+ 64.8%

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

      5. pow2 64.8%

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

      6. distribute-lft-neg-in 64.8%

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

      7. *-commutative 64.8%

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

      8. distribute-rgt-neg-in 64.8%

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

      9. metadata-eval 64.8%

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

      10. associate-*r* 64.8%

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

      11. *-commutative 64.8%

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

      12. *-commutative 64.8%

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

      13. fma-udef 64.8%

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

   5. Applied egg-rr 64.8%

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

      1. fma-def 64.8%

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

      2. fma-def 64.8%

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

      3. associate-*l* 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in b around -inf 82.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + -0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b}} \]
   9. Step-by-step derivation

      1. +-commutative 82.9%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 82.9%

         \[\leadsto -0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      3. unsub-neg 82.9%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} - \frac{b}{a}} \]

      4. *-commutative 82.9%

         \[\leadsto \color{blue}{\frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} \cdot -0.25} - \frac{b}{a} \]

      5. associate-*r* 82.9%

         \[\leadsto \frac{\color{blue}{\left(-8 \cdot a\right) \cdot c} + 4 \cdot \left(a \cdot c\right)}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      6. associate-*r* 82.9%

         \[\leadsto \frac{\left(-8 \cdot a\right) \cdot c + \color{blue}{\left(4 \cdot a\right) \cdot c}}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      7. distribute-rgt-in 82.9%

         \[\leadsto \frac{\color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right)}}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      8. times-frac 76.3%

         \[\leadsto \color{blue}{\left(\frac{c}{a} \cdot \frac{-8 \cdot a + 4 \cdot a}{b}\right)} \cdot -0.25 - \frac{b}{a} \]

      9. distribute-rgt-out 76.3%

         \[\leadsto \left(\frac{c}{a} \cdot \frac{\color{blue}{a \cdot \left(-8 + 4\right)}}{b}\right) \cdot -0.25 - \frac{b}{a} \]

      10. metadata-eval 76.3%

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

   10. Simplified 76.3%

       \[\leadsto \color{blue}{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}} \]
   11. Step-by-step derivation

       1. add-cube-cbrt 74.9%

          \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}} \cdot \sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}\right) \cdot \sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}} \]

       2. pow3 75.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}\right)}^{3}} \]

       3. associate-*l* 75.0%

          \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{c}{a} \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right)} - \frac{b}{a}}\right)}^{3} \]

   12. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{c}{a} \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - \frac{b}{a}}\right)}^{3}} \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 76.3%

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

       2. sub-neg 76.3%

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

       3. *-commutative 76.3%

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

       4. *-un-lft-identity 76.3%

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

       5. times-frac 76.3%

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

       6. metadata-eval 76.3%

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

   14. Applied egg-rr 76.3%

       \[\leadsto \color{blue}{\frac{c}{a} \cdot \left(\left(-4 \cdot \frac{a}{b}\right) \cdot -0.25\right) + \left(-\frac{b}{a}\right)} \]
   15. Step-by-step derivation

       1. sub-neg 76.3%

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

       2. associate-*l/ 88.7%

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

       3. div-sub 88.8%

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

       4. associate-*r/ 88.8%

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

       5. *-commutative 88.8%

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

   16. Simplified 88.8%

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

   if -1.5999999999999999e-48 < b < 4.19999999999999987e-39

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

   3. Simplified 73.5%

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

      1. add-sqr-sqrt 73.1%

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

      2. pow2 73.1%

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

      3. pow1/2 73.1%

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

      4. sqrt-pow1 73.2%

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

      5. sub-neg 73.2%

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

      6. +-commutative 73.2%

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

      7. distribute-lft-neg-in 73.2%

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

      8. *-commutative 73.2%

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

      9. distribute-rgt-neg-in 73.2%

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

      10. metadata-eval 73.2%

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

      11. associate-*r* 71.8%

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

      12. *-commutative 71.8%

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

      13. fma-udef 71.8%

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

      14. pow2 71.8%

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

      15. metadata-eval 71.8%

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

   5. Applied egg-rr 71.8%

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

   6. Taylor expanded in c around -inf 48.1%

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

      1. expm1-log1p-u 48.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2}\right)\right)}}{a} \]

      2. expm1-udef 26.5%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2}\right)} - 1}}{a} \]

   8. Applied egg-rr 38.1%

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25}\right)}^{2}\right)} - 1\right)} - b}{a \cdot 2} \]
   9. Step-by-step derivation

      1. expm1-def 69.0%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25}\right)}^{2}\right)\right)}}{a} \]

      2. expm1-log1p 71.6%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{{\left({\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25}\right)}^{2}}}{a} \]

      3. unpow2 71.6%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{{\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25} \cdot {\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25}}}{a} \]

      4. pow-sqr 71.9%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{{\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{\left(2 \cdot 0.25\right)}}}{a} \]

