Quadratic roots, narrow range

Percentage Accurate: 55.7% → 91.9%

Time: 13.7s

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: 55.7% 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: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)\\ t_1 := c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_1 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_1}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \mathsf{fma}\left(-0.25, \frac{t_0 \cdot t_0 + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-1, \frac{c}{b}, -2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) (* (* a a) -2.0))) (t_1 (* c (* a 4.0))))
   (if (<= b 0.82)
     (/
      (/ (+ (pow (- b) 2.0) (- t_1 (* b b))) (- (- b) (sqrt (- (* b b) t_1))))
      (* 2.0 a))
     (fma
      -1.0
      (/ (* a (* c c)) (pow b 3.0))
      (fma
       -0.25
       (/
        (+ (* t_0 t_0) (* 16.0 (* (pow c 4.0) (pow a 4.0))))
        (* a (pow b 7.0)))
       (fma -1.0 (/ c b) (* -2.0 (/ (* (* a a) (pow c 3.0)) (pow b 5.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * ((a * a) * -2.0);
	double t_1 = c * (a * 4.0);
	double tmp;
	if (b <= 0.82) {
		tmp = ((pow(-b, 2.0) + (t_1 - (b * b))) / (-b - sqrt(((b * b) - t_1)))) / (2.0 * a);
	} else {
		tmp = fma(-1.0, ((a * (c * c)) / pow(b, 3.0)), fma(-0.25, (((t_0 * t_0) + (16.0 * (pow(c, 4.0) * pow(a, 4.0)))) / (a * pow(b, 7.0))), fma(-1.0, (c / b), (-2.0 * (((a * a) * pow(c, 3.0)) / pow(b, 5.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * Float64(Float64(a * a) * -2.0))
	t_1 = Float64(c * Float64(a * 4.0))
	tmp = 0.0
	if (b <= 0.82)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_1 - Float64(b * b))) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_1)))) / Float64(2.0 * a));
	else
		tmp = fma(-1.0, Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)), fma(-0.25, Float64(Float64(Float64(t_0 * t_0) + Float64(16.0 * Float64((c ^ 4.0) * (a ^ 4.0)))) / Float64(a * (b ^ 7.0))), fma(-1.0, Float64(c / b), Float64(-2.0 * Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 5.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.82], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$1 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(16.0 * N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(c / b), $MachinePrecision] + N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.819999999999999951

   1. Initial program 84.4%

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

      1. flip-+ 85.3%

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

      2. pow2 85.3%

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

      3. add-sqr-sqrt 86.6%

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

      4. *-commutative 86.6%

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

      5. *-commutative 86.6%

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

      6. *-commutative 86.6%

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

      7. *-commutative 86.6%

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

   3. Applied egg-rr 86.6%

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

   if 0.819999999999999951 < b

   1. Initial program 52.0%

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

      1. neg-sub0 52.0%

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

      2. associate-+l- 52.0%

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

      3. sub0-neg 52.0%

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

      4. neg-mul-1 52.0%

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

      5. associate-*l/ 52.0%

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

      6. *-commutative 52.0%

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

      7. associate-/r* 52.0%

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

      8. /-rgt-identity 52.0%

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

      9. metadata-eval 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in b around inf 93.7%

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

      1. fma-def 93.7%

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

      2. *-commutative 93.7%

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

      3. unpow2 93.7%

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

      4. fma-def 93.7%

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

   6. Simplified 93.7%

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

      1. unpow2 93.7%

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

      2. associate-*l* 93.7%

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

      3. associate-*l* 93.7%

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

   8. Applied egg-rr 93.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(a \cdot 4\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \mathsf{fma}\left(-0.25, \frac{\left(\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-1, \frac{c}{b}, -2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right)\right)\\ \end{array} \]

