Quadratic roots, narrow range

Percentage Accurate: 55.6% → 91.2%

Time: 14.4s

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: 55.6% 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: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}}\\ \frac{1}{\left(\mathsf{fma}\left(-2, t_0 \cdot \left(-0.5 \cdot \left(a \cdot a\right)\right), \frac{a}{b}\right) - \frac{b}{c}\right) + \mathsf{fma}\left(-0.125, \frac{b}{c} \cdot \frac{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, \frac{4 \cdot {c}^{4}}{{b}^{6}}\right)}{c}, \frac{c \cdot c}{{b}^{5}} - \frac{c}{\frac{b}{t_0} \cdot \frac{b}{-0.5}}\right) \cdot \left(-2 \cdot {a}^{3}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 3.0))))
   (/
    1.0
    (+
     (- (fma -2.0 (* t_0 (* -0.5 (* a a))) (/ a b)) (/ b c))
     (*
      (fma
       -0.125
       (*
        (/ b c)
        (/
         (fma
          16.0
          (/ (pow c 4.0) (pow b 6.0))
          (/ (* 4.0 (pow c 4.0)) (pow b 6.0)))
         c))
       (- (/ (* c c) (pow b 5.0)) (/ c (* (/ b t_0) (/ b -0.5)))))
      (* -2.0 (pow a 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 3.0);
	return 1.0 / ((fma(-2.0, (t_0 * (-0.5 * (a * a))), (a / b)) - (b / c)) + (fma(-0.125, ((b / c) * (fma(16.0, (pow(c, 4.0) / pow(b, 6.0)), ((4.0 * pow(c, 4.0)) / pow(b, 6.0))) / c)), (((c * c) / pow(b, 5.0)) - (c / ((b / t_0) * (b / -0.5))))) * (-2.0 * pow(a, 3.0))));
}

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 3.0))
	return Float64(1.0 / Float64(Float64(fma(-2.0, Float64(t_0 * Float64(-0.5 * Float64(a * a))), Float64(a / b)) - Float64(b / c)) + Float64(fma(-0.125, Float64(Float64(b / c) * Float64(fma(16.0, Float64((c ^ 4.0) / (b ^ 6.0)), Float64(Float64(4.0 * (c ^ 4.0)) / (b ^ 6.0))) / c)), Float64(Float64(Float64(c * c) / (b ^ 5.0)) - Float64(c / Float64(Float64(b / t_0) * Float64(b / -0.5))))) * Float64(-2.0 * (a ^ 3.0)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(N[(-2.0 * N[(t$95$0 * N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.125 * N[(N[(b / c), $MachinePrecision] * N[(N[(16.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / N[(N[(b / t$95$0), $MachinePrecision] * N[(b / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. unsub-neg 54.4%