      5. metadata-eval 71.9%

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

      6. unpow1/2 71.9%

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

      7. associate-*l* 70.5%

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

      8. associate-/r/ 70.5%

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

      9. metadata-eval 70.5%

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

      10. associate-*r* 70.5%

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

      11. metadata-eval 70.5%

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

      12. *-commutative 70.5%

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

      13. associate-*r* 71.8%

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

      14. *-commutative 71.8%

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

   10. Simplified 72.0%

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

   if 4.19999999999999987e-39 < b

   1. Initial program 13.5%

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

      1. *-commutative 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in b around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. distribute-neg-frac 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{-4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-47)
   (/ (- (* c (* (/ (* a -4.0) b) -0.25)) b) a)
   (if (<= b 2.9e-39) (* 0.5 (/ (sqrt (* -4.0 (* c a))) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-47) {
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	} else if (b <= 2.9e-39) {
		tmp = 0.5 * (sqrt((-4.0 * (c * a))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.1d-47)) then
        tmp = ((c * (((a * (-4.0d0)) / b) * (-0.25d0))) - b) / a
    else if (b <= 2.9d-39) then
        tmp = 0.5d0 * (sqrt(((-4.0d0) * (c * a))) / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-47) {
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	} else if (b <= 2.9e-39) {
		tmp = 0.5 * (Math.sqrt((-4.0 * (c * a))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.1e-47:
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a
	elif b <= 2.9e-39:
		tmp = 0.5 * (math.sqrt((-4.0 * (c * a))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-47)
		tmp = Float64(Float64(Float64(c * Float64(Float64(Float64(a * -4.0) / b) * -0.25)) - b) / a);
	elseif (b <= 2.9e-39)
		tmp = Float64(0.5 * Float64(sqrt(Float64(-4.0 * Float64(c * a))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.1e-47)
		tmp = ((c * (((a * -4.0) / b) * -0.25)) - b) / a;
	elseif (b <= 2.9e-39)
		tmp = 0.5 * (sqrt((-4.0 * (c * a))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-47], N[(N[(N[(c * N[(N[(N[(a * -4.0), $MachinePrecision] / b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.9e-39], N[(0.5 * N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.10000000000000009e-47

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

   3. Simplified 65.0%

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

      1. prod-diff 64.8%

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

      2. *-commutative 64.8%

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

      3. fma-def 64.8%

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

      4. associate-+l+ 64.8%

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

      5. pow2 64.8%

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

      6. distribute-lft-neg-in 64.8%

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

      7. *-commutative 64.8%

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

      8. distribute-rgt-neg-in 64.8%

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

      9. metadata-eval 64.8%

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

      10. associate-*r* 64.8%

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

      11. *-commutative 64.8%

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

      12. *-commutative 64.8%

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

      13. fma-udef 64.8%

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

   5. Applied egg-rr 64.8%

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

      1. fma-def 64.8%

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

      2. fma-def 64.8%

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

      3. associate-*l* 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in b around -inf 82.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + -0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b}} \]
   9. Step-by-step derivation

      1. +-commutative 82.9%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 82.9%

         \[\leadsto -0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      3. unsub-neg 82.9%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} - \frac{b}{a}} \]

      4. *-commutative 82.9%

         \[\leadsto \color{blue}{\frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b} \cdot -0.25} - \frac{b}{a} \]

      5. associate-*r* 82.9%

         \[\leadsto \frac{\color{blue}{\left(-8 \cdot a\right) \cdot c} + 4 \cdot \left(a \cdot c\right)}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      6. associate-*r* 82.9%

         \[\leadsto \frac{\left(-8 \cdot a\right) \cdot c + \color{blue}{\left(4 \cdot a\right) \cdot c}}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      7. distribute-rgt-in 82.9%

         \[\leadsto \frac{\color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right)}}{a \cdot b} \cdot -0.25 - \frac{b}{a} \]

      8. times-frac 76.3%

         \[\leadsto \color{blue}{\left(\frac{c}{a} \cdot \frac{-8 \cdot a + 4 \cdot a}{b}\right)} \cdot -0.25 - \frac{b}{a} \]

      9. distribute-rgt-out 76.3%

         \[\leadsto \left(\frac{c}{a} \cdot \frac{\color{blue}{a \cdot \left(-8 + 4\right)}}{b}\right) \cdot -0.25 - \frac{b}{a} \]

      10. metadata-eval 76.3%

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

   10. Simplified 76.3%

       \[\leadsto \color{blue}{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}} \]
   11. Step-by-step derivation

       1. add-cube-cbrt 74.9%

          \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}} \cdot \sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}\right) \cdot \sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}} \]

       2. pow3 75.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{b}\right) \cdot -0.25 - \frac{b}{a}}\right)}^{3}} \]

       3. associate-*l* 75.0%

          \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{c}{a} \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right)} - \frac{b}{a}}\right)}^{3} \]

   12. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{c}{a} \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - \frac{b}{a}}\right)}^{3}} \]
   13. Step-by-step derivation

