Quadratic roots, narrow range

Percentage Accurate: 55.1% → 91.1%

Time: 15.1s

Alternatives: 11

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: 55.1% 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: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\left({c}^{4} \cdot 20\right) \cdot \frac{a}{{b}^{6}}}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (+
      (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
      (* -0.25 (/ (* (* (pow c 4.0) 20.0) (/ a (pow b 6.0))) b))))
    (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-0.25 * (((pow(c, 4.0) * 20.0) * (a / pow(b, 6.0))) / b)))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-0.25d0) * ((((c ** 4.0d0) * 20.0d0) * (a / (b ** 6.0d0))) / b)))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-0.25 * (((Math.pow(c, 4.0) * 20.0) * (a / Math.pow(b, 6.0))) / b)))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((a * ((-2.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-0.25 * (((math.pow(c, 4.0) * 20.0) * (a / math.pow(b, 6.0))) / b)))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-0.25 * Float64(Float64(Float64((c ^ 4.0) * 20.0) * Float64(a / (b ^ 6.0))) / b)))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((a * ((-2.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-0.25 * ((((c ^ 4.0) * 20.0) * (a / (b ^ 6.0))) / b)))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision] * N[(a / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

   1. *-commutative 55.6%

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

3. Simplified 55.6%

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

5. Taylor expanded in a around 0 92.1%

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

6. Taylor expanded in b around 0 92.1%

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

   1. *-commutative 92.1%

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

   2. associate-/l* 92.1%

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

   3. distribute-rgt-out 92.1%

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

   4. metadata-eval 92.1%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\left({c}^{4} \cdot \color{blue}{20}\right) \cdot \frac{a}{{b}^{6}}}{b}\right)\right) \]

8. Simplified 92.1%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot 20\right) \cdot \frac{a}{{b}^{6}}}}{b}\right)\right) \]

9. Final simplification 92.1%

   \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\left({c}^{4} \cdot 20\right) \cdot \frac{a}{{b}^{6}}}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 2: 90.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. *-commutative 55.6%

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

   2. +-commutative 55.6%

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

   3. sqr-neg 55.6%

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

   4. unsub-neg 55.6%

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

   5. sqr-neg 55.6%

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

   6. fma-neg 55.7%

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

   7. distribute-lft-neg-in 55.7%

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

   8. *-commutative 55.7%

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

   9. *-commutative 55.7%

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

   10. distribute-rgt-neg-in 55.7%

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

   11. metadata-eval 55.7%

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

3. Simplified 55.7%

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

5. Taylor expanded in a around inf 55.3%

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

6. Taylor expanded in c around 0 92.0%

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

   1. fma-neg 92.0%

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

8. Simplified 92.0%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}}}{b} \cdot \frac{20}{a}\right)\right)\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right)} \]

9. Taylor expanded in a around inf 92.0%

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

Alternative 3: 88.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. *-commutative 55.6%

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

3. Simplified 55.6%

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

5. Taylor expanded in c around 0 88.4%

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

   1. clear-num 88.4%

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

   2. inv-pow 88.4%

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

7. Applied egg-rr 88.4%

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

   1. unpow-1 88.4%

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

   2. times-frac 88.3%

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

   3. associate-*r* 88.3%

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

   4. associate-*r/ 88.3%

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

9. Simplified 88.3%

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

10. Taylor expanded in c around 0 89.2%

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

12. Simplified 89.2%

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

13. Final simplification 89.2%

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

Alternative 4: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.295:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.295)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ 1.0 (/ (- (/ (* c a) b) b) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.295) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.295)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * a) / b) - b) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.295], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2949999999999999

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. +-commutative 82.0%

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

      3. sqr-neg 82.0%

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

      4. unsub-neg 82.0%

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

      5. sqr-neg 82.0%

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

      6. fma-neg 82.0%

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

      7. distribute-lft-neg-in 82.0%

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

      8. *-commutative 82.0%

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

      9. *-commutative 82.0%

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

      10. distribute-rgt-neg-in 82.0%

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

      11. metadata-eval 82.0%

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

   3. Simplified 82.0%

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

   if 2.2949999999999999 < b

   1. Initial program 50.0%

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

      1. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in c around 0 91.8%

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

      1. clear-num 91.8%

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

      2. inv-pow 91.8%

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

   7. Applied egg-rr 91.8%

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

      1. unpow-1 91.8%

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

      2. times-frac 91.7%

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

      3. associate-*r* 91.7%

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

      4. associate-*r/ 91.7%

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

   9. Simplified 91.7%

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

   10. Taylor expanded in c around 0 87.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.295:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. *-commutative 55.6%