Alternative 2: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 0.8:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))))
   (if (<= b 0.8)
     (/
      (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) (sqrt (- (* b b) t_0))))
      (* 2.0 a))
     (-
      (-
       (fma
        -0.25
        (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 20.0)))
        (* -2.0 (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))))
       (/ c b))
      (/ (* c (* c a)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (b <= 0.8) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0 * a);
	} else {
		tmp = (fma(-0.25, (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0))), (-2.0 * ((a * a) / (pow(b, 5.0) / pow(c, 3.0))))) - (c / b)) - ((c * (c * a)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	tmp = 0.0
	if (b <= 0.8)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_0)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(-0.25, Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 20.0))), Float64(-2.0 * Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))))) - Float64(c / b)) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.8], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 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[(-2.0 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.80000000000000004

   1. Initial program 84.4%

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

      1. flip-+ 85.3%

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

      2. pow2 85.3%

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

      3. add-sqr-sqrt 86.6%

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

      4. *-commutative 86.6%

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

      5. *-commutative 86.6%

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

      6. *-commutative 86.6%

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

      7. *-commutative 86.6%

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

   3. Applied egg-rr 86.6%

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

   if 0.80000000000000004 < b

   1. Initial program 52.0%

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

      1. neg-sub0 52.0%

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

      2. associate-+l- 52.0%

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

      3. sub0-neg 52.0%

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

      4. neg-mul-1 52.0%

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

      5. associate-*l/ 52.0%

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

      6. *-commutative 52.0%

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

      7. associate-/r* 52.0%

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

      8. /-rgt-identity 52.0%

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

      9. metadata-eval 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in a around 0 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in b around 0 93.7%

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

      1. associate-/l* 93.7%

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

      2. distribute-rgt-out 93.7%

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

      3. metadata-eval 93.7%

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

   9. Simplified 93.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.8:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(a \cdot 4\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 3: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))))
   (if (<= b 1.8)
     (/
      (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) (sqrt (- (* b b) t_0))))
      (* 2.0 a))
     (-
      (- (* -2.0 (* (* a a) (/ (pow c 3.0) (pow b 5.0)))) (/ c b))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (b <= 1.8) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0 * a);
	} else {
		tmp = ((-2.0 * ((a * a) * (pow(c, 3.0) / pow(b, 5.0)))) - (c / b)) - ((c * c) / (pow(b, 3.0) / a));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = c * (a * 4.0d0)
    if (b <= 1.8d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0d0 * a)
    else
        tmp = (((-2.0d0) * ((a * a) * ((c ** 3.0d0) / (b ** 5.0d0)))) - (c / b)) - ((c * c) / ((b ** 3.0d0) / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (b <= 1.8) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - Math.sqrt(((b * b) - t_0)))) / (2.0 * a);
	} else {
		tmp = ((-2.0 * ((a * a) * (Math.pow(c, 3.0) / Math.pow(b, 5.0)))) - (c / b)) - ((c * c) / (Math.pow(b, 3.0) / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 4.0)
	tmp = 0
	if b <= 1.8:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - math.sqrt(((b * b) - t_0)))) / (2.0 * a)
	else:
		tmp = ((-2.0 * ((a * a) * (math.pow(c, 3.0) / math.pow(b, 5.0)))) - (c / b)) - ((c * c) / (math.pow(b, 3.0) / a))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	tmp = 0.0
	if (b <= 1.8)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_0)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(a * a) * Float64((c ^ 3.0) / (b ^ 5.0)))) - Float64(c / b)) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 4.0);
	tmp = 0.0;
	if (b <= 1.8)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0 * a);
	else
		tmp = ((-2.0 * ((a * a) * ((c ^ 3.0) / (b ^ 5.0)))) - (c / b)) - ((c * c) / ((b ^ 3.0) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.8], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 83.9%

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

      1. flip-+ 84.7%

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

      2. pow2 84.7%

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

      3. add-sqr-sqrt 85.9%

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

      4. *-commutative 85.9%

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

      5. *-commutative 85.9%

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

      6. *-commutative 85.9%

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

      7. *-commutative 85.9%

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

   3. Applied egg-rr 85.9%

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

   if 1.80000000000000004 < b

   1. Initial program 51.4%

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

      1. neg-sub0 51.4%

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

      2. associate-+l- 51.4%

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

      3. sub0-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-*l/ 51.4%

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

      6. *-commutative 51.4%

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

      7. associate-/r* 51.4%

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

      8. /-rgt-identity 51.4%

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

      9. metadata-eval 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in b around inf 90.9%