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

   4. fma-neg 54.5%

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

   5. associate-*l* 54.5%

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

   6. *-commutative 54.5%

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

   7. distribute-rgt-neg-in 54.5%

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

   8. metadata-eval 54.5%

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

3. Simplified 54.5%

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

   1. fma-udef 54.4%

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

   2. associate-*l* 54.4%

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

5. Applied egg-rr 54.4%

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

   1. add-cbrt-cube 54.4%

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

   2. fma-def 54.6%

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

   3. fma-def 54.6%

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

7. Applied egg-rr 54.5%

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

9. Simplified 54.5%

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

    1. rem-cbrt-cube 54.5%

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

    2. clear-num 54.5%

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

11. Applied egg-rr 54.5%

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

12. Taylor expanded in a around 0 92.9%

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

14. Simplified 93.0%

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

15. Final simplification 93.0%

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

Alternative 2: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot {a}^{3}\\ t_1 := {\left(c \cdot a\right)}^{4}\\ \frac{1}{\frac{a}{b} + \mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot -0.5\right)}{{b}^{3}}, \frac{-2 \cdot \left(\mathsf{fma}\left(-0.125, \frac{\mathsf{fma}\left(16, t_1, 4 \cdot t_1\right)}{c \cdot \left(c \cdot a\right)}, c \cdot t_0\right) - c \cdot \left(-0.5 \cdot t_0\right)\right)}{{b}^{5}} - \frac{b}{c}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (pow a 3.0))) (t_1 (pow (* c a) 4.0)))
   (/
    1.0
    (+
     (/ a b)
     (fma
      -2.0
      (/ (* (* a a) (* c -0.5)) (pow b 3.0))
      (-
       (/
        (*
         -2.0
         (-
          (fma -0.125 (/ (fma 16.0 t_1 (* 4.0 t_1)) (* c (* c a))) (* c t_0))
          (* c (* -0.5 t_0))))
        (pow b 5.0))
       (/ b c)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * pow(a, 3.0);
	double t_1 = pow((c * a), 4.0);
	return 1.0 / ((a / b) + fma(-2.0, (((a * a) * (c * -0.5)) / pow(b, 3.0)), (((-2.0 * (fma(-0.125, (fma(16.0, t_1, (4.0 * t_1)) / (c * (c * a))), (c * t_0)) - (c * (-0.5 * t_0)))) / pow(b, 5.0)) - (b / c))));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * (a ^ 3.0))
	t_1 = Float64(c * a) ^ 4.0
	return Float64(1.0 / Float64(Float64(a / b) + fma(-2.0, Float64(Float64(Float64(a * a) * Float64(c * -0.5)) / (b ^ 3.0)), Float64(Float64(Float64(-2.0 * Float64(fma(-0.125, Float64(fma(16.0, t_1, Float64(4.0 * t_1)) / Float64(c * Float64(c * a))), Float64(c * t_0)) - Float64(c * Float64(-0.5 * t_0)))) / (b ^ 5.0)) - Float64(b / c)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]}, N[(1.0 / N[(N[(a / b), $MachinePrecision] + N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * N[(N[(-0.125 * N[(N[(16.0 * t$95$1 + N[(4.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(c * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. unsub-neg 54.4%

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

   4. fma-neg 54.5%

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

   5. associate-*l* 54.5%

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

   6. *-commutative 54.5%

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

   7. distribute-rgt-neg-in 54.5%

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

   8. metadata-eval 54.5%

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

3. Simplified 54.5%

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

   1. fma-udef 54.4%

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

   2. associate-*l* 54.4%

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

5. Applied egg-rr 54.4%

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

   1. add-cbrt-cube 54.4%

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

   2. fma-def 54.6%

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

   3. fma-def 54.6%

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

7. Applied egg-rr 54.5%

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

9. Simplified 54.5%

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

    1. rem-cbrt-cube 54.5%

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

    2. clear-num 54.5%

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

11. Applied egg-rr 54.5%

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

12. Taylor expanded in b around inf 92.9%

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

14. Simplified 92.9%

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

15. Final simplification 92.9%

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

Alternative 3: 90.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. /-rgt-identity 54.4%

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

   2. metadata-eval 54.4%

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

   3. associate-/l* 54.4%

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

   4. associate-*r/ 54.4%

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

   5. +-commutative 54.4%

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

   6. unsub-neg 54.4%

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

   7. fma-neg 54.5%

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

   8. associate-*l* 54.5%

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

   9. *-commutative 54.5%

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

   10. distribute-rgt-neg-in 54.5%

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

   11. metadata-eval 54.5%

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

   12. associate-/r* 54.5%

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

   13. metadata-eval 54.5%

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

   14. metadata-eval 54.5%

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

3. Simplified 54.5%

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

4. Taylor expanded in a around 0 92.7%

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

6. Simplified 92.7%

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

7. Taylor expanded in b around 0 92.7%

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

   1. associate-/l* 92.7%

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

   2. distribute-rgt-out 92.7%

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

   3. metadata-eval 92.7%

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

9. Simplified 92.7%

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

10. Final simplification 92.7%

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

Alternative 4: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.033:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.033)
   (/ 1.0 (/ (* a 2.0) (- (sqrt (fma b b (* a (* c -4.0)))) b)))
   (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.033) {
		tmp = 1.0 / ((a * 2.0) / (sqrt(fma(b, b, (a * (c * -4.0)))) - b));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.033)
		tmp = Float64(1.0 / Float64(Float64(a * 2.0) / Float64(sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))) - b)));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.033], N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.033000000000000002

