Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.7%

Time: 12.1s

Alternatives: 8

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: 18.0% 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: 97.7% accurate, 0.2× speedup

Math

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

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)))
  (/ (* c (* c a)) (pow b 3.0))))

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))) - ((c * (c * a)) / pow(b, 3.0));
}

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(Float64(c * Float64(c * a)) / (b ^ 3.0)))
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[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.7%

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

   1. neg-sub0 19.7%

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

   2. associate-+l- 19.7%

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

   3. sub0-neg 19.7%

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

   4. neg-mul-1 19.7%

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

   5. associate-*l/ 19.7%

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

   6. *-commutative 19.7%

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

   7. associate-/r* 19.7%

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

   8. /-rgt-identity 19.7%

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

   9. metadata-eval 19.7%

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

3. Simplified 19.7%

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

4. Taylor expanded in b around inf 97.8%

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

   1. +-commutative 97.8%

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

   2. mul-1-neg 97.8%

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

   3. unsub-neg 97.8%

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

6. Simplified 97.8%

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

7. Taylor expanded in b around 0 97.8%

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

   1. distribute-rgt-out 97.8%

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

   2. metadata-eval 97.8%

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

   3. pow-sqr 97.8%

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

   4. metadata-eval 97.8%

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

   5. pow-sqr 97.8%

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

   6. unswap-sqr 97.8%

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

   7. unpow2 97.8%

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

   8. unpow2 97.8%

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

   9. unswap-sqr 97.8%

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

   10. unpow2 97.8%

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

   11. unpow2 97.8%

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

   12. unpow2 97.8%

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

   13. unswap-sqr 97.8%

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

   14. unpow2 97.8%

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

   15. pow-sqr 97.8%

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

   16. metadata-eval 97.8%

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

   17. metadata-eval 97.8%

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

9. Simplified 97.8%

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

10. Final simplification 97.8%

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

Alternative 2: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{\left(c \cdot c\right) \cdot \left(-{a}^{3}\right)}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (pow
  (+
   (/ a b)
   (fma
    -2.0
    (/ (* (* c (* a a)) -0.5) (pow b 3.0))
    (- (* -2.0 (/ (* (* c c) (- (pow a 3.0))) (pow b 5.0))) (/ b c))))
  -1.0))

C

double code(double a, double b, double c) {
	return pow(((a / b) + fma(-2.0, (((c * (a * a)) * -0.5) / pow(b, 3.0)), ((-2.0 * (((c * c) * -pow(a, 3.0)) / pow(b, 5.0))) - (b / c)))), -1.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(a / b) + fma(-2.0, Float64(Float64(Float64(c * Float64(a * a)) * -0.5) / (b ^ 3.0)), Float64(Float64(-2.0 * Float64(Float64(Float64(c * c) * Float64(-(a ^ 3.0))) / (b ^ 5.0))) - Float64(b / c)))) ^ -1.0
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(a / b), $MachinePrecision] + N[(-2.0 * N[(N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[(c * c), $MachinePrecision] * (-N[Power[a, 3.0], $MachinePrecision])), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 19.7%

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

   1. clear-num 19.7%

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

   2. inv-pow 19.7%

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

   3. *-commutative 19.7%

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

   4. neg-mul-1 19.7%

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

   5. fma-def 19.7%

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

   6. *-commutative 19.7%

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

   7. *-commutative 19.7%

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

3. Applied egg-rr 19.7%

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

4. Taylor expanded in b around inf 97.6%

   \[\leadsto {\color{blue}{\left(\frac{a}{b} + \left(-2 \cdot \frac{0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + \left(-1 \cdot \frac{b}{c} + -2 \cdot \frac{-1 \cdot \left(c \cdot \left(\left(0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)\right) \cdot a\right)\right) + \left({c}^{2} \cdot {a}^{3} + -0.125 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{c}^{2} \cdot a}\right)}{{b}^{5}}\right)\right)\right)}}^{-1} \]
5. Step-by-step derivation

   1. fma-def 97.6%

      \[\leadsto {\left(\frac{a}{b} + \color{blue}{\mathsf{fma}\left(-2, \frac{0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}}, -1 \cdot \frac{b}{c} + -2 \cdot \frac{-1 \cdot \left(c \cdot \left(\left(0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)\right) \cdot a\right)\right) + \left({c}^{2} \cdot {a}^{3} + -0.125 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{c}^{2} \cdot a}\right)}{{b}^{5}}\right)}\right)}^{-1} \]

6. Simplified 97.6%

   \[\leadsto {\color{blue}{\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{\mathsf{fma}\left(c \cdot c, {a}^{3}, -0.125 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{c \cdot \left(c \cdot a\right)}\right) - c \cdot \left(a \cdot \left(\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5\right)\right)}{{b}^{5}} - \frac{b}{c}\right)\right)}}^{-1} \]