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

      1. associate-+r+ 90.9%

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

      2. distribute-lft-out 90.9%

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

      3. fma-def 90.9%

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

      4. *-commutative 90.9%

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

      5. associate-/l* 90.8%

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

      6. *-commutative 90.8%

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

      7. associate-/l* 90.8%

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

      8. unpow2 90.8%

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

      9. unpow2 90.8%

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

      10. cube-prod 90.8%

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

   6. Simplified 90.8%

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

      1. unpow3 91.1%

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

      2. unswap-sqr 91.1%

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

      3. associate-*r* 91.1%

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

      4. *-commutative 91.1%

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

      5. associate-*r* 91.1%

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

      6. *-commutative 91.1%

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

   8. Applied egg-rr 90.8%

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

   9. Taylor expanded in a around 0 91.3%

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

       1. +-commutative 91.3%

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

       2. mul-1-neg 91.3%

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

       3. unsub-neg 91.3%

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

       4. +-commutative 91.3%

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

       5. mul-1-neg 91.3%

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

       6. unsub-neg 91.3%

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

       7. unpow2 91.3%

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

       8. associate-/l* 91.3%

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

       9. associate-/r/ 91.3%

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

       10. unpow2 91.3%

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

       11. associate-/l* 91.3%

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

   11. Simplified 91.3%

       \[\leadsto \color{blue}{\left(-2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(a \cdot 4\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 4: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(c \cdot \left(\frac{a}{b} + c \cdot \left(a \cdot \frac{a}{{b}^{3}}\right)\right)\right) + -4 \cdot \frac{\left(c \cdot a\right) \cdot \left(a \cdot \left(c \cdot \left(c \cdot a\right)\right)\right)}{{b}^{5}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))))
   (if (<= b 1.8)
     (/
      (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) (sqrt (- (* b b) t_0))))
      (* 2.0 a))
     (/
      (+
       (* -2.0 (* c (+ (/ a b) (* c (* a (/ a (pow b 3.0)))))))
       (* -4.0 (/ (* (* c a) (* a (* c (* c a)))) (pow b 5.0))))
      (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (b <= 1.8) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0 * a);
	} else {
		tmp = ((-2.0 * (c * ((a / b) + (c * (a * (a / pow(b, 3.0))))))) + (-4.0 * (((c * a) * (a * (c * (c * a)))) / pow(b, 5.0)))) / (2.0 * a);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = c * (a * 4.0d0)
    if (b <= 1.8d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0d0 * a)
    else
        tmp = (((-2.0d0) * (c * ((a / b) + (c * (a * (a / (b ** 3.0d0))))))) + ((-4.0d0) * (((c * a) * (a * (c * (c * a)))) / (b ** 5.0d0)))) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (b <= 1.8) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - Math.sqrt(((b * b) - t_0)))) / (2.0 * a);
	} else {
		tmp = ((-2.0 * (c * ((a / b) + (c * (a * (a / Math.pow(b, 3.0))))))) + (-4.0 * (((c * a) * (a * (c * (c * a)))) / Math.pow(b, 5.0)))) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 4.0)
	tmp = 0
	if b <= 1.8:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - math.sqrt(((b * b) - t_0)))) / (2.0 * a)
	else:
		tmp = ((-2.0 * (c * ((a / b) + (c * (a * (a / math.pow(b, 3.0))))))) + (-4.0 * (((c * a) * (a * (c * (c * a)))) / math.pow(b, 5.0)))) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	tmp = 0.0
	if (b <= 1.8)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_0)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-2.0 * Float64(c * Float64(Float64(a / b) + Float64(c * Float64(a * Float64(a / (b ^ 3.0))))))) + Float64(-4.0 * Float64(Float64(Float64(c * a) * Float64(a * Float64(c * Float64(c * a)))) / (b ^ 5.0)))) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 4.0);
	tmp = 0.0;
	if (b <= 1.8)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0 * a);
	else
		tmp = ((-2.0 * (c * ((a / b) + (c * (a * (a / (b ^ 3.0))))))) + (-4.0 * (((c * a) * (a * (c * (c * a)))) / (b ^ 5.0)))) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.8], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(c * N[(N[(a / b), $MachinePrecision] + N[(c * N[(a * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(N[(c * a), $MachinePrecision] * N[(a * N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 83.9%

