The quadratic formula (r2)

Percentage Accurate: 52.5% → 90.9%

Time: 14.1s

Alternatives: 12

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+104}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-275}:\\ \;\;\;\;\frac{c}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot -0.5}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+104)
   (- (/ c b))
   (if (<= b -7e-275)
     (/ c (* (- b (sqrt (fma -4.0 (* c a) (* b b)))) -0.5))
     (if (<= b 1.8e+127)
       (/ (+ b (sqrt (fma b b (* c (* -4.0 a))))) (* a -2.0))
       (- (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+104) {
		tmp = -(c / b);
	} else if (b <= -7e-275) {
		tmp = c / ((b - sqrt(fma(-4.0, (c * a), (b * b)))) * -0.5);
	} else if (b <= 1.8e+127) {
		tmp = (b + sqrt(fma(b, b, (c * (-4.0 * a))))) / (a * -2.0);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+104)
		tmp = Float64(-Float64(c / b));
	elseif (b <= -7e-275)
		tmp = Float64(c / Float64(Float64(b - sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) * -0.5));
	elseif (b <= 1.8e+127)
		tmp = Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(-4.0 * a))))) / Float64(a * -2.0));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+104], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, -7e-275], N[(c / N[(N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e+127], N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if b < -1e104

   1. Initial program 4.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.0

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

   5. Simplified 89.0%

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

   if -1e104 < b < -6.99999999999999938e-275

   1. Initial program 53.5%

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

   4. Applied egg-rr 53.6%

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

   6. Applied egg-rr 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

   8. Applied egg-rr 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   10. Applied egg-rr 89.3%

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

   if -6.99999999999999938e-275 < b < 1.79999999999999989e127

   1. Initial program 87.1%

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

   4. Applied egg-rr 87.2%

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

   if 1.79999999999999989e127 < b

   1. Initial program 51.6%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      3. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+104}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-275}:\\ \;\;\;\;\frac{c}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot -0.5}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+104}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-234}:\\ \;\;\;\;\frac{c}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot -0.5}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+127}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+104)
   (- (/ c b))
   (if (<= b -7e-234)
     (/ c (* (- b (sqrt (fma -4.0 (* c a) (* b b)))) -0.5))
     (if (<= b 1.7e+127)
       (* (+ b (sqrt (fma b b (* c (* -4.0 a))))) (/ -0.5 a))
       (- (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+104) {
		tmp = -(c / b);
	} else if (b <= -7e-234) {
		tmp = c / ((b - sqrt(fma(-4.0, (c * a), (b * b)))) * -0.5);
	} else if (b <= 1.7e+127) {
		tmp = (b + sqrt(fma(b, b, (c * (-4.0 * a))))) * (-0.5 / a);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+104)
		tmp = Float64(-Float64(c / b));
	elseif (b <= -7e-234)
		tmp = Float64(c / Float64(Float64(b - sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) * -0.5));
	elseif (b <= 1.7e+127)
		tmp = Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(-4.0 * a))))) * Float64(-0.5 / a));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+104], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, -7e-234], N[(c / N[(N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e+127], N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if b < -1e104

   1. Initial program 7.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 96.2

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

   5. Simplified 96.2%

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

   if -1e104 < b < -7.0000000000000003e-234

   1. Initial program 46.6%

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

   4. Applied egg-rr 46.5%

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

   6. Applied egg-rr 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

   8. Applied egg-rr 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   10. Applied egg-rr 88.6%

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

   if -7.0000000000000003e-234 < b < 1.69999999999999989e127

   1. Initial program 86.1%

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

   4. Applied egg-rr 85.8%

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

   if 1.69999999999999989e127 < b

   1. Initial program 48.5%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      3. lower-/.f64 96.9

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

   5. Simplified 96.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+104}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-234}:\\ \;\;\;\;\frac{c}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot -0.5}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+127}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024218 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

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