Quadratic roots, wide range

Percentage Accurate: 17.6% → 99.9%

Time: 13.4s

Alternatives: 4

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: 17.6% 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: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

4. Applied egg-rr 15.1%

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

   1. associate-*r* N/A

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

   2. difference-of-squares N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

6. Applied egg-rr 15.4%

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

   1. associate-/l* N/A

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

   2. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in c around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 99.9

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

11. Simplified 99.9%

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

Alternative 2: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

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

   5. /-lowering-/.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. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 95.4

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

5. Simplified 95.4%

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

6. Final simplification 95.4%

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

Alternative 3: 90.5% 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 15.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 92.1

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

5. Simplified 92.1%

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

Alternative 4: 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 15.0%

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

4. Applied egg-rr 15.1%

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. +-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 16.5

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

6. Applied egg-rr 16.5%

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

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 2024207 
(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)))