7. Taylor expanded in c around 0 97.6%

   \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{\color{blue}{{c}^{2} \cdot \left(\left({a}^{3} + -2.5 \cdot {a}^{3}\right) - -0.5 \cdot {a}^{3}\right)}}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]
8. Step-by-step derivation

   1. distribute-rgt1-in 97.6%

      \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{{c}^{2} \cdot \left(\color{blue}{\left(-2.5 + 1\right) \cdot {a}^{3}} - -0.5 \cdot {a}^{3}\right)}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]

   2. distribute-rgt-out-- 97.6%

      \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{{c}^{2} \cdot \color{blue}{\left({a}^{3} \cdot \left(\left(-2.5 + 1\right) - -0.5\right)\right)}}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]

   3. metadata-eval 97.6%

      \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{{c}^{2} \cdot \left({a}^{3} \cdot \left(\color{blue}{-1.5} - -0.5\right)\right)}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]

   4. metadata-eval 97.6%

      \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{{c}^{2} \cdot \left({a}^{3} \cdot \color{blue}{-1}\right)}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]

   5. *-commutative 97.6%

      \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{{c}^{2} \cdot \color{blue}{\left(-1 \cdot {a}^{3}\right)}}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]

   6. unpow2 97.6%

      \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{\color{blue}{\left(c \cdot c\right)} \cdot \left(-1 \cdot {a}^{3}\right)}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]

   7. mul-1-neg 97.6%

      \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{\left(c \cdot c\right) \cdot \color{blue}{\left(-{a}^{3}\right)}}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]

9. Simplified 97.6%

   \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{\color{blue}{\left(c \cdot c\right) \cdot \left(-{a}^{3}\right)}}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]

10. Final simplification 97.6%

    \[\leadsto {\left(\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}}, -2 \cdot \frac{\left(c \cdot c\right) \cdot \left(-{a}^{3}\right)}{{b}^{5}} - \frac{b}{c}\right)\right)}^{-1} \]

Alternative 3: 96.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-2.0d0) / ((b ** 5.0d0) / (a * (a * (c ** 3.0d0))))) - (c / b)) - ((c * (c * a)) / (b ** 3.0d0))
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)) - ((c * (c * a)) / Math.pow(b, 3.0));
}

Python

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

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(Float64(c * Float64(c * a)) / (b ^ 3.0)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-2.0 / ((b ^ 5.0) / (a * (a * (c ^ 3.0))))) - (c / b)) - ((c * (c * a)) / (b ^ 3.0));
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[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.7%

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

   1. neg-sub0 19.7%

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

   2. associate-+l- 19.7%

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

   3. sub0-neg 19.7%

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

   4. neg-mul-1 19.7%

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

   5. associate-*l/ 19.7%

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

   6. *-commutative 19.7%

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

   7. associate-/r* 19.7%

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

   8. /-rgt-identity 19.7%

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

   9. metadata-eval 19.7%

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

3. Simplified 19.7%

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

4. Taylor expanded in b around inf 96.9%

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

   1. +-commutative 96.9%

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

   2. mul-1-neg 96.9%

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

   3. unsub-neg 96.9%

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

   4. +-commutative 96.9%

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

   5. mul-1-neg 96.9%

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

   6. unsub-neg 96.9%

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

   7. associate-*r/ 96.9%

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

   8. associate-/l* 96.9%

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

   9. *-commutative 96.9%

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

   10. unpow2 96.9%

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

   11. associate-*l* 96.9%

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

6. Simplified 96.9%

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

7. Final simplification 96.9%

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

Alternative 4: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (pow
  (+ (/ a b) (- (* -2.0 (/ (* (* c (* a a)) -0.5) (pow b 3.0))) (/ b c)))
  -1.0))

C

double code(double a, double b, double c) {
	return pow(((a / b) + ((-2.0 * (((c * (a * a)) * -0.5) / pow(b, 3.0))) - (b / c))), -1.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((a / b) + (((-2.0d0) * (((c * (a * a)) * (-0.5d0)) / (b ** 3.0d0))) - (b / c))) ** (-1.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return Math.pow(((a / b) + ((-2.0 * (((c * (a * a)) * -0.5) / Math.pow(b, 3.0))) - (b / c))), -1.0);
}

Python

def code(a, b, c):
	return math.pow(((a / b) + ((-2.0 * (((c * (a * a)) * -0.5) / math.pow(b, 3.0))) - (b / c))), -1.0)

