Quadratic roots, wide range

Percentage Accurate: 18.3% → 97.6%

Time: 14.3s

Alternatives: 5

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: 18.3% 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: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (fma
   -0.25
   (/ (* (pow (* c a) 4.0) 20.0) (* a (pow b 7.0)))
   (- (/ -2.0 (/ (pow b 5.0) (* a (* a (pow c 3.0))))) (/ c b)))
  (/ a (/ (pow b 3.0) (* c c)))))

C

double code(double a, double b, double c) {
	return fma(-0.25, ((pow((c * a), 4.0) * 20.0) / (a * pow(b, 7.0))), ((-2.0 / (pow(b, 5.0) / (a * (a * pow(c, 3.0))))) - (c / b))) - (a / (pow(b, 3.0) / (c * c)));
}

Julia

function code(a, b, c)
	return Float64(fma(-0.25, Float64(Float64((Float64(c * a) ^ 4.0) * 20.0) / Float64(a * (b ^ 7.0))), Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64(a * Float64(a * (c ^ 3.0))))) - Float64(c / b))) - Float64(a / Float64((b ^ 3.0) / Float64(c * c))))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.25 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

   1. neg-sub0 19.5%

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

   2. associate-+l- 19.5%

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

   3. sub0-neg 19.5%

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

   4. neg-mul-1 19.5%

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

   5. associate-*l/ 19.5%

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

   6. *-commutative 19.5%

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

   7. associate-/r* 19.5%

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

   8. /-rgt-identity 19.5%

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

   9. metadata-eval 19.5%

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

3. Simplified 19.5%

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

4. Taylor expanded in b around inf 97.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

6. Simplified 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(16, {c}^{4} \cdot {a}^{4}, 4 \cdot \left({c}^{4} \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}} \]

7. Taylor expanded in b around 0 97.4%

   \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]
8. Step-by-step derivation

   1. distribute-rgt-out 97.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   2. *-commutative 97.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   3. metadata-eval 97.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({a}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {c}^{4}\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   4. pow-sqr 97.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{\left({a}^{2} \cdot {a}^{2}\right)} \cdot {c}^{4}\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   5. metadata-eval 97.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left({a}^{2} \cdot {a}^{2}\right) \cdot {c}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   6. pow-sqr 97.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left({a}^{2} \cdot {a}^{2}\right) \cdot \color{blue}{\left({c}^{2} \cdot {c}^{2}\right)}\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   7. unswap-sqr 97.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   8. *-commutative 97.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{\left({c}^{2} \cdot {a}^{2}\right)} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   9. unpow2 97.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   10. unpow2 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left(\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   11. unswap-sqr 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   12. unpow2 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{{\left(c \cdot a\right)}^{2}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   13. *-commutative 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \color{blue}{\left({c}^{2} \cdot {a}^{2}\right)}\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   14. unpow2 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}\right)\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   15. unpow2 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   16. unswap-sqr 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   17. unpow2 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \color{blue}{{\left(c \cdot a\right)}^{2}}\right) \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   18. pow-sqr 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{{\left(c \cdot a\right)}^{\left(2 \cdot 2\right)}} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   19. metadata-eval 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{\color{blue}{4}} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   20. metadata-eval 97.4%

       \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4} \cdot \color{blue}{20}}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

9. Simplified 97.4%

   \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

10. Final simplification 97.4%

    \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

Alternative 2: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (- (/ -2.0 (/ (pow b 5.0) (* a (* a (pow c 3.0))))) (/ c b))
  (/ a (/ (pow b 3.0) (* c c)))))

C

double code(double a, double b, double c) {
	return ((-2.0 / (pow(b, 5.0) / (a * (a * pow(c, 3.0))))) - (c / b)) - (a / (pow(b, 3.0) / (c * c)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-2.0d0) / ((b ** 5.0d0) / (a * (a * (c ** 3.0d0))))) - (c / b)) - (a / ((b ** 3.0d0) / (c * c)))
end function

Java

public static double code(double a, double b, double c) {
	return ((-2.0 / (Math.pow(b, 5.0) / (a * (a * Math.pow(c, 3.0))))) - (c / b)) - (a / (Math.pow(b, 3.0) / (c * c)));
}

Python

def code(a, b, c):
	return ((-2.0 / (math.pow(b, 5.0) / (a * (a * math.pow(c, 3.0))))) - (c / b)) - (a / (math.pow(b, 3.0) / (c * c)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64(a * Float64(a * (c ^ 3.0))))) - Float64(c / b)) - Float64(a / Float64((b ^ 3.0) / Float64(c * c))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-2.0 / ((b ^ 5.0) / (a * (a * (c ^ 3.0))))) - (c / b)) - (a / ((b ^ 3.0) / (c * c)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

