Quadratic roots, narrow range

Percentage Accurate: 55.2% → 91.5%

Time: 13.7s

Alternatives: 7

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.2% 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: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}} \cdot -0.5\\ \frac{-1}{\frac{b}{c} - a \cdot \mathsf{fma}\left(a, -2 \cdot \left(t\_0 + a \cdot \left(\mathsf{fma}\left(-0.125, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{c}^{2}}, \frac{{c}^{2}}{{b}^{5}}\right) - c \cdot \frac{t\_0}{{b}^{2}}\right)\right), \frac{1}{b}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ c (pow b 3.0)) -0.5)))
   (/
    -1.0
    (-
     (/ b c)
     (*
      a
      (fma
       a
       (*
        -2.0
        (+
         t_0
         (*
          a
          (-
           (fma
            -0.125
            (* b (/ (* (/ (pow c 4.0) (pow b 6.0)) 20.0) (pow c 2.0)))
            (/ (pow c 2.0) (pow b 5.0)))
           (* c (/ t_0 (pow b 2.0)))))))
       (/ 1.0 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / pow(b, 3.0)) * -0.5;
	return -1.0 / ((b / c) - (a * fma(a, (-2.0 * (t_0 + (a * (fma(-0.125, (b * (((pow(c, 4.0) / pow(b, 6.0)) * 20.0) / pow(c, 2.0))), (pow(c, 2.0) / pow(b, 5.0))) - (c * (t_0 / pow(b, 2.0))))))), (1.0 / b))));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / (b ^ 3.0)) * -0.5)
	return Float64(-1.0 / Float64(Float64(b / c) - Float64(a * fma(a, Float64(-2.0 * Float64(t_0 + Float64(a * Float64(fma(-0.125, Float64(b * Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 20.0) / (c ^ 2.0))), Float64((c ^ 2.0) / (b ^ 5.0))) - Float64(c * Float64(t_0 / (b ^ 2.0))))))), Float64(1.0 / b)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a * N[(a * N[(-2.0 * N[(t$95$0 + N[(a * N[(N[(-0.125 * N[(b * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * N[(t$95$0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 55.1%

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

   1. *-commutative 55.1%

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

   2. +-commutative 55.1%

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

   3. sqr-neg 55.1%

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

   4. unsub-neg 55.1%

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

   5. sqr-neg 55.1%

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

   6. fma-neg 55.2%

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

   7. distribute-lft-neg-in 55.2%

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

   8. *-commutative 55.2%

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

   9. *-commutative 55.2%

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

   10. distribute-rgt-neg-in 55.2%

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

   11. metadata-eval 55.2%

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

3. Simplified 55.2%

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

5. Taylor expanded in a around inf 55.0%

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

   1. clear-num 55.0%

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

   2. inv-pow 55.0%

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

   3. *-commutative 55.0%

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

   4. fma-define 55.0%

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

7. Applied egg-rr 55.0%

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

   1. unpow-1 55.0%

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

   2. associate-/l* 55.0%

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

9. Simplified 55.0%

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

10. Taylor expanded in a around 0 92.1%

    \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + 0.5 \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(-0.125 \cdot \frac{b \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) + -2 \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + 0.5 \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}} \]

12. Simplified 92.1%

    \[\leadsto \frac{1}{\color{blue}{a \cdot \mathsf{fma}\left(a, -2 \cdot \left(a \cdot \left(\mathsf{fma}\left(-0.125, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{c}^{2}}, \frac{{c}^{2}}{{b}^{5}}\right) - c \cdot \frac{\frac{c}{{b}^{3}} \cdot -0.5}{{b}^{2}}\right) + \frac{c}{{b}^{3}} \cdot -0.5\right), \frac{1}{b}\right) - \frac{b}{c}}} \]

13. Final simplification 92.1%

    \[\leadsto \frac{-1}{\frac{b}{c} - a \cdot \mathsf{fma}\left(a, -2 \cdot \left(\frac{c}{{b}^{3}} \cdot -0.5 + a \cdot \left(\mathsf{fma}\left(-0.125, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{c}^{2}}, \frac{{c}^{2}}{{b}^{5}}\right) - c \cdot \frac{\frac{c}{{b}^{3}} \cdot -0.5}{{b}^{2}}\right)\right), \frac{1}{b}\right)} \]
14. Add Preprocessing

Alternative 2: 91.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c ** 4.0d0) * (((-5.0d0) * ((a ** 3.0d0) / (b ** 7.0d0))) - (((2.0d0 * ((a ** 2.0d0) / (b ** 5.0d0))) + (a / (c * (b ** 3.0d0)))) / c))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((c ^ 4.0) * ((-5.0 * ((a ^ 3.0) / (b ^ 7.0))) - (((2.0 * ((a ^ 2.0) / (b ^ 5.0))) + (a / (c * (b ^ 3.0)))) / c))) - (c / b);
end

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. *-commutative 55.1%

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

3. Simplified 55.1%

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

5. Taylor expanded in a around 0 91.8%

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

6. Taylor expanded in c around -inf 91.8%

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

7. Final simplification 91.8%

   \[\leadsto {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b} \]
8. Add Preprocessing

Alternative 3: 88.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. *-commutative 55.1%