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

      1. flip-+ 84.7%

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

      2. pow2 84.7%

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

      3. add-sqr-sqrt 85.9%

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

      4. *-commutative 85.9%

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

      5. *-commutative 85.9%

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

      6. *-commutative 85.9%

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

      7. *-commutative 85.9%

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

   3. Applied egg-rr 85.9%

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

   if 1.80000000000000004 < b

   1. Initial program 51.4%

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

      1. *-commutative 51.4%

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

      2. +-commutative 51.4%

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

      3. unsub-neg 51.4%

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

      4. fma-neg 51.5%

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

      5. associate-*l* 51.5%

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

      6. *-commutative 51.5%

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

      7. distribute-rgt-neg-in 51.5%

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

      8. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

      1. fma-udef 51.4%

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

      2. *-commutative 51.4%

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

   5. Applied egg-rr 51.4%

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

   6. Taylor expanded in b around inf 90.9%

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

      1. +-commutative 90.9%

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

      2. associate-+r+ 90.9%

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

   8. Simplified 91.1%

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

      1. unpow3 91.1%

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

      2. unswap-sqr 91.1%

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

      3. associate-*r* 91.1%

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

      4. *-commutative 91.1%

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

      5. associate-*r* 91.1%

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

      6. *-commutative 91.1%

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

   10. Applied egg-rr 91.1%

       \[\leadsto \frac{-2 \cdot \left(c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a}{{b}^{3}} \cdot a\right)\right)\right) + -4 \cdot \frac{\color{blue}{\left(\left(\left(c \cdot a\right) \cdot c\right) \cdot a\right) \cdot \left(c \cdot a\right)}}{{b}^{5}}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(a \cdot 4\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(c \cdot \left(\frac{a}{b} + c \cdot \left(a \cdot \frac{a}{{b}^{3}}\right)\right)\right) + -4 \cdot \frac{\left(c \cdot a\right) \cdot \left(a \cdot \left(c \cdot \left(c \cdot a\right)\right)\right)}{{b}^{5}}}{2 \cdot a}\\ \end{array} \]

Alternative 5: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))))
   (if (<= b 1.8)
     (/
      (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) (sqrt (- (* b b) t_0))))
      (* 2.0 a))
     (- (/ (- c) b) (/ (* c (* c a)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (b <= 1.8) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0 * a);
	} else {
		tmp = (-c / b) - ((c * (c * a)) / pow(b, 3.0));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = c * (a * 4.0d0)
    if (b <= 1.8d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0d0 * a)
    else
        tmp = (-c / b) - ((c * (c * a)) / (b ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (b <= 1.8) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - Math.sqrt(((b * b) - t_0)))) / (2.0 * a);
	} else {
		tmp = (-c / b) - ((c * (c * a)) / Math.pow(b, 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 4.0)
	tmp = 0
	if b <= 1.8:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - math.sqrt(((b * b) - t_0)))) / (2.0 * a)
	else:
		tmp = (-c / b) - ((c * (c * a)) / math.pow(b, 3.0))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	tmp = 0.0
	if (b <= 1.8)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_0)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 4.0);
	tmp = 0.0;
	if (b <= 1.8)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (2.0 * a);
	else
		tmp = (-c / b) - ((c * (c * a)) / (b ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.8], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 83.9%

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

      1. flip-+ 84.7%

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

      2. pow2 84.7%

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

      3. add-sqr-sqrt 85.9%

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

      4. *-commutative 85.9%

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

      5. *-commutative 85.9%

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

      6. *-commutative 85.9%

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

      7. *-commutative 85.9%

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

   3. Applied egg-rr 85.9%

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

   if 1.80000000000000004 < b

   1. Initial program 51.4%

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

      1. neg-sub0 51.4%

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

      2. associate-+l- 51.4%

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

      3. sub0-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-*l/ 51.4%

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

      6. *-commutative 51.4%

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

      7. associate-/r* 51.4%

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

      8. /-rgt-identity 51.4%

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

      9. metadata-eval 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in b around inf 85.8%

