Quadratic roots, narrow range

Percentage Accurate: 55.3% → 90.8%

Time: 12.2s

Alternatives: 7

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% 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: 90.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

2. Taylor expanded in a around 0 92.0%

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

4. Simplified 92.0%

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

5. Taylor expanded in c around 0 92.0%

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

   1. associate-*r/ 92.0%

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

   2. *-commutative 92.0%

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

   3. associate-*r* 92.0%

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

7. Simplified 92.0%

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

   1. add-log-exp 82.7%

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

9. Applied egg-rr 82.7%

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

    1. add-log-exp 92.0%

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

    2. associate-*l/ 92.0%

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

    3. unpow3 92.0%

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

    4. times-frac 92.0%

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

11. Applied egg-rr 92.0%

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

12. Final simplification 92.0%

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

Alternative 2: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 22:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left(a \cdot a\right)}{\frac{{b}^{5}}{{c}^{3}}} - \frac{c}{b}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{b} \cdot \frac{-1}{b \cdot b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 22.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (+
    (- (/ (* -2.0 (* a a)) (/ (pow b 5.0) (pow c 3.0))) (/ c b))
    (* (* c c) (* (/ a b) (/ -1.0 (* b b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 22.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (a * a)) / (pow(b, 5.0) / pow(c, 3.0))) - (c / b)) + ((c * c) * ((a / b) * (-1.0 / (b * b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 22.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64(a * a)) / Float64((b ^ 5.0) / (c ^ 3.0))) - Float64(c / b)) + Float64(Float64(c * c) * Float64(Float64(a / b) * Float64(-1.0 / Float64(b * b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 22.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(N[(a / b), $MachinePrecision] * N[(-1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 22

   1. Initial program 81.7%

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

   3. Simplified 81.8%

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

   if 22 < b

   1. Initial program 41.1%

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

   2. Taylor expanded in b around inf 95.1%

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

      1. associate-+r+ 95.1%

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

      2. mul-1-neg 95.1%

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

      3. unsub-neg 95.1%

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

      4. mul-1-neg 95.1%

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

      5. unsub-neg 95.1%

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

      6. associate-/l* 95.1%

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

      7. associate-*r/ 95.1%

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

      8. unpow2 95.1%

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

      9. associate-/l* 95.1%

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

      10. associate-/r/ 95.1%

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

      11. unpow2 95.1%

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

   4. Simplified 95.1%

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

      1. add-log-exp 85.9%

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

   6. Applied egg-rr 85.9%

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

      1. add-log-exp 95.1%

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

      2. *-un-lft-identity 95.1%

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

      3. unpow3 95.1%

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

      4. times-frac 95.1%

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

   8. Applied egg-rr 95.1%

      \[\leadsto \left(\frac{-2 \cdot \left(a \cdot a\right)}{\frac{{b}^{5}}{{c}^{3}}} - \frac{c}{b}\right) - \color{blue}{\left(\frac{1}{b \cdot b} \cdot \frac{a}{b}\right)} \cdot \left(c \cdot c\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 22:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left(a \cdot a\right)}{\frac{{b}^{5}}{{c}^{3}}} - \frac{c}{b}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{b} \cdot \frac{-1}{b \cdot b}\right)\\ \end{array} \]

Alternative 3: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 22:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{c}{b}\right) - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 22.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (- (/ c b)) (* (* c c) (/ a (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 22.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b) - ((c * c) * (a / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 22.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-Float64(c / b)) - Float64(Float64(c * c) * Float64(a / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 22.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-N[(c / b), $MachinePrecision]) - N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 22

   1. Initial program 81.7%

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

   3. Simplified 81.8%

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

   if 22 < b

   1. Initial program 41.1%

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

   2. Taylor expanded in b around inf 91.0%

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

      1. mul-1-neg 91.0%

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

      2. unsub-neg 91.0%

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

      3. mul-1-neg 91.0%

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

      4. distribute-neg-frac 91.0%

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

      5. associate-/l* 91.0%

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

      6. associate-/r/ 91.0%

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

      7. unpow2 91.0%

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

   4. Simplified 91.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 22:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{c}{b}\right) - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\\ \end{array} \]

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 22:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{c}{b}\right) - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 22.0)
   (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
   (- (- (/ c b)) (* (* c c) (/ a (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 22.0) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b) - ((c * c) * (a / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 22.0d0) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = -(c / b) - ((c * c) * (a / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 22.0) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b) - ((c * c) * (a / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 22.0:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -(c / b) - ((c * c) * (a / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 22.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-Float64(c / b)) - Float64(Float64(c * c) * Float64(a / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 22.0)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -(c / b) - ((c * c) * (a / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 22.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-N[(c / b), $MachinePrecision]) - N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 22

   1. Initial program 81.7%

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

   3. Simplified 81.8%

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

      1. *-commutative 81.8%

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

      2. metadata-eval 81.8%

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

      3. distribute-lft-neg-in 81.8%

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

      4. distribute-rgt-neg-in 81.8%

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

      5. *-commutative 81.8%

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

      6. fma-neg 81.7%

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

      7. associate-*l* 81.7%

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

   5. Applied egg-rr 81.7%

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

   if 22 < b

   1. Initial program 41.1%

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

   2. Taylor expanded in b around inf 91.0%

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

      1. mul-1-neg 91.0%

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

      2. unsub-neg 91.0%

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

      3. mul-1-neg 91.0%

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

      4. distribute-neg-frac 91.0%

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

      5. associate-/l* 91.0%

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

      6. associate-/r/ 91.0%

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

      7. unpow2 91.0%

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

   4. Simplified 91.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 22:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{c}{b}\right) - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\\ \end{array} \]

Alternative 5: 81.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

2. Taylor expanded in b around inf 82.7%

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

   1. mul-1-neg 82.7%

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

   2. unsub-neg 82.7%

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

   3. mul-1-neg 82.7%

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

   4. distribute-neg-frac 82.7%

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

   5. associate-/l* 82.7%

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

   6. associate-/r/ 82.7%

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

   7. unpow2 82.7%

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

4. Simplified 82.7%

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

5. Final simplification 82.7%

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

Alternative 6: 81.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b, c):
	return ((-2.0 * ((a * c) / b)) + (-2.0 * (((c * c) / (b * b)) * ((a * a) / b)))) / (a * 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

2. Taylor expanded in b around inf 82.6%

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

   1. pow-prod-down 82.6%

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

   2. *-commutative 82.6%

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

4. Applied egg-rr 82.6%

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

   1. unpow2 82.6%

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

   2. swap-sqr 82.6%

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

   3. unpow3 82.6%

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

   4. times-frac 82.6%

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

6. Applied egg-rr 82.6%

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

7. Final simplification 82.6%

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

Alternative 7: 64.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(-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 52.1%

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

2. Taylor expanded in b around inf 67.1%

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

   1. mul-1-neg 67.1%

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

   2. distribute-neg-frac 67.1%

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

4. Simplified 67.1%

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

5. Final simplification 67.1%

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

Reproduce

herbie shell --seed 2023272 
(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)))