Quadratic roots, wide range

Percentage Accurate: 17.7% → 99.7%

Time: 11.3s

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: 17.7% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a} \cdot 0.5}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (/ (fma (* c a) -4.0 0.0) a) 0.5)
  (+ (sqrt (fma (* c a) -4.0 (* b b))) b)))

C

double code(double a, double b, double c) {
	return ((fma((c * a), -4.0, 0.0) / a) * 0.5) / (sqrt(fma((c * a), -4.0, (b * b))) + b);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(fma(Float64(c * a), -4.0, 0.0) / a) * 0.5) / Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(c * a), $MachinePrecision] * -4.0 + 0.0), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 17.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 17.5

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

4. Applied rewrites 17.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

6. Applied rewrites 17.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{-0.25}{c} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.5 (* (/ -0.25 c) (+ (sqrt (fma (* c a) -4.0 (* b b))) b))))

C

double code(double a, double b, double c) {
	return 0.5 / ((-0.25 / c) * (sqrt(fma((c * a), -4.0, (b * b))) + b));
}

Julia

function code(a, b, c)
	return Float64(0.5 / Float64(Float64(-0.25 / c) * Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b)))
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(N[(-0.25 / c), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 17.7

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 17.7

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

4. Applied rewrites 17.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

6. Applied rewrites 18.1%

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

7. Taylor expanded in a around 0

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

   1. lower-/.f64 99.4

      \[\leadsto \frac{0.5}{\color{blue}{\frac{-0.25}{c}} \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)} \]

9. Applied rewrites 99.4%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{-0.25}{c}} \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)} \]

10. Final simplification 99.4%

    \[\leadsto \frac{0.5}{\frac{-0.25}{c} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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)))