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. +-commutative 77.5%

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

      3. unsub-neg 77.5%

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

      4. fma-neg 77.8%

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

      5. associate-*l* 77.8%

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

      6. *-commutative 77.8%

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

      7. distribute-rgt-neg-in 77.8%

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

      8. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

      1. fma-udef 77.5%

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

      2. associate-*l* 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. add-cbrt-cube 77.5%

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

      2. fma-def 77.8%

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

      3. fma-def 77.9%

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

   7. Applied egg-rr 77.8%

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

   9. Simplified 77.8%

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

       1. rem-cbrt-cube 77.8%

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

       2. clear-num 77.8%

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

   11. Applied egg-rr 77.8%

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

   if -0.033000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 45.1%

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

      1. *-commutative 45.1%

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

      2. +-commutative 45.1%

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

      3. unsub-neg 45.1%

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

      4. fma-neg 45.2%

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

      5. associate-*l* 45.2%

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

      6. *-commutative 45.2%

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

      7. distribute-rgt-neg-in 45.2%

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

      8. metadata-eval 45.2%

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

   3. Simplified 45.2%

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

      1. fma-udef 45.1%

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

      2. associate-*l* 45.1%

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

   5. Applied egg-rr 45.1%

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

      1. add-cbrt-cube 45.1%

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

      2. fma-def 45.3%

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

      3. fma-def 45.4%

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

   7. Applied egg-rr 45.2%

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

   9. Simplified 45.2%

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

       1. rem-cbrt-cube 45.2%

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

       2. clear-num 45.2%

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

   11. Applied egg-rr 45.2%

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

   12. Taylor expanded in a around 0 90.0%

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

       1. mul-1-neg 90.0%

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

       2. unsub-neg 90.0%

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

   14. Simplified 90.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.033:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 5: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.033:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.033)
   (* (- (sqrt (fma b b (* (* c a) -4.0))) b) (/ 0.5 a))
   (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.033) {
		tmp = (sqrt(fma(b, b, ((c * a) * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.033)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.033], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.033000000000000002