       1. rem-cube-cbrt 76.3%

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

       2. sub-neg 76.3%

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

       3. *-commutative 76.3%

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

       4. *-un-lft-identity 76.3%

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

       5. times-frac 76.3%

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

       6. metadata-eval 76.3%

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

   14. Applied egg-rr 76.3%

       \[\leadsto \color{blue}{\frac{c}{a} \cdot \left(\left(-4 \cdot \frac{a}{b}\right) \cdot -0.25\right) + \left(-\frac{b}{a}\right)} \]
   15. Step-by-step derivation

       1. sub-neg 76.3%

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

       2. associate-*l/ 88.7%

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

       3. div-sub 88.8%

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

       4. associate-*r/ 88.8%

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

       5. *-commutative 88.8%

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

   16. Simplified 88.8%

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

   if -1.10000000000000009e-47 < b < 2.89999999999999988e-39

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

   3. Simplified 73.5%

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

      1. add-sqr-sqrt 73.1%

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

      2. pow2 73.1%

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

      3. pow1/2 73.1%

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

      4. sqrt-pow1 73.2%

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

      5. sub-neg 73.2%

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

      6. +-commutative 73.2%

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

      7. distribute-lft-neg-in 73.2%

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

      8. *-commutative 73.2%

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

      9. distribute-rgt-neg-in 73.2%

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

      10. metadata-eval 73.2%

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

      11. associate-*r* 71.8%

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

      12. *-commutative 71.8%

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

      13. fma-udef 71.8%

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

      14. pow2 71.8%

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

      15. metadata-eval 71.8%

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

   5. Applied egg-rr 71.8%

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

   6. Taylor expanded in c around -inf 48.1%

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

   7. Taylor expanded in b around 0 48.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{\left(e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2}}{a}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 48.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2}\right)\right)}}{a} \]

      2. expm1-udef 26.5%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2}\right)} - 1}}{a} \]

   9. Applied egg-rr 37.4%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left({\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25}\right)}^{2}\right)} - 1}}{a} \]
   10. Step-by-step derivation

       1. expm1-def 69.0%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25}\right)}^{2}\right)\right)}}{a} \]

       2. expm1-log1p 71.6%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{{\left({\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25}\right)}^{2}}}{a} \]

       3. unpow2 71.6%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{{\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25} \cdot {\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{0.25}}}{a} \]

       4. pow-sqr 71.9%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{{\left(\left(a \cdot 4\right) \cdot \frac{1}{\frac{-1}{c}}\right)}^{\left(2 \cdot 0.25\right)}}}{a} \]

       5. metadata-eval 71.9%

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

       6. unpow1/2 71.9%

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

       7. associate-*l* 70.5%

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

       8. associate-/r/ 70.5%

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

       9. metadata-eval 70.5%

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

       10. associate-*r* 70.5%

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

       11. metadata-eval 70.5%

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

       12. *-commutative 70.5%

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

       13. associate-*r* 71.8%

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

       14. *-commutative 71.8%

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

   11. Simplified 71.8%

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

   if 2.89999999999999988e-39 < b

   1. Initial program 13.5%

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

      1. *-commutative 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in b around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. distribute-neg-frac 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a \cdot -4}{b} \cdot -0.25\right) - b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{-4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.2% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in b around -inf 69.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   5. 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%

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

   6. Simplified 69.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

   3. Simplified 26.9%

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

   4. Taylor expanded in b around inf 70.6%

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

      1. mul-1-neg 70.6%

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

      2. distribute-neg-frac 70.6%

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

   6. Simplified 70.6%

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

4. Final simplification 70.1%

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

Alternative 6: 43.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.9e+44) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.9e+44) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.9d+44) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.9e+44) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.9e+44:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.9e+44)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.9e+44)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.9e+44], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9000000000000001e44

   1. Initial program 64.9%

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

      1. *-commutative 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in b around -inf 48.9%

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

      1. associate-*r/ 48.9%

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

      2. mul-1-neg 48.9%

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

   6. Simplified 48.9%

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

   if 1.9000000000000001e44 < b

   1. Initial program 8.3%

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

      1. *-commutative 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in b around -inf 2.7%

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

   5. Taylor expanded in b around 0 28.7%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7: 68.1% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in b around -inf 69.4%

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

      1. associate-*r/ 69.4%

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

      2. mul-1-neg 69.4%

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

   6. Simplified 69.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

   3. Simplified 26.9%

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

   4. Taylor expanded in b around inf 70.6%

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

      1. mul-1-neg 70.6%

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

      2. distribute-neg-frac 70.6%

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

   6. Simplified 70.6%

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

4. Final simplification 70.0%

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

Alternative 8: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.5%

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

   1. *-commutative 48.5%

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

3. Simplified 48.5%

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

5. Applied egg-rr 29.5%

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

6. Taylor expanded in a around 0 2.5%

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

7. Final simplification 2.5%

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

Alternative 9: 10.8% 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 48.5%

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

   1. *-commutative 48.5%

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

3. Simplified 48.5%

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

4. Taylor expanded in b around -inf 33.4%

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

5. Taylor expanded in b around 0 10.5%

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

6. Final simplification 10.5%

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

Developer target: 70.3% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $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 2023322 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))