Quadratic roots, wide range

Percentage Accurate: 17.8% → 99.7%

Time: 34.2s

Alternatives: 7

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: 17.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. *-commutative 20.0%

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

   2. +-commutative 20.0%

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

   3. sqr-neg 20.0%

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

   4. unsub-neg 20.0%

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

   5. sqr-neg 20.0%

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

   6. fma-neg 20.1%

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

   7. distribute-lft-neg-in 20.1%

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

   8. *-commutative 20.1%

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

   9. *-commutative 20.1%

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

   10. distribute-rgt-neg-in 20.1%

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

   11. metadata-eval 20.1%

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

3. Simplified 20.1%

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

5. Taylor expanded in a around inf 20.1%

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

   1. flip-- 20.0%

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

   2. add-sqr-sqrt 20.5%

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

   3. *-commutative 20.5%

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

   4. fma-define 20.5%

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

   5. unpow2 20.5%

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

   6. *-commutative 20.5%

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

   7. fma-define 20.5%

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

7. Applied egg-rr 20.5%

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

8. Taylor expanded in a around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

10. Simplified 99.4%

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

    1. *-un-lft-identity 99.4%

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

    2. associate-/l/ 99.3%

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

    3. associate-*l* 99.3%

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

    4. sqrt-prod 99.2%

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

    5. fma-define 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-lft-identity 99.3%

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

    2. associate-/r* 99.7%

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

    3. associate-*r* 99.7%

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

    4. *-commutative 99.7%

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

    5. times-frac 99.7%

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

    6. associate-*r/ 99.7%

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

    7. metadata-eval 99.7%

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

14. Simplified 99.7%

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

    1. associate-/l* 99.5%

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

    2. fma-undefine 99.4%

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

    3. sqrt-prod 99.5%

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

16. Applied egg-rr 99.5%

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

    1. associate-*r/ 99.7%

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

    2. associate-*r* 99.7%

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

    3. *-commutative 99.7%

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

    4. associate-*r* 99.7%

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

    5. +-commutative 99.7%

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

18. Simplified 99.7%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. *-commutative 20.0%

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

   2. +-commutative 20.0%

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

   3. sqr-neg 20.0%

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

   4. unsub-neg 20.0%

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

   5. sqr-neg 20.0%

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

   6. fma-neg 20.1%

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

   7. distribute-lft-neg-in 20.1%

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

   8. *-commutative 20.1%

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

   9. *-commutative 20.1%

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

   10. distribute-rgt-neg-in 20.1%

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

   11. metadata-eval 20.1%

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

3. Simplified 20.1%

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

5. Taylor expanded in a around inf 20.1%

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

   1. flip-- 20.0%

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

   2. add-sqr-sqrt 20.5%

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

   3. *-commutative 20.5%

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

   4. fma-define 20.5%

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

   5. unpow2 20.5%

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

   6. *-commutative 20.5%

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

   7. fma-define 20.5%

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

7. Applied egg-rr 20.5%

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

8. Taylor expanded in a around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

10. Simplified 99.4%

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

    1. *-un-lft-identity 99.4%

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

    2. associate-/l/ 99.3%

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

    3. associate-*l* 99.3%

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

    4. sqrt-prod 99.2%

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

    5. fma-define 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-lft-identity 99.3%

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

    2. associate-/r* 99.7%

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

    3. associate-*r* 99.7%

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

    4. *-commutative 99.7%

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

    5. times-frac 99.7%

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

    6. associate-*r/ 99.7%

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

    7. metadata-eval 99.7%

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

14. Simplified 99.7%

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

    1. associate-/l* 99.5%

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

    2. fma-undefine 99.4%

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

    3. sqrt-prod 99.5%

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

16. Applied egg-rr 99.5%

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

17. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \left(a \cdot c\right)\\ \frac{\frac{t\_0}{b + \sqrt{{b}^{2} + t\_0}}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -4.0 (* a c))))
   (/ (/ t_0 (+ b (sqrt (+ (pow b 2.0) t_0)))) (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = -4.0 * (a * c);
	return (t_0 / (b + sqrt((pow(b, 2.0) + t_0)))) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = (-4.0d0) * (a * c)
    code = (t_0 / (b + sqrt(((b ** 2.0d0) + t_0)))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -4.0 * (a * c);
	return (t_0 / (b + Math.sqrt((Math.pow(b, 2.0) + t_0)))) / (a * 2.0);
}

Python

def code(a, b, c):
	t_0 = -4.0 * (a * c)
	return (t_0 / (b + math.sqrt((math.pow(b, 2.0) + t_0)))) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(-4.0 * Float64(a * c))
	return Float64(Float64(t_0 / Float64(b + sqrt(Float64((b ^ 2.0) + t_0)))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = -4.0 * (a * c);
	tmp = (t_0 / (b + sqrt(((b ^ 2.0) + t_0)))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 20.0%

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

   1. *-commutative 20.0%

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

   2. +-commutative 20.0%

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

   3. sqr-neg 20.0%

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

   4. unsub-neg 20.0%

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

   5. sqr-neg 20.0%

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

   6. fma-neg 20.1%

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

   7. distribute-lft-neg-in 20.1%

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

   8. *-commutative 20.1%

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

   9. *-commutative 20.1%

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

   10. distribute-rgt-neg-in 20.1%

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

   11. metadata-eval 20.1%

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

3. Simplified 20.1%

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

5. Taylor expanded in a around inf 20.1%

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

   1. flip-- 20.0%

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

   2. add-sqr-sqrt 20.5%

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

   3. *-commutative 20.5%

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

   4. fma-define 20.5%

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

   5. unpow2 20.5%

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

   6. *-commutative 20.5%

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

   7. fma-define 20.5%

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

7. Applied egg-rr 20.5%

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

8. Taylor expanded in a around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in a around 0 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 95.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. *-commutative 20.0%