   1. Initial program 77.5%

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

      1. /-rgt-identity 77.5%

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

      2. metadata-eval 77.5%

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

      3. associate-/l* 77.5%

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

      4. associate-*r/ 77.6%

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

      5. +-commutative 77.6%

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

      6. unsub-neg 77.6%

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

      7. fma-neg 77.8%

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

      8. associate-*l* 77.8%

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

      9. *-commutative 77.8%

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

      10. distribute-rgt-neg-in 77.8%

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

      11. metadata-eval 77.8%

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

      12. associate-/r* 77.8%

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

      13. metadata-eval 77.8%

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

      14. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

   if -0.033000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 45.1%

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

      1. *-commutative 45.1%

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

      2. +-commutative 45.1%

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

      3. unsub-neg 45.1%

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

      4. fma-neg 45.2%

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

      5. associate-*l* 45.2%

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

      6. *-commutative 45.2%

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

      7. distribute-rgt-neg-in 45.2%

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

      8. metadata-eval 45.2%

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

   3. Simplified 45.2%

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

      1. fma-udef 45.1%

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

      2. associate-*l* 45.1%

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

   5. Applied egg-rr 45.1%

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

      1. add-cbrt-cube 45.1%

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

      2. fma-def 45.3%

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

      3. fma-def 45.4%

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

   7. Applied egg-rr 45.2%

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

   9. Simplified 45.2%

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

       1. rem-cbrt-cube 45.2%

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

       2. clear-num 45.2%

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

   11. Applied egg-rr 45.2%

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

   12. Taylor expanded in a around 0 90.0%

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

       1. mul-1-neg 90.0%

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

       2. unsub-neg 90.0%

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

   14. Simplified 90.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.033:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 6: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.033:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.033)
   (* (/ 0.5 a) (- (sqrt (+ (* b b) (* a (* c -4.0)))) b))
   (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.033) {
		tmp = (0.5 / a) * (sqrt(((b * b) + (a * (c * -4.0)))) - b);
	} else {
		tmp = 1.0 / ((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 (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-0.033d0)) then
        tmp = (0.5d0 / a) * (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b)
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.033) {
		tmp = (0.5 / a) * (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.033:
		tmp = (0.5 / a) * (math.sqrt(((b * b) + (a * (c * -4.0)))) - b)
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.033)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.033)
		tmp = (0.5 / a) * (sqrt(((b * b) + (a * (c * -4.0)))) - b);
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.033], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.033000000000000002

   1. Initial program 77.5%

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

      1. /-rgt-identity 77.5%

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

      2. metadata-eval 77.5%

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

      3. associate-/l* 77.5%

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

      4. associate-*r/ 77.6%

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

      5. +-commutative 77.6%

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

      6. unsub-neg 77.6%

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

      7. fma-neg 77.8%

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

      8. associate-*l* 77.8%

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

      9. *-commutative 77.8%

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

      10. distribute-rgt-neg-in 77.8%

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

      11. metadata-eval 77.8%

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

      12. associate-/r* 77.8%

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

      13. metadata-eval 77.8%

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

      14. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

      1. fma-udef 77.5%

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

      2. associate-*l* 77.5%

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

   5. Applied egg-rr 77.6%

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

   if -0.033000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 45.1%

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

      1. *-commutative 45.1%

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

      2. +-commutative 45.1%

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

      3. unsub-neg 45.1%

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

      4. fma-neg 45.2%

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

      5. associate-*l* 45.2%

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

      6. *-commutative 45.2%

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

      7. distribute-rgt-neg-in 45.2%

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

      8. metadata-eval 45.2%

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

   3. Simplified 45.2%

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

      1. fma-udef 45.1%

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

      2. associate-*l* 45.1%

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

   5. Applied egg-rr 45.1%

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

      1. add-cbrt-cube 45.1%

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

      2. fma-def 45.3%

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

      3. fma-def 45.4%

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

   7. Applied egg-rr 45.2%

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

   9. Simplified 45.2%

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

       1. rem-cbrt-cube 45.2%

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

       2. clear-num 45.2%

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

   11. Applied egg-rr 45.2%

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

   12. Taylor expanded in a around 0 90.0%

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

       1. mul-1-neg 90.0%

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

       2. unsub-neg 90.0%

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

   14. Simplified 90.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.033:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 7: 88.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. unsub-neg 54.4%

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

   4. fma-neg 54.5%

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

   5. associate-*l* 54.5%

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

   6. *-commutative 54.5%

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

   7. distribute-rgt-neg-in 54.5%

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

   8. metadata-eval 54.5%

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

3. Simplified 54.5%

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

   1. fma-udef 54.4%

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

   2. associate-*l* 54.4%

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

5. Applied egg-rr 54.4%

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

   1. add-cbrt-cube 54.4%

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

   2. fma-def 54.6%

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

   3. fma-def 54.6%

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

7. Applied egg-rr 54.5%

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

9. Simplified 54.5%

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

    1. rem-cbrt-cube 54.5%

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

    2. clear-num 54.5%

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

11. Applied egg-rr 54.5%

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

12. Taylor expanded in a around 0 89.7%

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

    1. associate-+r+ 89.7%

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

    2. mul-1-neg 89.7%

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

    3. unsub-neg 89.7%

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

    4. fma-def 89.7%

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

    5. distribute-rgt-out 89.7%

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

    6. metadata-eval 89.7%

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

    7. associate-*l* 89.7%

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

    8. unpow2 89.7%

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

14. Simplified 89.7%

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

15. Final simplification 89.7%

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

Alternative 8: 88.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. unsub-neg 54.4%

