Quadratic roots, wide range

Percentage Accurate: 17.5% → 99.3%

Time: 10.8s

Alternatives: 5

Speedup: 3.6×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.5% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -4\right)\\ t_1 := \mathsf{fma}\left(b, b, t\_0\right)\\ t_2 := b \cdot b - t\_1\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{t\_2 \cdot \frac{t\_0}{b + \sqrt{t\_1}}}{t\_2}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c -4.0))) (t_1 (fma b b t_0)) (t_2 (- (* b b) t_1)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -5e-21)
     (/ (/ (* t_2 (/ t_0 (+ b (sqrt t_1)))) t_2) (* a 2.0))
     (/ (fma a (/ (* c c) (* b b)) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * -4.0);
	double t_1 = fma(b, b, t_0);
	double t_2 = (b * b) - t_1;
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -5e-21) {
		tmp = ((t_2 * (t_0 / (b + sqrt(t_1)))) / t_2) / (a * 2.0);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -4.0))
	t_1 = fma(b, b, t_0)
	t_2 = Float64(Float64(b * b) - t_1)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -5e-21)
		tmp = Float64(Float64(Float64(t_2 * Float64(t_0 / Float64(b + sqrt(t_1)))) / t_2) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * b + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -5e-21], N[(N[(N[(t$95$2 * N[(t$95$0 / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -4.99999999999999973e-21

   1. Initial program 60.7%

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

   4. Applied egg-rr 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

   6. Applied egg-rr 60.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 N/A

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

   8. Applied egg-rr 99.1%

      \[\leadsto \frac{\frac{\left(b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right) \cdot \left(\color{blue}{\frac{0 - a \cdot \left(c \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}} \cdot -1\right)}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{2 \cdot a} \]

   if -4.99999999999999973e-21 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 3.8%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\mathsf{fma}\left(a, \frac{{c}^{2}}{{b}^{2}}, c\right)}}{b}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(a, \color{blue}{\frac{{c}^{2}}{{b}^{2}}}, c\right)}{b}\right) \]

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right) \cdot \frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

5. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot -2\right) \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}} \]
6. Add Preprocessing

Alternative 3: 95.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (fma a (/ (* c c) (* b b)) c) (- b)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\mathsf{fma}\left(a, \frac{{c}^{2}}{{b}^{2}}, c\right)}}{b}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(a, \color{blue}{\frac{{c}^{2}}{{b}^{2}}}, c\right)}{b}\right) \]

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 95.7

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

5. Simplified 95.7%

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

6. Final simplification 95.7%

   \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b} \]
7. Add Preprocessing

Alternative 4: 90.7% accurate, 3.6× 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 / Float64(-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 18.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. lower-neg.f64 N/A

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

   3. lower-/.f64 90.4

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

5. Simplified 90.4%

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

6. Final simplification 90.4%

   \[\leadsto \frac{c}{-b} \]
7. Add Preprocessing

Alternative 5: 3.3% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 18.0%

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

4. Applied egg-rr 18.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-lft-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 19.3

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

   11. lift-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 19.3

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 19.3

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

6. Applied egg-rr 19.3%

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

7. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{b}{a}} \]
8. Step-by-step derivation

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

   3. mul0-rgt 3.3

      \[\leadsto \color{blue}{0} \]

9. Simplified 3.3%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))