Quadratic roots, wide range

Percentage Accurate: 18.1% → 99.4%

Time: 9.8s

Alternatives: 3

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: 18.1% 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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 17.6%

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

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}{\left(2 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)}} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

Alternative 2: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

3. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-sub N/A

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

   8. unsub-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-out N/A

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

   11. associate-/l* N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 95.2%

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

7. Applied rewrites 95.2%

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

8. Final simplification 95.2%

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

Alternative 3: 90.2% 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(Float64(-c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 90.1

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

5. Applied rewrites 90.1%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024309 
(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)))