Quadratic roots, narrow range

Percentage Accurate: 55.4% → 96.6%

Time: 17.1s

Alternatives: 11

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% 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: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1550:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1550.0)
   (/
    (log1p
     (expm1 (/ (* c (* 4.0 a)) (- (- b) (sqrt (fma b b (* c (* a -4.0))))))))
    (* a 2.0))
   (-
    (-
     (fma
      -0.25
      (* 20.0 (/ (pow c 4.0) (/ (pow b 7.0) (pow a 3.0))))
      (* -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 tmp;
	if (b <= 1550.0) {
		tmp = log1p(expm1(((c * (4.0 * a)) / (-b - sqrt(fma(b, b, (c * (a * -4.0)))))))) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, (20.0 * (pow(c, 4.0) / (pow(b, 7.0) / pow(a, 3.0)))), (-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)
	tmp = 0.0
	if (b <= 1550.0)
		tmp = Float64(log1p(expm1(Float64(Float64(c * Float64(4.0 * a)) / Float64(Float64(-b) - sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64(20.0 * Float64((c ^ 4.0) / Float64((b ^ 7.0) / (a ^ 3.0)))), 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_] := If[LessEqual[b, 1550.0], N[(N[Log[1 + N[(Exp[N[(N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(20.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[a, 3.0], $MachinePrecision]), $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 < 1550

   1. Initial program 75.2%

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

      1. log1p-expm1-u 73.9%

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

      2. neg-mul-1 73.9%

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

      3. fma-def 73.9%

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

      4. *-commutative 73.9%

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

      5. *-commutative 73.9%

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

   3. Applied egg-rr 73.9%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)\right)\right)}}{2 \cdot a} \]
   4. Step-by-step derivation

      1. fma-udef 73.9%

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

      2. *-commutative 73.9%

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

   5. Applied egg-rr 73.9%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-1 \cdot b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\right)\right)}{2 \cdot a} \]
   6. Step-by-step derivation

      1. neg-mul-1 73.9%

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

      2. flip-+ 73.5%

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

      3. add-sqr-sqrt 75.0%

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

   7. Applied egg-rr 75.0%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\right)\right)}{2 \cdot a} \]
   8. Step-by-step derivation

      1. sqr-neg 75.0%

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

      2. unpow2 75.0%

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

      3. unpow2 75.0%

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

      4. associate--r- 97.8%

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

      5. +-inverses 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{0} + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\right)\right)}{2 \cdot a} \]

      6. fma-neg 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right)}}}\right)\right)}{2 \cdot a} \]

      7. distribute-rgt-neg-in 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)}}\right)\right)}{2 \cdot a} \]

      8. distribute-lft-neg-in 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)}}\right)\right)}{2 \cdot a} \]

      9. metadata-eval 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)}}\right)\right)}{2 \cdot a} \]

      10. *-commutative 97.8%

          \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot -4\right)}\right)}}\right)\right)}{2 \cdot a} \]

   9. Simplified 97.8%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}\right)\right)}{2 \cdot a} \]

   if 1550 < b

   1. Initial program 45.4%

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

      1. neg-sub0 45.4%

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

      2. associate-+l- 45.4%

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

      3. sub0-neg 45.4%

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

      4. neg-mul-1 45.4%

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

      5. associate-*l/ 45.3%

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

      6. *-commutative 45.3%

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

      7. associate-/r* 45.3%

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

      8. /-rgt-identity 45.3%

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

      9. metadata-eval 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in a around 0 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in c around 0 97.0%

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

      1. associate-/l* 97.0%

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

   9. Simplified 97.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1550:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 2: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1600:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{20}{a} \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1600.0)
   (/
    (log1p
     (expm1 (/ (* c (* 4.0 a)) (- (- b) (sqrt (fma b b (* c (* a -4.0))))))))
    (* a 2.0))
   (-
    (fma
     -0.25
     (* (/ 20.0 a) (/ (pow (* c a) 4.0) (pow b 7.0)))
     (- (* -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))) (/ c b)))
    (/ (* c c) (/ (pow b 3.0) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1600.0) {
		tmp = log1p(expm1(((c * (4.0 * a)) / (-b - sqrt(fma(b, b, (c * (a * -4.0)))))))) / (a * 2.0);
	} else {
		tmp = fma(-0.25, ((20.0 / a) * (pow((c * a), 4.0) / pow(b, 7.0))), ((-2.0 * (pow(c, 3.0) / (pow(b, 5.0) / (a * a)))) - (c / b))) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1600.0)
		tmp = Float64(log1p(expm1(Float64(Float64(c * Float64(4.0 * a)) / Float64(Float64(-b) - sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))))))) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-0.25, Float64(Float64(20.0 / a) * Float64((Float64(c * a) ^ 4.0) / (b ^ 7.0))), Float64(Float64(-2.0 * Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a)))) - Float64(c / b))) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1600.0], N[(N[Log[1 + N[(Exp[N[(N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(20.0 / a), $MachinePrecision] * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $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 < 1600