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

      1. +-commutative 85.8%

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

      2. mul-1-neg 85.8%

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

      3. unsub-neg 85.8%

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

      4. associate-*r/ 85.8%

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

      5. neg-mul-1 85.8%

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

      6. unpow2 85.8%

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

      7. associate-*l* 85.8%

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

   6. Simplified 85.8%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(a \cdot 4\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8)
   (* (- (sqrt (fma b b (* (* c a) -4.0))) b) (/ 0.5 a))
   (- (/ (- c) b) (/ (* c (* c a)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8) {
		tmp = (sqrt(fma(b, b, ((c * a) * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = (-c / b) - ((c * (c * a)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8], 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[(N[((-c) / b), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 83.9%

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

      1. /-rgt-identity 83.9%

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

      2. metadata-eval 83.9%

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

      3. associate-/l* 83.9%

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

      4. associate-*r/ 83.9%

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

      5. +-commutative 83.9%

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

      6. unsub-neg 83.9%

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

      7. fma-neg 84.1%

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

      8. associate-*l* 84.1%

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

      9. *-commutative 84.1%

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

      10. distribute-rgt-neg-in 84.1%

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

      11. metadata-eval 84.1%

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

      12. associate-/r* 84.1%

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

      13. metadata-eval 84.1%

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

      14. metadata-eval 84.1%

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

   3. Simplified 84.1%

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

   if 1.80000000000000004 < b

   1. Initial program 51.4%

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

      1. neg-sub0 51.4%

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

      2. associate-+l- 51.4%

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

      3. sub0-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-*l/ 51.4%

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

      6. *-commutative 51.4%

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

      7. associate-/r* 51.4%

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

      8. /-rgt-identity 51.4%

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

      9. metadata-eval 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in b around inf 85.8%

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

      1. +-commutative 85.8%

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

      2. mul-1-neg 85.8%

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

      3. unsub-neg 85.8%

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

      4. associate-*r/ 85.8%

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

      5. neg-mul-1 85.8%

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

      6. unpow2 85.8%

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

      7. associate-*l* 85.8%

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

   6. Simplified 85.8%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 7: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8)
   (/ (- (sqrt (fma b b (* (* c a) -4.0))) b) (* 2.0 a))
   (- (/ (- c) b) (/ (* c (* c a)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8) {
		tmp = (sqrt(fma(b, b, ((c * a) * -4.0))) - b) / (2.0 * a);
	} else {
		tmp = (-c / b) - ((c * (c * a)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -4.0))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. +-commutative 83.9%

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

      3. unsub-neg 83.9%

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

      4. fma-neg 84.1%

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

      5. associate-*l* 84.1%

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

      6. *-commutative 84.1%

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

      7. distribute-rgt-neg-in 84.1%

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

      8. metadata-eval 84.1%

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

   3. Simplified 84.1%

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

   if 1.80000000000000004 < b

   1. Initial program 51.4%

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

      1. neg-sub0 51.4%

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

      2. associate-+l- 51.4%

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

      3. sub0-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-*l/ 51.4%

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

      6. *-commutative 51.4%

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

      7. associate-/r* 51.4%

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

      8. /-rgt-identity 51.4%

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

      9. metadata-eval 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in b around inf 85.8%

