Quadratic roots, medium range

Percentage Accurate: 31.4% → 95.5%

Time: 11.0s

Alternatives: 5

Speedup: 29.0×

Specification

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

Math

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

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: 31.4% 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: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, \mathsf{fma}\left(-1, \frac{c}{b}, -2 \cdot \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -1.0
  (/ (* a (* c c)) (pow b 3.0))
  (fma
   -0.25
   (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0)))
   (fma -1.0 (/ c b) (* -2.0 (/ (* (pow c 3.0) (* a a)) (pow b 5.0)))))))

C

double code(double a, double b, double c) {
	return fma(-1.0, ((a * (c * c)) / pow(b, 3.0)), fma(-0.25, ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0))), fma(-1.0, (c / b), (-2.0 * ((pow(c, 3.0) * (a * a)) / pow(b, 5.0))))));
}

Julia

function code(a, b, c)
	return fma(-1.0, Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)), fma(-0.25, Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0))), fma(-1.0, Float64(c / b), Float64(-2.0 * Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0))))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(c / b), $MachinePrecision] + N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.8%

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

   1. neg-sub0 32.8%

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

   2. associate-+l- 32.8%

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

   3. sub0-neg 32.8%

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

   4. neg-mul-1 32.8%

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

   5. associate-*l/ 32.8%

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

   6. *-commutative 32.8%

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

   7. associate-/r* 32.8%

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

   8. /-rgt-identity 32.8%

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

   9. metadata-eval 32.8%

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

3. Simplified 32.8%

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

4. Taylor expanded in b around inf 94.9%

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

   1. fma-def 94.9%

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

   2. *-commutative 94.9%

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

   3. unpow2 94.9%

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

   4. fma-def 94.9%

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

6. Simplified 94.9%

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

7. Taylor expanded in c around 0 94.9%

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

   1. +-commutative 94.9%

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

   2. distribute-rgt-out 94.9%

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

   3. associate-*r* 94.9%

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

   4. times-frac 94.9%

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

9. Simplified 94.9%

   \[\leadsto \mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}}, \mathsf{fma}\left(-1, \frac{c}{b}, -2 \cdot \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}\right)\right)\right) \]

10. Final simplification 94.9%

    \[\leadsto \mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, \mathsf{fma}\left(-1, \frac{c}{b}, -2 \cdot \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}\right)\right)\right) \]

Alternative 2: 95.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

   1. neg-sub0 32.8%

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

   2. associate-+l- 32.8%

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

   3. sub0-neg 32.8%

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

   4. neg-mul-1 32.8%

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

   5. associate-*l/ 32.8%

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

   6. *-commutative 32.8%

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

   7. associate-/r* 32.8%

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

   8. /-rgt-identity 32.8%

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

   9. metadata-eval 32.8%

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

3. Simplified 32.8%

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

4. Taylor expanded in a around 0 94.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{\left({\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right) \cdot {a}^{3}}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

6. Simplified 94.9%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b} \cdot {a}^{3}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]

7. Taylor expanded in c around 0 94.9%

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

   1. associate-/l* 94.9%

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

9. Simplified 94.9%

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

10. Final simplification 94.9%

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

Alternative 3: 93.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((-2.0 * pow(c, 3.0)) / (pow(b, 5.0) / (a * a))) - (c / b)) - (c * ((a * c) / 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) * (c ** 3.0d0)) / ((b ** 5.0d0) / (a * a))) - (c / b)) - (c * ((a * c) / (b ** 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

   1. neg-sub0 32.8%

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

   2. associate-+l- 32.8%

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

   3. sub0-neg 32.8%

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

   4. neg-mul-1 32.8%

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

   5. associate-*l/ 32.8%

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

   6. *-commutative 32.8%

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

   7. associate-/r* 32.8%

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

   8. /-rgt-identity 32.8%

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

   9. metadata-eval 32.8%

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

3. Simplified 32.8%

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

4. Taylor expanded in b around inf 92.9%

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

   1. associate-+r+ 92.9%

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

   2. distribute-lft-out 92.9%

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

   3. fma-def 92.9%

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

   4. *-commutative 92.9%

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

   5. associate-/l* 92.9%

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

   6. *-commutative 92.9%

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

   7. associate-/l* 92.9%

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

   8. unpow2 92.9%

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

   9. unpow2 92.9%

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

   10. cube-prod 92.9%

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

6. Simplified 92.9%

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

7. Taylor expanded in a around 0 93.4%

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

   1. +-commutative 93.4%

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

   2. mul-1-neg 93.4%

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

   3. unpow2 93.4%

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

   4. associate-*r* 93.4%

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

   5. unsub-neg 93.4%

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

9. Simplified 93.4%

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

10. Final simplification 93.4%

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

Alternative 4: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.8%

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

   1. *-commutative 32.8%

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

   2. +-commutative 32.8%

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

   3. unsub-neg 32.8%

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

   4. fma-neg 32.8%

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

   5. associate-*l* 32.8%

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

   6. *-commutative 32.8%

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

   7. distribute-rgt-neg-in 32.8%

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

   8. metadata-eval 32.8%

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

3. Simplified 32.8%

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

   1. fma-udef 32.8%

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

   2. *-commutative 32.8%

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

5. Applied egg-rr 32.8%

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

6. Taylor expanded in b around inf 90.5%

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

   1. +-commutative 90.5%

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

   2. mul-1-neg 90.5%

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

   3. unsub-neg 90.5%

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

   4. mul-1-neg 90.5%

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

   5. unpow2 90.5%

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

   6. associate-*r* 90.5%

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

   7. *-rgt-identity 90.5%

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

   8. associate-*r/ 90.5%

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

   9. associate-*l* 90.5%

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

   10. associate-*r/ 90.5%

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

   11. *-commutative 90.5%

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

   12. *-lft-identity 90.5%

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

8. Simplified 90.5%

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

9. Final simplification 90.5%

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

Alternative 5: 81.3% 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 32.8%

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

   1. neg-sub0 32.8%

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

   2. associate-+l- 32.8%

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

   3. sub0-neg 32.8%

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

   4. neg-mul-1 32.8%

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

   5. associate-*l/ 32.8%

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

   6. *-commutative 32.8%

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

   7. associate-/r* 32.8%

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

   8. /-rgt-identity 32.8%

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

   9. metadata-eval 32.8%

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

3. Simplified 32.8%

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

4. Taylor expanded in b around inf 80.4%

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

   1. associate-*r/ 80.4%

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

   2. neg-mul-1 80.4%

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

6. Simplified 80.4%

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

7. Final simplification 80.4%

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

Reproduce

herbie shell --seed 2023193 
(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)))