   1. Initial program 75.2%

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

      1. log1p-expm1-u 73.9%

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

      2. neg-mul-1 73.9%

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

      3. fma-def 73.9%

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

      4. *-commutative 73.9%

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

      5. *-commutative 73.9%

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

   3. Applied egg-rr 73.9%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)\right)\right)}}{2 \cdot a} \]
   4. Step-by-step derivation

      1. fma-udef 73.9%

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

      2. *-commutative 73.9%

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

   5. Applied egg-rr 73.9%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-1 \cdot b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\right)\right)}{2 \cdot a} \]
   6. Step-by-step derivation

      1. neg-mul-1 73.9%

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

      2. flip-+ 73.5%

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

      3. add-sqr-sqrt 75.0%

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

   7. Applied egg-rr 75.0%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\right)\right)}{2 \cdot a} \]
   8. Step-by-step derivation

      1. sqr-neg 75.0%

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

      2. unpow2 75.0%

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

      3. unpow2 75.0%

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

      4. associate--r- 97.8%

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

      5. +-inverses 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{0} + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\right)\right)}{2 \cdot a} \]

      6. fma-neg 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right)}}}\right)\right)}{2 \cdot a} \]

      7. distribute-rgt-neg-in 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)}}\right)\right)}{2 \cdot a} \]

      8. distribute-lft-neg-in 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)}}\right)\right)}{2 \cdot a} \]

      9. metadata-eval 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)}}\right)\right)}{2 \cdot a} \]

      10. *-commutative 97.8%

          \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot -4\right)}\right)}}\right)\right)}{2 \cdot a} \]

   9. Simplified 97.8%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}\right)\right)}{2 \cdot a} \]

   if 1600 < b

   1. Initial program 45.4%

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

      1. log1p-expm1-u 36.5%

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

      2. neg-mul-1 36.5%

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

      3. fma-def 36.5%

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

      4. *-commutative 36.5%

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

      5. *-commutative 36.5%

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

   3. Applied egg-rr 36.5%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)\right)\right)}}{2 \cdot a} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 36.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)\right)\right)}{2 \cdot a} \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)\right)\right)}{2 \cdot a}\right) \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)\right)\right)}{2 \cdot a}}} \]

