Quadratic roots, medium range

Average Accuracy: 31.8% → 95.2%

Time: 17.1s

Precision: binary64

Cost: 61696

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

\[0.5 \cdot \left(\left({c}^{3} \cdot \frac{a \cdot a}{\frac{{b}^{5}}{-4}} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (-
  (*
   0.5
   (+
    (+
     (* (pow c 3.0) (/ (* a a) (/ (pow b 5.0) -4.0)))
     (*
      (* c c)
      (+
       (/ a (pow b 3.0))
       (* 2.0 (/ (* (/ (* a a) (pow b 4.0)) -1.5) (/ a b))))))
    (*
     (pow c 4.0)
     (+
      (/ (* (/ (pow a 4.0) (pow b 8.0)) 2.25) (/ a b))
      (* -12.25 (/ (pow a 3.0) (pow b 7.0)))))))
  (/ c b)))

C

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

↓

double code(double a, double b, double c) {
	return (0.5 * (((pow(c, 3.0) * ((a * a) / (pow(b, 5.0) / -4.0))) + ((c * c) * ((a / pow(b, 3.0)) + (2.0 * ((((a * a) / pow(b, 4.0)) * -1.5) / (a / b)))))) + (pow(c, 4.0) * ((((pow(a, 4.0) / pow(b, 8.0)) * 2.25) / (a / b)) + (-12.25 * (pow(a, 3.0) / pow(b, 7.0))))))) - (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 = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (0.5d0 * ((((c ** 3.0d0) * ((a * a) / ((b ** 5.0d0) / (-4.0d0)))) + ((c * c) * ((a / (b ** 3.0d0)) + (2.0d0 * ((((a * a) / (b ** 4.0d0)) * (-1.5d0)) / (a / b)))))) + ((c ** 4.0d0) * (((((a ** 4.0d0) / (b ** 8.0d0)) * 2.25d0) / (a / b)) + ((-12.25d0) * ((a ** 3.0d0) / (b ** 7.0d0))))))) - (c / b)
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);
}

↓

public static double code(double a, double b, double c) {
	return (0.5 * (((Math.pow(c, 3.0) * ((a * a) / (Math.pow(b, 5.0) / -4.0))) + ((c * c) * ((a / Math.pow(b, 3.0)) + (2.0 * ((((a * a) / Math.pow(b, 4.0)) * -1.5) / (a / b)))))) + (Math.pow(c, 4.0) * ((((Math.pow(a, 4.0) / Math.pow(b, 8.0)) * 2.25) / (a / b)) + (-12.25 * (Math.pow(a, 3.0) / Math.pow(b, 7.0))))))) - (c / b);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

↓

def code(a, b, c):
	return (0.5 * (((math.pow(c, 3.0) * ((a * a) / (math.pow(b, 5.0) / -4.0))) + ((c * c) * ((a / math.pow(b, 3.0)) + (2.0 * ((((a * a) / math.pow(b, 4.0)) * -1.5) / (a / b)))))) + (math.pow(c, 4.0) * ((((math.pow(a, 4.0) / math.pow(b, 8.0)) * 2.25) / (a / b)) + (-12.25 * (math.pow(a, 3.0) / math.pow(b, 7.0))))))) - (c / b)

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

↓

function code(a, b, c)
	return Float64(Float64(0.5 * Float64(Float64(Float64((c ^ 3.0) * Float64(Float64(a * a) / Float64((b ^ 5.0) / -4.0))) + Float64(Float64(c * c) * Float64(Float64(a / (b ^ 3.0)) + Float64(2.0 * Float64(Float64(Float64(Float64(a * a) / (b ^ 4.0)) * -1.5) / Float64(a / b)))))) + Float64((c ^ 4.0) * Float64(Float64(Float64(Float64((a ^ 4.0) / (b ^ 8.0)) * 2.25) / Float64(a / b)) + Float64(-12.25 * Float64((a ^ 3.0) / (b ^ 7.0))))))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