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

      1. +-commutative 85.8%

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

      2. mul-1-neg 85.8%

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

      3. unsub-neg 85.8%

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

      4. associate-*r/ 85.8%

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

      5. neg-mul-1 85.8%

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

      6. unpow2 85.8%

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

      7. associate-*l* 85.8%

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

   6. Simplified 85.8%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 8: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8)
   (* (/ 0.5 a) (- (sqrt (+ (* b b) (* (* c a) -4.0))) b))
   (- (/ (- c) b) (/ (* c (* c a)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8) {
		tmp = (0.5 / a) * (sqrt(((b * b) + ((c * a) * -4.0))) - b);
	} else {
		tmp = (-c / b) - ((c * (c * a)) / pow(b, 3.0));
	}
	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 <= 1.8d0) then
        tmp = (0.5d0 / a) * (sqrt(((b * b) + ((c * a) * (-4.0d0)))) - b)
    else
        tmp = (-c / b) - ((c * (c * a)) / (b ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8) {
		tmp = (0.5 / a) * (Math.sqrt(((b * b) + ((c * a) * -4.0))) - b);
	} else {
		tmp = (-c / b) - ((c * (c * a)) / Math.pow(b, 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.8:
		tmp = (0.5 / a) * (math.sqrt(((b * b) + ((c * a) * -4.0))) - b)
	else:
		tmp = (-c / b) - ((c * (c * a)) / math.pow(b, 3.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -4.0))) - b));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.8)
		tmp = (0.5 / a) * (sqrt(((b * b) + ((c * a) * -4.0))) - b);
	else
		tmp = (-c / b) - ((c * (c * a)) / (b ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 83.9%

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

      1. /-rgt-identity 83.9%

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

      2. metadata-eval 83.9%

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

      3. associate-/l* 83.9%

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

      4. associate-*r/ 83.9%

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

      5. +-commutative 83.9%

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

      6. unsub-neg 83.9%

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

      7. fma-neg 84.1%

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

      8. associate-*l* 84.1%

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

      9. *-commutative 84.1%

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

      10. distribute-rgt-neg-in 84.1%

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

      11. metadata-eval 84.1%

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

      12. associate-/r* 84.1%

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

      13. metadata-eval 84.1%

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

      14. metadata-eval 84.1%

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

   3. Simplified 84.1%

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

      1. fma-udef 83.9%

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

      2. *-commutative 83.9%

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

   5. Applied egg-rr 83.9%

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

   if 1.80000000000000004 < b

   1. Initial program 51.4%

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

      1. neg-sub0 51.4%

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

      2. associate-+l- 51.4%

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

      3. sub0-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-*l/ 51.4%

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

      6. *-commutative 51.4%

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

      7. associate-/r* 51.4%

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

      8. /-rgt-identity 51.4%

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

      9. metadata-eval 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in b around inf 85.8%

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

      1. +-commutative 85.8%

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

      2. mul-1-neg 85.8%

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

      3. unsub-neg 85.8%

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

      4. associate-*r/ 85.8%

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

      5. neg-mul-1 85.8%

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

      6. unpow2 85.8%

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

      7. associate-*l* 85.8%

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

   6. Simplified 85.8%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 9: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.6%

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

   1. neg-sub0 57.6%

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

   2. associate-+l- 57.6%

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

   3. sub0-neg 57.6%

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

   4. neg-mul-1 57.6%

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

   5. associate-*l/ 57.6%

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

   6. *-commutative 57.6%

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

   7. associate-/r* 57.6%

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

   8. /-rgt-identity 57.6%

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

   9. metadata-eval 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in b around inf 79.7%

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

   1. +-commutative 79.7%

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

   2. mul-1-neg 79.7%

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

   3. unsub-neg 79.7%

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

   4. associate-*r/ 79.7%

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

   5. neg-mul-1 79.7%

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

   6. unpow2 79.7%

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

   7. associate-*l* 79.7%

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

6. Simplified 79.7%

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

7. Final simplification 79.7%

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

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 57.6%

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

   1. neg-sub0 57.6%

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

   2. associate-+l- 57.6%

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

   3. sub0-neg 57.6%

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

   4. neg-mul-1 57.6%

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

   5. associate-*l/ 57.6%

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

   6. *-commutative 57.6%

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

   7. associate-/r* 57.6%

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

   8. /-rgt-identity 57.6%

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

   9. metadata-eval 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in b around inf 62.4%

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

   1. associate-*r/ 62.4%

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

   2. neg-mul-1 62.4%

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

6. Simplified 62.4%

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

7. Final simplification 62.4%

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

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 57.6%

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

   1. add-log-exp 53.0%

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

   2. neg-mul-1 53.0%

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

   3. fma-def 53.0%

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

   4. *-commutative 53.0%

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

   5. *-commutative 53.0%

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

   6. *-commutative 53.0%

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

3. Applied egg-rr 53.0%

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

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 2023200 
(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)))