   5. Applied egg-rr 45.4%

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

      1. associate-*l* 45.4%

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

      2. cube-unmult 45.4%

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

   7. Simplified 45.4%

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

   8. Taylor expanded in b around inf 96.9%

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

      1. +-commutative 96.9%

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

      2. mul-1-neg 96.9%

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

      3. unsub-neg 96.9%

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

   10. Simplified 96.9%

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

   11. Taylor expanded in b around 0 96.9%

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

       1. distribute-rgt-out 96.9%

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

       2. *-commutative 96.9%

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

       3. metadata-eval 96.9%

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

       4. pow-sqr 96.9%

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

       5. metadata-eval 96.9%

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

       6. pow-sqr 96.9%

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

       7. unswap-sqr 96.9%

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

       8. *-commutative 96.9%

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

       9. unpow2 96.9%

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

       10. unpow2 96.9%

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

       11. unswap-sqr 96.9%

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

       12. unpow2 96.9%

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

       13. *-commutative 96.9%

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

       14. unpow2 96.9%

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

       15. unpow2 96.9%

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

       16. unswap-sqr 96.9%

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

       17. unpow2 96.9%

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

       18. pow-sqr 96.9%

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

       19. metadata-eval 96.9%

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

       20. metadata-eval 96.9%

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

   13. Simplified 96.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1600:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{20}{a} \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 3: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1600:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot a}{\frac{{b}^{3}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1600.0)
   (/
    (log1p
     (expm1 (/ (* c (* 4.0 a)) (- (- b) (sqrt (fma b b (* c (* a -4.0))))))))
    (* a 2.0))
   (-
    (- (/ (* -2.0 (pow c 3.0)) (/ (pow b 5.0) (* a a))) (/ c b))
    (/ (* c a) (/ (pow b 3.0) c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1600.0) {
		tmp = log1p(expm1(((c * (4.0 * a)) / (-b - sqrt(fma(b, b, (c * (a * -4.0)))))))) / (a * 2.0);
	} else {
		tmp = (((-2.0 * pow(c, 3.0)) / (pow(b, 5.0) / (a * a))) - (c / b)) - ((c * a) / (pow(b, 3.0) / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1600.0)
		tmp = Float64(log1p(expm1(Float64(Float64(c * Float64(4.0 * a)) / Float64(Float64(-b) - sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * (c ^ 3.0)) / Float64((b ^ 5.0) / Float64(a * a))) - Float64(c / b)) - Float64(Float64(c * a) / Float64((b ^ 3.0) / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1600.0], N[(N[Log[1 + N[(Exp[N[(N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1600

   1. Initial program 75.2%

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

      1. log1p-expm1-u 73.9%

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

      2. neg-mul-1 73.9%

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

      3. fma-def 73.9%

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

      4. *-commutative 73.9%

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

      5. *-commutative 73.9%

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

   3. Applied egg-rr 73.9%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)\right)\right)}}{2 \cdot a} \]
   4. Step-by-step derivation

      1. fma-udef 73.9%

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

      2. *-commutative 73.9%

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

   5. Applied egg-rr 73.9%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-1 \cdot b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\right)\right)}{2 \cdot a} \]
   6. Step-by-step derivation

      1. neg-mul-1 73.9%

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

      2. flip-+ 73.5%

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

      3. add-sqr-sqrt 75.0%

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

   7. Applied egg-rr 75.0%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\right)\right)}{2 \cdot a} \]
   8. Step-by-step derivation

      1. sqr-neg 75.0%

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

      2. unpow2 75.0%

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

      3. unpow2 75.0%

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

      4. associate--r- 97.8%

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

      5. +-inverses 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{0} + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\right)\right)}{2 \cdot a} \]

      6. fma-neg 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right)}}}\right)\right)}{2 \cdot a} \]

      7. distribute-rgt-neg-in 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)}}\right)\right)}{2 \cdot a} \]

      8. distribute-lft-neg-in 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)}}\right)\right)}{2 \cdot a} \]

      9. metadata-eval 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)}}\right)\right)}{2 \cdot a} \]

      10. *-commutative 97.8%

          \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot -4\right)}\right)}}\right)\right)}{2 \cdot a} \]

   9. Simplified 97.8%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{0 + c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}\right)\right)}{2 \cdot a} \]

   if 1600 < b

   1. Initial program 45.4%

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

      1. log1p-expm1-u 36.5%

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

      2. neg-mul-1 36.5%

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

      3. fma-def 36.5%

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

      4. *-commutative 36.5%

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

      5. *-commutative 36.5%

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

   3. Applied egg-rr 36.5%

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

   4. Taylor expanded in b around inf 95.4%

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

      1. +-commutative 95.4%

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

      2. mul-1-neg 95.4%

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

      3. unpow2 95.4%

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

      4. associate-*r* 95.4%

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

      5. unsub-neg 95.4%

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

   6. Simplified 95.4%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1600:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot a}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 4: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot a}{\frac{{b}^{3}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))))
   (if (<= b 6.15)
     (/
      (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) (sqrt (- (* b b) t_0))))
      (* a 2.0))
     (-
      (- (/ (* -2.0 (pow c 3.0)) (/ (pow b 5.0) (* a a))) (/ c b))
      (/ (* c a) (/ (pow b 3.0) c))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double tmp;
	if (b <= 6.15) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 2.0);
	} else {
		tmp = (((-2.0 * pow(c, 3.0)) / (pow(b, 5.0) / (a * a))) - (c / b)) - ((c * a) / (pow(b, 3.0) / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if (b <= 6.15d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 2.0d0)
    else
        tmp = ((((-2.0d0) * (c ** 3.0d0)) / ((b ** 5.0d0) / (a * a))) - (c / b)) - ((c * a) / ((b ** 3.0d0) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double tmp;
	if (b <= 6.15) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - Math.sqrt(((b * b) - t_0)))) / (a * 2.0);
	} else {
		tmp = (((-2.0 * Math.pow(c, 3.0)) / (Math.pow(b, 5.0) / (a * a))) - (c / b)) - ((c * a) / (Math.pow(b, 3.0) / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	tmp = 0
	if b <= 6.15:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - math.sqrt(((b * b) - t_0)))) / (a * 2.0)
	else:
		tmp = (((-2.0 * math.pow(c, 3.0)) / (math.pow(b, 5.0) / (a * a))) - (c / b)) - ((c * a) / (math.pow(b, 3.0) / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	tmp = 0.0
	if (b <= 6.15)
		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(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * (c ^ 3.0)) / Float64((b ^ 5.0) / Float64(a * a))) - Float64(c / b)) - Float64(Float64(c * a) / Float64((b ^ 3.0) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (4.0 * a);
	tmp = 0.0;
	if (b <= 6.15)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 2.0);
	else
		tmp = (((-2.0 * (c ^ 3.0)) / ((b ^ 5.0) / (a * a))) - (c / b)) - ((c * a) / ((b ^ 3.0) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 6.15], 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[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.1500000000000004