Julia

function code(a, b, c)
	return Float64(Float64(a / b) + Float64(Float64(-2.0 * Float64(Float64(Float64(c * Float64(a * a)) * -0.5) / (b ^ 3.0))) - Float64(b / c))) ^ -1.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a / b) + ((-2.0 * (((c * (a * a)) * -0.5) / (b ^ 3.0))) - (b / c))) ^ -1.0;
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(a / b), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 19.7%

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

   1. clear-num 19.7%

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

   2. inv-pow 19.7%

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

   3. *-commutative 19.7%

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

   4. neg-mul-1 19.7%

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

   5. fma-def 19.7%

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

   6. *-commutative 19.7%

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

   7. *-commutative 19.7%

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

3. Applied egg-rr 19.7%

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

4. Taylor expanded in b around inf 96.7%

   \[\leadsto {\color{blue}{\left(\frac{a}{b} + \left(-2 \cdot \frac{0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + -1 \cdot \frac{b}{c}\right)\right)}}^{-1} \]
5. Step-by-step derivation

   1. mul-1-neg 96.7%

      \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + \color{blue}{\left(-\frac{b}{c}\right)}\right)\right)}^{-1} \]

   2. unsub-neg 96.7%

      \[\leadsto {\left(\frac{a}{b} + \color{blue}{\left(-2 \cdot \frac{0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} - \frac{b}{c}\right)}\right)}^{-1} \]

   3. distribute-rgt-out 96.7%

      \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{\color{blue}{\left(c \cdot {a}^{2}\right) \cdot \left(0.5 + -1\right)}}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \]

   4. unpow2 96.7%

      \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(0.5 + -1\right)}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \]

   5. metadata-eval 96.7%

      \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{-0.5}}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \]

6. Simplified 96.7%

   \[\leadsto {\color{blue}{\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}} - \frac{b}{c}\right)\right)}}^{-1} \]

7. Final simplification 96.7%

   \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \]

Alternative 5: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

   1. neg-sub0 19.7%

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

   2. associate-+l- 19.7%

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

   3. sub0-neg 19.7%

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

   4. neg-mul-1 19.7%

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

   5. associate-*l/ 19.7%

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

   6. *-commutative 19.7%

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

   7. associate-/r* 19.7%

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

   8. /-rgt-identity 19.7%

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

   9. metadata-eval 19.7%

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

3. Simplified 19.7%

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

4. Taylor expanded in b around inf 95.1%

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

   1. distribute-lft-out 95.1%

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

   2. mul-1-neg 95.1%

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

   3. +-commutative 95.1%

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

   4. unpow2 95.1%

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

   5. associate-*l* 95.1%

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

6. Simplified 95.1%

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

7. Final simplification 95.1%

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

Alternative 6: 95.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (pow (- (/ a b) (/ b c)) -1.0))

C

double code(double a, double b, double c) {
	return pow(((a / b) - (b / c)), -1.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((a / b) - (b / c)) ** (-1.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return Math.pow(((a / b) - (b / c)), -1.0);
}

Python

def code(a, b, c):
	return math.pow(((a / b) - (b / c)), -1.0)

Julia

function code(a, b, c)
	return Float64(Float64(a / b) - Float64(b / c)) ^ -1.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a / b) - (b / c)) ^ -1.0;
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 19.7%

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

   1. clear-num 19.7%

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

   2. inv-pow 19.7%

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

   3. *-commutative 19.7%

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

   4. neg-mul-1 19.7%

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

   5. fma-def 19.7%

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

   6. *-commutative 19.7%

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

   7. *-commutative 19.7%

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

3. Applied egg-rr 19.7%

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

4. Taylor expanded in b around inf 95.0%

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

   1. mul-1-neg 95.0%

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

   2. unsub-neg 95.0%

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

6. Simplified 95.0%

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

7. Final simplification 95.0%

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

Alternative 7: 90.3% 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 19.7%

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

   1. neg-sub0 19.7%

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

   2. associate-+l- 19.7%

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

   3. sub0-neg 19.7%

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

   4. neg-mul-1 19.7%

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

   5. associate-*l/ 19.7%

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

   6. *-commutative 19.7%

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

   7. associate-/r* 19.7%

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

   8. /-rgt-identity 19.7%

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

   9. metadata-eval 19.7%

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

3. Simplified 19.7%

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

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 8: 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 19.7%

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

   1. clear-num 19.7%

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

   2. inv-pow 19.7%

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

   3. *-commutative 19.7%

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

   4. neg-mul-1 19.7%

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

   5. fma-def 19.7%

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

   6. *-commutative 19.7%

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

   7. *-commutative 19.7%

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

3. Applied egg-rr 19.7%

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

4. Taylor expanded in a 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 2023175 
(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)))