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

   2. +-commutative 55.1%

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

   3. sqr-neg 55.1%

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

   4. unsub-neg 55.1%

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

   5. sqr-neg 55.1%

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

   6. fma-neg 55.2%

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

   7. distribute-lft-neg-in 55.2%

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

   8. *-commutative 55.2%

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

   9. *-commutative 55.2%

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

   10. distribute-rgt-neg-in 55.2%

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

   11. metadata-eval 55.2%

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

3. Simplified 55.2%

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

5. Taylor expanded in a around inf 55.0%

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

   1. clear-num 55.0%

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

   2. inv-pow 55.0%

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

   3. *-commutative 55.0%

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

   4. fma-define 55.0%

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

7. Applied egg-rr 55.0%

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

   1. unpow-1 55.0%

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

   2. associate-/l* 55.0%

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

9. Simplified 55.0%

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

10. Taylor expanded in c around 0 89.0%

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

    1. neg-mul-1 89.0%

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

    2. +-commutative 89.0%

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

    3. unsub-neg 89.0%

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

    4. fma-define 89.0%

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

    5. distribute-rgt-out 89.0%

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

    6. metadata-eval 89.0%

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

12. Simplified 89.0%

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

13. Final simplification 89.0%

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

Alternative 4: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. *-commutative 55.1%

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

   2. +-commutative 55.1%

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

   3. sqr-neg 55.1%

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

   4. unsub-neg 55.1%

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

   5. sqr-neg 55.1%

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

   6. fma-neg 55.2%

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

   7. distribute-lft-neg-in 55.2%

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

   8. *-commutative 55.2%

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

   9. *-commutative 55.2%

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

   10. distribute-rgt-neg-in 55.2%

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

   11. metadata-eval 55.2%

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

3. Simplified 55.2%

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

5. Taylor expanded in a around inf 55.0%

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

   1. clear-num 55.0%

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

   2. inv-pow 55.0%

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

   3. *-commutative 55.0%

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

   4. fma-define 55.0%

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

7. Applied egg-rr 55.0%

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

   1. unpow-1 55.0%

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

   2. associate-/l* 55.0%

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

9. Simplified 55.0%

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

10. Taylor expanded in a around 0 89.0%

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

12. Simplified 89.0%

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

13. Final simplification 89.0%

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

Alternative 5: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 140:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 140.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ -1.0 (- (/ b c) (/ a b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 140.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

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) :: tmp
    if (b <= 140.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 140.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 140.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 140.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 140.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 140.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 140

   1. Initial program 77.8%

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

   if 140 < b

   1. Initial program 46.1%

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

      1. *-commutative 46.1%

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

      2. +-commutative 46.1%

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

      3. sqr-neg 46.1%

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

      4. unsub-neg 46.1%

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

      5. sqr-neg 46.1%

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

      6. fma-neg 46.1%

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

      7. distribute-lft-neg-in 46.1%

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

      8. *-commutative 46.1%

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

      9. *-commutative 46.1%

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

      10. distribute-rgt-neg-in 46.1%

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

      11. metadata-eval 46.1%

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

   3. Simplified 46.1%

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

   5. Taylor expanded in a around inf 46.0%

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

      1. clear-num 46.0%

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

      2. inv-pow 46.0%

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

      3. *-commutative 46.0%

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

      4. fma-define 46.0%

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

   7. Applied egg-rr 46.0%

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

      1. unpow-1 46.0%

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

      2. associate-/l* 46.0%

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

   9. Simplified 46.0%

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

   10. Taylor expanded in a around 0 88.8%

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

       1. neg-mul-1 88.8%

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

       2. distribute-frac-neg2 88.8%

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

       3. +-commutative 88.8%

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

       4. distribute-frac-neg2 88.8%

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

       5. unsub-neg 88.8%

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

   12. Simplified 88.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 140:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{b}{c} - \frac{a}{b}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. *-commutative 55.1%

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

   2. +-commutative 55.1%

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

   3. sqr-neg 55.1%

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

   4. unsub-neg 55.1%

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

   5. sqr-neg 55.1%

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

   6. fma-neg 55.2%

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

   7. distribute-lft-neg-in 55.2%

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

   8. *-commutative 55.2%

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

   9. *-commutative 55.2%

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

   10. distribute-rgt-neg-in 55.2%

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

   11. metadata-eval 55.2%

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

3. Simplified 55.2%

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

5. Taylor expanded in a around inf 55.0%

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

   1. clear-num 55.0%

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

   2. inv-pow 55.0%

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

   3. *-commutative 55.0%

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

   4. fma-define 55.0%

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

7. Applied egg-rr 55.0%

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

   1. unpow-1 55.0%

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

   2. associate-/l* 55.0%

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

9. Simplified 55.0%

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

10. Taylor expanded in a around 0 82.7%

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

    1. neg-mul-1 82.7%

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

    2. distribute-frac-neg2 82.7%

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

    3. +-commutative 82.7%

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

    4. distribute-frac-neg2 82.7%

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

    5. unsub-neg 82.7%

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

12. Simplified 82.7%

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

13. Final simplification 82.7%

    \[\leadsto \frac{-1}{\frac{b}{c} - \frac{a}{b}} \]
14. Add Preprocessing

Alternative 7: 64.5% 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 55.1%

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

   1. *-commutative 55.1%

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

3. Simplified 55.1%

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

5. Taylor expanded in b around inf 64.5%

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

   1. associate-*r/ 64.5%

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

   2. mul-1-neg 64.5%

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

7. Simplified 64.5%

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

8. Final simplification 64.5%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))