   1. Initial program 81.3%

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

      1. flip-+ 81.1%

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

      2. pow2 81.1%

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

      3. add-sqr-sqrt 82.3%

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

      4. *-commutative 82.3%

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

      5. *-commutative 82.3%

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

      6. *-commutative 82.3%

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

      7. *-commutative 82.3%

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

   3. Applied egg-rr 82.3%

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

   if 6.1500000000000004 < b

   1. Initial program 50.8%

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

      1. log1p-expm1-u 43.3%

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

      2. neg-mul-1 43.3%

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

      3. fma-def 43.3%

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

      4. *-commutative 43.3%

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

      5. *-commutative 43.3%

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

   3. Applied egg-rr 43.3%

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

   4. Taylor expanded in b around inf 93.1%

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

      1. +-commutative 93.1%

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

      2. mul-1-neg 93.1%

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

      3. unpow2 93.1%

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

      4. associate-*r* 93.1%

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

      5. unsub-neg 93.1%

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

   6. Simplified 93.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(4 \cdot a\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot a}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 5: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-a\right)\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))))
   (if (<= b 6.15)
     (/
      (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) (sqrt (- (* b b) t_0))))
      (* a 2.0))
     (- (/ (* c (* c (- a))) (pow b 3.0)) (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double tmp;
	if (b <= 6.15) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 2.0);
	} else {
		tmp = ((c * (c * -a)) / pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if (b <= 6.15d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 2.0d0)
    else
        tmp = ((c * (c * -a)) / (b ** 3.0d0)) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double tmp;
	if (b <= 6.15) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - Math.sqrt(((b * b) - t_0)))) / (a * 2.0);
	} else {
		tmp = ((c * (c * -a)) / Math.pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	tmp = 0
	if b <= 6.15:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - math.sqrt(((b * b) - t_0)))) / (a * 2.0)
	else:
		tmp = ((c * (c * -a)) / math.pow(b, 3.0)) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	tmp = 0.0
	if (b <= 6.15)
		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(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c * Float64(c * Float64(-a))) / (b ^ 3.0)) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (4.0 * a);
	tmp = 0.0;
	if (b <= 6.15)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 2.0);
	else
		tmp = ((c * (c * -a)) / (b ^ 3.0)) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 6.15], 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[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(c * (-a)), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.1500000000000004

   1. Initial program 81.3%

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

      1. flip-+ 81.1%

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

      2. pow2 81.1%

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

      3. add-sqr-sqrt 82.3%

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

      4. *-commutative 82.3%

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

      5. *-commutative 82.3%

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

      6. *-commutative 82.3%

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

      7. *-commutative 82.3%

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

   3. Applied egg-rr 82.3%

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

   if 6.1500000000000004 < b

   1. Initial program 50.8%

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

      1. neg-sub0 50.8%

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

      2. associate-+l- 50.8%

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

      3. sub0-neg 50.8%

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

      4. neg-mul-1 50.8%

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

      5. associate-*l/ 50.8%

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

      6. *-commutative 50.8%

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

      7. associate-/r* 50.8%

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

      8. /-rgt-identity 50.8%

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

      9. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in b around inf 87.7%

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

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. associate-*r/ 87.7%

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

      5. neg-mul-1 87.7%

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

      6. unpow2 87.7%

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

      7. associate-*l* 87.7%

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

   6. Simplified 87.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(4 \cdot a\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-a\right)\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 6: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-a\right)\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.15)
   (* (- (sqrt (fma b b (* -4.0 (* c a)))) b) (/ 0.5 a))
   (- (/ (* c (* c (- a))) (pow b 3.0)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.15) {
		tmp = (sqrt(fma(b, b, (-4.0 * (c * a)))) - b) * (0.5 / a);
	} else {
		tmp = ((c * (c * -a)) / pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.15)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(c * Float64(c * Float64(-a))) / (b ^ 3.0)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.15], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(c * (-a)), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.1500000000000004