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

3. Simplified 20.0%

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

5. Taylor expanded in c around 0 92.8%

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

   1. associate-*r/ 92.8%

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

   2. neg-mul-1 92.8%

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

   3. distribute-rgt-neg-in 92.8%

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

7. Simplified 92.8%

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

   1. frac-sub 92.7%

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

   2. distribute-rgt-neg-out 92.7%

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

   3. pow-plus 92.6%

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

   4. metadata-eval 92.6%

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

9. Applied egg-rr 92.6%

   \[\leadsto c \cdot \color{blue}{\frac{\left(-a \cdot c\right) \cdot b - {b}^{3} \cdot 1}{{b}^{4}}} \]
10. Step-by-step derivation

    1. *-commutative 92.6%

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

    2. distribute-rgt-neg-in 92.6%

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

    3. *-rgt-identity 92.6%

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

11. Simplified 92.6%

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

12. Taylor expanded in b around -inf 92.8%

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

    1. associate-*r/ 92.8%

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

    2. mul-1-neg 92.8%

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

    3. +-commutative 92.8%

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

    4. associate-/l* 92.8%

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

14. Simplified 92.8%

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

    1. associate-*r/ 93.1%

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

    2. fma-define 93.1%

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

    3. div-inv 93.1%

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

    4. pow-flip 93.1%

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

    5. metadata-eval 93.1%

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

16. Applied egg-rr 93.1%

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

Alternative 5: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. *-commutative 20.0%

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

3. Simplified 20.0%

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

5. Taylor expanded in c around 0 92.8%

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

   1. associate-*r/ 92.8%

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

   2. neg-mul-1 92.8%

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

   3. distribute-rgt-neg-in 92.8%

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

7. Simplified 92.8%

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

   1. frac-sub 92.7%

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

   2. distribute-rgt-neg-out 92.7%

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

   3. pow-plus 92.6%

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

   4. metadata-eval 92.6%

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

9. Applied egg-rr 92.6%

   \[\leadsto c \cdot \color{blue}{\frac{\left(-a \cdot c\right) \cdot b - {b}^{3} \cdot 1}{{b}^{4}}} \]
10. Step-by-step derivation

    1. *-commutative 92.6%

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

    2. distribute-rgt-neg-in 92.6%

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

    3. *-rgt-identity 92.6%

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

11. Simplified 92.6%

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

12. Taylor expanded in b around inf 93.1%

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

    1. neg-mul-1 93.1%

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

    2. mul-1-neg 93.1%

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

    3. unsub-neg 93.1%

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

    4. associate-/l* 93.1%

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

    5. unpow2 93.1%

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

    6. unpow2 93.1%

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

    7. times-frac 93.1%

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

    8. sqr-neg 93.1%

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

    9. distribute-frac-neg 93.1%

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

    10. distribute-frac-neg 93.1%

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

    11. unpow2 93.1%

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

14. Simplified 93.1%

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

15. Final simplification 93.1%

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

Alternative 6: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((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 * (((-1.0d0) / b) - ((a * c) * (b ** (-3.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. *-commutative 20.0%

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

3. Simplified 20.0%

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

5. Taylor expanded in c around 0 92.8%

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

   1. associate-*r/ 92.8%

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

   2. neg-mul-1 92.8%

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

   3. distribute-rgt-neg-in 92.8%

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

7. Simplified 92.8%

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

   1. div-inv 92.8%

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

   2. *-un-lft-identity 92.8%

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

   3. prod-diff 92.8%

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

   4. distribute-rgt-neg-out 92.8%

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

   5. pow-flip 92.8%

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

   6. metadata-eval 92.8%

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

9. Applied egg-rr 92.8%

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

    1. +-commutative 92.8%

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

    2. fma-undefine 92.8%

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

    3. *-rgt-identity 92.8%

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

    4. associate-+r+ 92.8%

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

    5. distribute-lft-neg-out 92.8%

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

    6. unsub-neg 92.8%

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

    7. fma-undefine 92.8%

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

    8. distribute-rgt-out 92.8%

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

    9. +-commutative 92.8%

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

    10. sub-neg 92.8%

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

    11. +-inverses 92.8%

        \[\leadsto c \cdot \left(\left(1 \cdot \color{blue}{0} - \left(a \cdot c\right) \cdot {b}^{-3}\right) + \left(-\frac{1}{b}\right)\right) \]

    12. metadata-eval 92.8%

        \[\leadsto c \cdot \left(\left(\color{blue}{0} - \left(a \cdot c\right) \cdot {b}^{-3}\right) + \left(-\frac{1}{b}\right)\right) \]

    13. neg-sub0 92.8%

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

11. Simplified 92.8%

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

12. Final simplification 92.8%

    \[\leadsto c \cdot \left(\frac{-1}{b} - \left(a \cdot c\right) \cdot {b}^{-3}\right) \]
13. Add Preprocessing

Alternative 7: 90.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. *-commutative 20.0%

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

3. Simplified 20.0%

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

5. Taylor expanded in b around inf 88.3%

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

   1. associate-*r/ 88.3%

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

   2. mul-1-neg 88.3%

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

7. Simplified 88.3%

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

8. Final simplification 88.3%

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

Reproduce

herbie shell --seed 2024105 
(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)))