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

   4. fma-neg 54.5%

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

   5. associate-*l* 54.5%

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

   6. *-commutative 54.5%

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

   7. distribute-rgt-neg-in 54.5%

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

   8. metadata-eval 54.5%

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

3. Simplified 54.5%

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

   1. fma-udef 54.4%

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

   2. associate-*l* 54.4%

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

5. Applied egg-rr 54.4%

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

   1. add-cbrt-cube 54.4%

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

   2. fma-def 54.6%

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

   3. fma-def 54.6%

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

7. Applied egg-rr 54.5%

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

9. Simplified 54.5%

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

    1. rem-cbrt-cube 54.5%

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

    2. clear-num 54.5%

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

11. Applied egg-rr 54.5%

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

12. Taylor expanded in a around 0 89.7%

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

    1. fma-def 89.7%

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

    2. *-commutative 89.7%

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

    3. unpow2 89.7%

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

    4. distribute-rgt-out 89.7%

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

    5. metadata-eval 89.7%

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

    6. +-commutative 89.7%

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

    7. associate-*r/ 89.7%

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

    8. mul-1-neg 89.7%

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

14. Simplified 89.7%

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

15. Final simplification 89.7%

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

Alternative 9: 81.9% 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 54.4%

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

   1. *-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. unsub-neg 54.4%

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

   4. fma-neg 54.5%

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

   5. associate-*l* 54.5%

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

   6. *-commutative 54.5%

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

   7. distribute-rgt-neg-in 54.5%

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

   8. metadata-eval 54.5%

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

3. Simplified 54.5%

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

   1. fma-udef 54.4%

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

   2. associate-*l* 54.4%

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

5. Applied egg-rr 54.4%

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

   1. add-cbrt-cube 54.4%

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

   2. fma-def 54.6%

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

   3. fma-def 54.6%

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

7. Applied egg-rr 54.5%

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

9. Simplified 54.5%

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

    1. rem-cbrt-cube 54.5%

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

    2. clear-num 54.5%

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

11. Applied egg-rr 54.5%

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

12. Taylor expanded in a around 0 83.3%

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

    1. mul-1-neg 83.3%

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

    2. unsub-neg 83.3%

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

14. Simplified 83.3%

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

15. Final simplification 83.3%

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

Alternative 10: 64.2% 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 54.4%

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

   1. /-rgt-identity 54.4%

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

   2. metadata-eval 54.4%

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

   3. associate-/l* 54.4%

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

   4. associate-*r/ 54.4%

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

   5. +-commutative 54.4%

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

   6. unsub-neg 54.4%

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

   7. fma-neg 54.5%

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

   8. associate-*l* 54.5%

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

   9. *-commutative 54.5%

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

   10. distribute-rgt-neg-in 54.5%

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

   11. metadata-eval 54.5%

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

   12. associate-/r* 54.5%

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

   13. metadata-eval 54.5%

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

   14. metadata-eval 54.5%

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

3. Simplified 54.5%

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

4. Taylor expanded in b around inf 65.5%

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

   1. associate-*r/ 65.5%

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

   2. neg-mul-1 65.5%

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

6. Simplified 65.5%

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

7. Final simplification 65.5%

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

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 54.4%

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

   1. add-log-exp 50.3%

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

   2. neg-mul-1 50.3%

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

   3. fma-def 50.3%

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

   4. *-commutative 50.3%

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

   5. *-commutative 50.3%

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

   6. *-commutative 50.3%

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

3. Applied egg-rr 50.3%

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

4. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
5. 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%

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

6. Simplified 3.2%

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

7. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023208 
(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)))