   1. Initial program 81.3%

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

      1. /-rgt-identity 81.3%

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

      2. metadata-eval 81.3%

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

      3. associate-/l* 81.3%

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

      4. associate-*r/ 81.3%

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

      5. +-commutative 81.3%

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

      6. unsub-neg 81.3%

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

      7. fma-neg 81.4%

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

      8. associate-*l* 81.4%

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

      9. *-commutative 81.4%

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

      10. distribute-rgt-neg-in 81.4%

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

      11. metadata-eval 81.4%

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

      12. associate-/r* 81.4%

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

      13. metadata-eval 81.4%

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

      14. metadata-eval 81.4%

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

   3. Simplified 81.4%

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

   if 6.1500000000000004 < b

   1. Initial program 50.8%

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

      1. neg-sub0 50.8%

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

      2. associate-+l- 50.8%

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

      3. sub0-neg 50.8%

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

      4. neg-mul-1 50.8%

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

      5. associate-*l/ 50.8%

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

      6. *-commutative 50.8%

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

      7. associate-/r* 50.8%

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

      8. /-rgt-identity 50.8%

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

      9. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in b around inf 87.7%

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

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. associate-*r/ 87.7%

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

      5. neg-mul-1 87.7%

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

      6. unpow2 87.7%

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

      7. associate-*l* 87.7%

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

   6. Simplified 87.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-a\right)\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 7: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-a\right)\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.15)
   (* (/ 0.5 a) (- (sqrt (+ (* b b) (* -4.0 (* c a)))) b))
   (- (/ (* c (* c (- a))) (pow b 3.0)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.15) {
		tmp = (0.5 / a) * (sqrt(((b * b) + (-4.0 * (c * a)))) - b);
	} else {
		tmp = ((c * (c * -a)) / pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6.15d0) then
        tmp = (0.5d0 / a) * (sqrt(((b * b) + ((-4.0d0) * (c * a)))) - b)
    else
        tmp = ((c * (c * -a)) / (b ** 3.0d0)) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.15) {
		tmp = (0.5 / a) * (Math.sqrt(((b * b) + (-4.0 * (c * a)))) - b);
	} else {
		tmp = ((c * (c * -a)) / Math.pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.15:
		tmp = (0.5 / a) * (math.sqrt(((b * b) + (-4.0 * (c * a)))) - b)
	else:
		tmp = ((c * (c * -a)) / math.pow(b, 3.0)) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.15)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a)))) - b));
	else
		tmp = Float64(Float64(Float64(c * Float64(c * Float64(-a))) / (b ^ 3.0)) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.15)
		tmp = (0.5 / a) * (sqrt(((b * b) + (-4.0 * (c * a)))) - b);
	else
		tmp = ((c * (c * -a)) / (b ^ 3.0)) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.15], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(c * (-a)), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.1500000000000004

