quadm (p42, negative)

Percentage Accurate: 52.5% → 90.9%

Time: 11.3s

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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024218 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

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