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

3. Simplified 55.6%

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

5. Taylor expanded in c around 0 88.4%

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

   1. clear-num 88.4%

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

   2. inv-pow 88.4%

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

7. Applied egg-rr 88.4%

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

   1. unpow-1 88.4%

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

   2. times-frac 88.3%

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

   3. associate-*r* 88.3%

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

   4. associate-*r/ 88.3%

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

9. Simplified 88.3%

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

10. Taylor expanded in a around 0 89.2%

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

    1. +-commutative 89.2%

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

    2. fma-define 89.2%

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

    3. +-commutative 89.2%

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

    4. *-commutative 89.2%

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

    5. associate-*l* 89.2%

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

    6. distribute-rgt-out 89.2%

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

    7. metadata-eval 89.2%

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

    8. associate-*l* 89.2%

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

    9. metadata-eval 89.2%

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

    10. mul-1-neg 89.2%

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

    11. distribute-frac-neg2 89.2%

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

12. Simplified 89.2%

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

13. Final simplification 89.2%

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

Alternative 6: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.295:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.295)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ 1.0 (/ (- (/ (* c a) b) b) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.295) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	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 <= 2.295d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / ((((c * a) / b) - b) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.295) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.295:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / ((((c * a) / b) - b) / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.295)
		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(Float64(Float64(c * a) / b) - b) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.295)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.295], 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[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2949999999999999

   1. Initial program 82.0%

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

   if 2.2949999999999999 < b

   1. Initial program 50.0%

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

      1. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in c around 0 91.8%

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

      1. clear-num 91.8%

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

      2. inv-pow 91.8%

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

   7. Applied egg-rr 91.8%

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

      1. unpow-1 91.8%

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

      2. times-frac 91.7%

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

      3. associate-*r* 91.7%

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

      4. associate-*r/ 91.7%

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

   9. Simplified 91.7%

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

   10. Taylor expanded in c around 0 87.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.295:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

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 / ((((c * a) / b) - b) / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. *-commutative 55.6%

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

3. Simplified 55.6%

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

5. Taylor expanded in c around 0 88.4%

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

   1. clear-num 88.4%

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

   2. inv-pow 88.4%

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

7. Applied egg-rr 88.4%

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

   1. unpow-1 88.4%

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

   2. times-frac 88.3%

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

   3. associate-*r* 88.3%

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

   4. associate-*r/ 88.3%

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

9. Simplified 88.3%

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

10. Taylor expanded in c around 0 82.6%

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

11. Final simplification 82.6%

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

Alternative 8: 82.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

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 / ((((c / b) * a) - b) / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. *-commutative 55.6%

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

3. Simplified 55.6%

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

5. Taylor expanded in c around 0 88.4%

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

   1. clear-num 88.4%

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

   2. inv-pow 88.4%

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

7. Applied egg-rr 88.4%

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

   1. unpow-1 88.4%

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

   2. times-frac 88.3%

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

   3. associate-*r* 88.3%

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

   4. associate-*r/ 88.3%

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

9. Simplified 88.3%

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

10. Taylor expanded in c around 0 82.6%

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

    1. associate-*r/ 82.6%

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

    2. +-commutative 82.6%

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

    3. mul-1-neg 82.6%

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

    4. unsub-neg 82.6%

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

12. Simplified 82.6%

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

13. Final simplification 82.6%

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

Alternative 9: 82.4% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

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) - (b / c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. *-commutative 55.6%

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

3. Simplified 55.6%

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

5. Taylor expanded in c around 0 88.4%

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

   1. clear-num 88.4%

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

   2. inv-pow 88.4%

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

7. Applied egg-rr 88.4%

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

   1. unpow-1 88.4%

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

   2. times-frac 88.3%

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

   3. associate-*r* 88.3%

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

   4. associate-*r/ 88.3%

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

9. Simplified 88.3%

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

10. Taylor expanded in a around 0 82.6%

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

    1. +-commutative 82.6%

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

    2. mul-1-neg 82.6%

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

    3. unsub-neg 82.6%

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

12. Simplified 82.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
13. Add Preprocessing

Alternative 10: 64.7% 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 55.6%

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

   1. *-commutative 55.6%

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

3. Simplified 55.6%

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

5. Taylor expanded in b around inf 64.2%

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

   1. associate-*r/ 64.2%

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

   2. mul-1-neg 64.2%

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

7. Simplified 64.2%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
8. Add Preprocessing

Alternative 11: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

   1. *-commutative 55.6%

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

3. Simplified 55.6%

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

5. Taylor expanded in b around inf 55.6%

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

   1. expm1-log1p-u 43.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{{b}^{2} \cdot \left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{a \cdot 2}\right)\right)} \]

   2. expm1-undefine 41.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{{b}^{2} \cdot \left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{a \cdot 2}\right)} - 1} \]

7. Applied egg-rr 41.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \mathsf{fma}\left(-4, a \cdot \frac{c}{{b}^{2}}, 1\right)}\right)}{2 \cdot a}\right)} - 1} \]

8. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

10. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(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)))