   1. neg-sub0 19.5%

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

   2. associate-+l- 19.5%

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

   3. sub0-neg 19.5%

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

   4. neg-mul-1 19.5%

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

   5. associate-*l/ 19.5%

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

   6. *-commutative 19.5%

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

   7. associate-/r* 19.5%

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

   8. /-rgt-identity 19.5%

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

   9. metadata-eval 19.5%

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

3. Simplified 19.5%

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

4. Taylor expanded in b around inf 96.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]
5. Step-by-step derivation

   1. +-commutative 96.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   2. mul-1-neg 96.4%

      \[\leadsto \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

   3. unsub-neg 96.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   4. +-commutative 96.4%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -1 \cdot \frac{c}{b}\right)} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   5. mul-1-neg 96.4%

      \[\leadsto \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(-\frac{c}{b}\right)}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   6. unsub-neg 96.4%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} - \frac{c}{b}\right)} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   7. associate-*r/ 96.4%

      \[\leadsto \left(\color{blue}{\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}}} - \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   8. associate-/l* 96.4%

      \[\leadsto \left(\color{blue}{\frac{-2}{\frac{{b}^{5}}{{c}^{3} \cdot {a}^{2}}}} - \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   9. *-commutative 96.4%

      \[\leadsto \left(\frac{-2}{\frac{{b}^{5}}{\color{blue}{{a}^{2} \cdot {c}^{3}}}} - \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   10. unpow2 96.4%

       \[\leadsto \left(\frac{-2}{\frac{{b}^{5}}{\color{blue}{\left(a \cdot a\right)} \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   11. associate-*l* 96.4%

       \[\leadsto \left(\frac{-2}{\frac{{b}^{5}}{\color{blue}{a \cdot \left(a \cdot {c}^{3}\right)}}} - \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   12. *-commutative 96.4%

       \[\leadsto \left(\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]

6. Simplified 96.4%

   \[\leadsto \color{blue}{\left(\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}} \]

7. Final simplification 96.4%

   \[\leadsto \left(\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

Alternative 3: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-a}{\frac{{b}^{3}}{c \cdot c}} - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (- (/ (- a) (/ (pow b 3.0) (* c c))) (/ c b)))

C

double code(double a, double b, double c) {
	return (-a / (pow(b, 3.0) / (c * c))) - (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 = (-a / ((b ** 3.0d0) / (c * c))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (-a / (Math.pow(b, 3.0) / (c * c))) - (c / b);
}

Python

def code(a, b, c):
	return (-a / (math.pow(b, 3.0) / (c * c))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-a) / Float64((b ^ 3.0) / Float64(c * c))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-a / ((b ^ 3.0) / (c * c))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[((-a) / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

   1. neg-sub0 19.5%

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

   2. associate-+l- 19.5%

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

   3. sub0-neg 19.5%

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

   4. neg-mul-1 19.5%

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

   5. associate-*l/ 19.5%

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

   6. *-commutative 19.5%

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

   7. associate-/r* 19.5%

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

   8. /-rgt-identity 19.5%

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

   9. metadata-eval 19.5%

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

3. Simplified 19.5%

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

4. Taylor expanded in b around inf 94.2%

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

   1. +-commutative 94.2%

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

   2. mul-1-neg 94.2%

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

   3. unsub-neg 94.2%

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

   4. associate-*r/ 94.2%

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

   5. neg-mul-1 94.2%

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

   6. *-commutative 94.2%

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

   7. associate-/l* 94.2%

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

   8. unpow2 94.2%

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

6. Simplified 94.2%

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

7. Final simplification 94.2%

   \[\leadsto \frac{-a}{\frac{{b}^{3}}{c \cdot c}} - \frac{c}{b} \]

Alternative 4: 90.2% 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(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 19.5%

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

   1. neg-sub0 19.5%

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

   2. associate-+l- 19.5%

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

   3. sub0-neg 19.5%

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

   4. neg-mul-1 19.5%

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

   5. associate-*l/ 19.5%

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

   6. *-commutative 19.5%

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

   7. associate-/r* 19.5%

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

   8. /-rgt-identity 19.5%

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

   9. metadata-eval 19.5%

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

3. Simplified 19.5%

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

4. Taylor expanded in b around inf 89.4%

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

   1. associate-*r/ 89.4%

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

   2. neg-mul-1 89.4%

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

6. Simplified 89.4%

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

7. Final simplification 89.4%

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

Alternative 5: 3.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

   1. add-cube-cbrt 19.5%

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

   2. pow3 19.5%

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

   3. neg-mul-1 19.5%

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

   4. fma-def 19.5%

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

   5. *-commutative 19.5%

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

   6. *-commutative 19.5%

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

3. Applied egg-rr 19.5%

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

4. Taylor expanded in c around 0 3.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
5. Step-by-step derivation

   1. associate-*r/ 3.3%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b + -1 \cdot b\right)}{a}} \]

   2. distribute-rgt1-in 3.3%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

   3. metadata-eval 3.3%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.3%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.3%

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

6. Simplified 3.3%

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

7. Final simplification 3.3%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023249 
(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)))