↓

function tmp = code(a, b, c)
	tmp = (0.5 * ((((c ^ 3.0) * ((a * a) / ((b ^ 5.0) / -4.0))) + ((c * c) * ((a / (b ^ 3.0)) + (2.0 * ((((a * a) / (b ^ 4.0)) * -1.5) / (a / b)))))) + ((c ^ 4.0) * (((((a ^ 4.0) / (b ^ 8.0)) * 2.25) / (a / b)) + (-12.25 * ((a ^ 3.0) / (b ^ 7.0))))))) - (c / b);
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]

↓

code[a_, b_, c_] := N[(N[(0.5 * N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision] * 2.25), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(-12.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.1%

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

2. Simplified 30.1%

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

3. Applied egg-rr 29.8%

   \[\leadsto \left(\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}\right)}^{2}} - b\right) \cdot \frac{0.5}{a} \]

4. Taylor expanded in c around 0 95.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(0.5 \cdot \left({c}^{4} \cdot \left(\frac{{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right)}^{2} \cdot b}{a} + \left(-2 \cdot \frac{-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)}{b} + 2 \cdot \frac{b \cdot \left(2 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-1 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(5.333333333333333 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-16 \cdot \frac{{a}^{4}}{{b}^{8}} + 0.041666666666666664 \cdot \frac{{a}^{4}}{{b}^{8}}\right)\right)\right)\right)}{a}\right)\right)\right) + \left(0.5 \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}}{b} + 2 \cdot \frac{\left(-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)\right) \cdot b}{a}\right)\right) + 0.5 \cdot \left({c}^{2} \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) \cdot b}{a}\right)\right)\right)\right)} \]

5. Simplified 95.3%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left({c}^{3} \cdot \mathsf{fma}\left(-2, \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{b}, 2 \cdot \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{\frac{a}{b}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + \mathsf{fma}\left(-2, \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{b}, \frac{\left(2 \cdot b\right) \cdot \left(\frac{{a}^{4}}{{b}^{8}} \cdot 1 + \frac{{a}^{4}}{{b}^{8}} \cdot -10.625\right)}{a}\right)\right)\right) - \frac{c}{b}} \]

6. Taylor expanded in a around 0 95.3%

   \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \mathsf{fma}\left(-2, \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{b}, 2 \cdot \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{\frac{a}{b}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + \color{blue}{-12.25 \cdot \frac{{a}^{3}}{{b}^{7}}}\right)\right) - \frac{c}{b} \]

7. Taylor expanded in c around 0 95.3%

   \[\leadsto 0.5 \cdot \left(\left(\color{blue}{{c}^{3} \cdot \left(3 \cdot \frac{{a}^{2}}{{b}^{5}} + -7 \cdot \frac{{a}^{2}}{{b}^{5}}\right)} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]

8. Simplified 95.3%

   \[\leadsto 0.5 \cdot \left(\left(\color{blue}{{c}^{3} \cdot \frac{a \cdot a}{\frac{{b}^{5}}{-4}}} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]

9. Final simplification 95.3%

   \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \frac{a \cdot a}{\frac{{b}^{5}}{-4}} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]

Alternatives

Alternative 1
Accuracy	95.2%
Cost	46976

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

Alternative 2
Accuracy	93.6%
Cost	41984

\[\begin{array}{l} t_0 := -1.5 \cdot \frac{c \cdot c}{{b}^{4}}\\ 0.5 \cdot \left(a \cdot \left(\frac{c \cdot c}{{b}^{3}} + 2 \cdot \left(b \cdot t_0\right)\right) + \left(a \cdot a\right) \cdot \mathsf{fma}\left(2, b \cdot \left(\frac{{c}^{3}}{{b}^{6}} \cdot -3.5\right), \frac{-2 \cdot \left(c \cdot t_0\right)}{b}\right)\right) - \frac{c}{b} \end{array} \]

Alternative 3
Accuracy	93.6%
Cost	20736

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

Alternative 4
Accuracy	90.4%
Cost	1024

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

Alternative 5
Accuracy	80.9%
Cost	256

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

Alternative 6
Accuracy	3.2%
Cost	192

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

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))