   1. Initial program 81.3%

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

      1. /-rgt-identity 81.3%

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

      2. metadata-eval 81.3%

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

      3. associate-/l* 81.3%

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

      4. associate-*r/ 81.3%

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

      5. +-commutative 81.3%

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

      6. unsub-neg 81.3%

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

      7. fma-neg 81.4%

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

      8. associate-*l* 81.4%

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

      9. *-commutative 81.4%

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

      10. distribute-rgt-neg-in 81.4%

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

      11. metadata-eval 81.4%

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

      12. associate-/r* 81.4%

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

      13. metadata-eval 81.4%

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

      14. metadata-eval 81.4%

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

   3. Simplified 81.4%

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

      1. fma-udef 81.3%

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

      2. *-commutative 81.3%

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

   5. Applied egg-rr 81.3%

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

   if 6.1500000000000004 < b

   1. Initial program 50.8%

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

      1. neg-sub0 50.8%

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

      2. associate-+l- 50.8%

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

      3. sub0-neg 50.8%

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

      4. neg-mul-1 50.8%

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

      5. associate-*l/ 50.8%

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

      6. *-commutative 50.8%

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

      7. associate-/r* 50.8%

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

      8. /-rgt-identity 50.8%

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

      9. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in b around inf 87.7%

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

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. associate-*r/ 87.7%

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

      5. neg-mul-1 87.7%

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

      6. unpow2 87.7%

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

      7. associate-*l* 87.7%

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

   6. Simplified 87.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-a\right)\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 8: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-a\right)\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.15)
   (/ (- (sqrt (+ (* b b) (* -4.0 (* c a)))) b) (* a 2.0))
   (- (/ (* c (* c (- a))) (pow b 3.0)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.15) {
		tmp = (sqrt(((b * b) + (-4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * (c * -a)) / pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6.15d0) then
        tmp = (sqrt(((b * b) + ((-4.0d0) * (c * a)))) - b) / (a * 2.0d0)
    else
        tmp = ((c * (c * -a)) / (b ** 3.0d0)) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.15) {
		tmp = (Math.sqrt(((b * b) + (-4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * (c * -a)) / Math.pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.15:
		tmp = (math.sqrt(((b * b) + (-4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = ((c * (c * -a)) / math.pow(b, 3.0)) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.15)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c * Float64(c * Float64(-a))) / (b ^ 3.0)) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.15)
		tmp = (sqrt(((b * b) + (-4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = ((c * (c * -a)) / (b ^ 3.0)) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.15], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(c * (-a)), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.1500000000000004

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

      2. +-commutative 81.3%

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

      3. unsub-neg 81.3%

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

      4. fma-neg 81.3%

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

      5. associate-*l* 81.3%

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

      6. *-commutative 81.3%

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

      7. distribute-rgt-neg-in 81.3%

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

      8. metadata-eval 81.3%

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

   3. Simplified 81.3%

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

      1. fma-udef 81.3%

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

      2. *-commutative 81.3%

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

   5. Applied egg-rr 81.3%

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

   if 6.1500000000000004 < b

   1. Initial program 50.8%

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

      1. neg-sub0 50.8%

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

      2. associate-+l- 50.8%

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

      3. sub0-neg 50.8%

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

      4. neg-mul-1 50.8%

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

      5. associate-*l/ 50.8%

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

      6. *-commutative 50.8%

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

      7. associate-/r* 50.8%

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

      8. /-rgt-identity 50.8%

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

      9. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in b around inf 87.7%

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

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. associate-*r/ 87.7%

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

      5. neg-mul-1 87.7%

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

      6. unpow2 87.7%

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

      7. associate-*l* 87.7%

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

   6. Simplified 87.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.15:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-a\right)\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 9: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

   1. neg-sub0 57.3%

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

   2. associate-+l- 57.3%

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

   3. sub0-neg 57.3%

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

   4. neg-mul-1 57.3%

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

   5. associate-*l/ 57.2%

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

   6. *-commutative 57.2%

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

   7. associate-/r* 57.2%

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

   8. /-rgt-identity 57.2%

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

   9. metadata-eval 57.2%

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

3. Simplified 57.3%

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

4. Taylor expanded in b around inf 81.8%

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

   1. +-commutative 81.8%

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

   2. mul-1-neg 81.8%

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

   3. unsub-neg 81.8%

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

   4. associate-*r/ 81.8%

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

   5. neg-mul-1 81.8%

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

   6. unpow2 81.8%

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

   7. associate-*l* 81.8%

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

6. Simplified 81.8%

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

7. Final simplification 81.8%

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

Alternative 10: 64.3% 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 57.3%

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

   1. neg-sub0 57.3%

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

   2. associate-+l- 57.3%

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

   3. sub0-neg 57.3%

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

   4. neg-mul-1 57.3%

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

   5. associate-*l/ 57.2%

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

   6. *-commutative 57.2%

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

   7. associate-/r* 57.2%

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

   8. /-rgt-identity 57.2%

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

   9. metadata-eval 57.2%

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

3. Simplified 57.3%

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

4. Taylor expanded in b around inf 63.4%

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

   1. associate-*r/ 63.4%

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

   2. neg-mul-1 63.4%

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

6. Simplified 63.4%

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

7. Final simplification 63.4%

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

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 57.3%

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

   1. log1p-expm1-u 51.4%

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

   2. neg-mul-1 51.4%

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

   3. fma-def 51.4%

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

   4. *-commutative 51.4%

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

   5. *-commutative 51.4%

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

3. Applied egg-rr 51.4%

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

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 2023229 
(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)))