Quadratic roots, wide range

Percentage Accurate: 18.4% → 99.4%

Time: 15.3s

Alternatives: 3

Speedup: 29.0×

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: 18.4% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.9%

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

   1. *-commutative 15.9%

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

3. Simplified 15.9%

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

   1. add-cube-cbrt 16.0%

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

   2. pow3 16.0%

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

6. Applied egg-rr 16.0%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}}{a \cdot -2}}\right)}^{3}} \]
7. Step-by-step derivation

   1. rem-cube-cbrt 15.9%

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

   2. div-inv 15.9%

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

   3. flip-- 15.8%

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

   4. frac-times 15.8%

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

   5. unpow2 15.8%

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

   6. add-sqr-sqrt 16.2%

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

8. Applied egg-rr 16.2%

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

   1. *-rgt-identity 16.2%

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

10. Simplified 16.2%

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

11. Taylor expanded in b around 0 99.5%

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

    1. *-commutative 99.5%

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

    2. *-commutative 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c + (a * pow((c / -b), 2.0))) / -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 + (a * ((c / -b) ** 2.0d0))) / -b
end function

Java

public static double code(double a, double b, double c) {
	return (c + (a * Math.pow((c / -b), 2.0))) / -b;
}

Python

def code(a, b, c):
	return (c + (a * math.pow((c / -b), 2.0))) / -b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (c + (a * ((c / -b) ^ 2.0))) / -b;
end

Wolfram

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

Derivation

1. Initial program 15.9%

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

   1. *-commutative 15.9%

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

3. Simplified 15.9%

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

5. Taylor expanded in c around 0 95.9%

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

   1. associate-*r/ 95.9%

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

   2. neg-mul-1 95.9%

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

   3. distribute-rgt-neg-in 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in c around 0 95.9%

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

   1. sub-neg 95.9%

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

   2. associate-*r/ 95.9%

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

   3. mul-1-neg 95.9%

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

   4. distribute-rgt-neg-out 95.9%

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

   5. associate-*r/ 95.9%

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

   6. +-commutative 95.9%

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

   7. distribute-frac-neg 95.9%

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

   8. distribute-rgt-neg-in 95.9%

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

   9. associate-/l* 95.9%

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

   10. unsub-neg 95.9%

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

   11. distribute-neg-frac 95.9%

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

   12. metadata-eval 95.9%

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

   13. associate-/l* 95.9%

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

10. Simplified 95.9%

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

11. Taylor expanded in b around inf 96.2%

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

    1. distribute-lft-out 96.2%

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

    2. associate-*r/ 96.2%

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

    3. mul-1-neg 96.2%

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

13. Simplified 96.2%

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

14. Final simplification 96.2%

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

Alternative 3: 89.9% accurate, 29.0× 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 15.9%

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

   1. *-commutative 15.9%

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

3. Simplified 15.9%

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

5. Taylor expanded in b around inf 91.8%

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

   1. associate-*r/ 91.8%

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

   2. neg-mul-1 91.8%

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

7. Simplified 91.8%

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

8. Final simplification 91.8%

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

Reproduce

herbie shell --seed 2024080 
(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)))