Cubic critical, wide range

Percentage Accurate: 17.9% → 98.8%

Time: 16.4s

Alternatives: 9

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot a\right) \cdot c\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_0} - b}{3 \cdot a} \leq -2 \cdot 10^{-27}:\\ \;\;\;\;0 - \frac{\frac{\left({b}^{2} - {b}^{2}\right) + t\_0}{b + \sqrt{{b}^{2} - t\_0}}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* 3.0 a) c)))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* 3.0 a)) -2e-27)
     (-
      0.0
      (/
       (/ (+ (- (pow b 2.0) (pow b 2.0)) t_0) (+ b (sqrt (- (pow b 2.0) t_0))))
       (* (pow (pow (* 3.0 a) 2.0) 0.3333333333333333) (cbrt (* 3.0 a)))))
     (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = (3.0 * a) * c;
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (3.0 * a)) <= -2e-27) {
		tmp = 0.0 - ((((pow(b, 2.0) - pow(b, 2.0)) + t_0) / (b + sqrt((pow(b, 2.0) - t_0)))) / (pow(pow((3.0 * a), 2.0), 0.3333333333333333) * cbrt((3.0 * a))));
	} else {
		tmp = fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(3.0 * a) * c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(3.0 * a)) <= -2e-27)
		tmp = Float64(0.0 - Float64(Float64(Float64(Float64((b ^ 2.0) - (b ^ 2.0)) + t_0) / Float64(b + sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(((Float64(3.0 * a) ^ 2.0) ^ 0.3333333333333333) * cbrt(Float64(3.0 * a)))));
	else
		tmp = Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2e-27], N[(0.0 - N[(N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Power[N[(3.0 * a), $MachinePrecision], 2.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Power[N[(3.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.0000000000000001e-27

   1. Initial program 63.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cbrt-cube 63.2%

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

      2. pow3 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. pow1/3 63.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]

      2. unpow3 63.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left(\left(\left(3 \cdot a\right) \cdot \left(3 \cdot a\right)\right) \cdot \left(3 \cdot a\right)\right)}}^{0.3333333333333333}} \]

      3. unpow-prod-down 63.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left(\left(3 \cdot a\right) \cdot \left(3 \cdot a\right)\right)}^{0.3333333333333333} \cdot {\left(3 \cdot a\right)}^{0.3333333333333333}}} \]

      4. pow2 63.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left({\left(3 \cdot a\right)}^{2}\right)}}^{0.3333333333333333} \cdot {\left(3 \cdot a\right)}^{0.3333333333333333}} \]

      5. pow1/3 63.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{3 \cdot a}}} \]

   6. Applied egg-rr 63.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 63.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt{b \cdot b} \cdot \sqrt{b \cdot b}} - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

      2. sqrt-unprod 63.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}} - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

      3. pow2 63.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right)} - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

      4. pow2 63.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt{{b}^{2} \cdot \color{blue}{{b}^{2}}} - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

      5. pow-prod-up 62.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt{\color{blue}{{b}^{\left(2 + 2\right)}}} - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

      6. metadata-eval 62.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt{{b}^{\color{blue}{4}}} - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

   8. Applied egg-rr 62.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt{{b}^{4}}} - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]
   9. Step-by-step derivation

      1. flip-+ 62.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt{{b}^{4}} - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt{{b}^{4}} - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt{{b}^{4}} - \left(3 \cdot a\right) \cdot c}}}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

   10. Applied egg-rr 64.9%

       \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(3 \cdot a\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(3 \cdot a\right)}}}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]
   11. Step-by-step derivation

       1. associate--r- 96.1%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(3 \cdot a\right)}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(3 \cdot a\right)}}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

       2. unpow2 96.1%

          \[\leadsto \frac{\frac{\left(\color{blue}{\left(-b\right) \cdot \left(-b\right)} - {b}^{2}\right) + c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(3 \cdot a\right)}}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

       3. sqr-neg 96.1%

          \[\leadsto \frac{\frac{\left(\color{blue}{b \cdot b} - {b}^{2}\right) + c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(3 \cdot a\right)}}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

       4. unpow2 96.1%

          \[\leadsto \frac{\frac{\left(\color{blue}{{b}^{2}} - {b}^{2}\right) + c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(3 \cdot a\right)}}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

   12. Simplified 96.1%

       \[\leadsto \frac{\color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(3 \cdot a\right)}}}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}} \]

   if -2.0000000000000001e-27 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 3.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 100.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

   4. Taylor expanded in b around inf 100.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.5 \cdot c}}{b} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{c \cdot -0.5}}{b} \]

      3. fma-define 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, c \cdot -0.5\right)}}{b} \]

      4. associate-/l* 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}, c \cdot -0.5\right)}{b} \]

      5. unpow2 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, c \cdot -0.5\right)}{b} \]

      6. unpow2 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}, c \cdot -0.5\right)}{b} \]

      7. times-frac 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}, c \cdot -0.5\right)}{b} \]

      8. unpow1 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right), c \cdot -0.5\right)}{b} \]

      9. pow-plus 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}, c \cdot -0.5\right)}{b} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}, c \cdot -0.5\right)}{b} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2 \cdot 10^{-27}:\\ \;\;\;\;0 - \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{b + \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{3 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 + a \cdot \left(-0.375 \cdot \frac{c \cdot c}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{\left(a \cdot -1.0546875\right) \cdot {c}^{4}}{{b}^{7}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* (/ c b) -0.5)
  (*
   a
   (+
    (* -0.375 (/ (* c c) (pow b 3.0)))
    (*
     a
     (+
      (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
      (/ (* (* a -1.0546875) (pow c 4.0)) (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	return ((c / b) * -0.5) + (a * ((-0.375 * ((c * c) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (((a * -1.0546875) * pow(c, 4.0)) / pow(b, 7.0))))));
}

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) * (-0.5d0)) + (a * (((-0.375d0) * ((c * c) / (b ** 3.0d0))) + (a * (((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + (((a * (-1.0546875d0)) * (c ** 4.0d0)) / (b ** 7.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c / b) * -0.5) + (a * ((-0.375 * ((c * c) / Math.pow(b, 3.0))) + (a * ((-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (((a * -1.0546875) * Math.pow(c, 4.0)) / Math.pow(b, 7.0))))));
}

Python

def code(a, b, c):
	return ((c / b) * -0.5) + (a * ((-0.375 * ((c * c) / math.pow(b, 3.0))) + (a * ((-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (((a * -1.0546875) * math.pow(c, 4.0)) / math.pow(b, 7.0))))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c / b) * -0.5) + Float64(a * Float64(Float64(-0.375 * Float64(Float64(c * c) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(Float64(Float64(a * -1.0546875) * (c ^ 4.0)) / (b ^ 7.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c / b) * -0.5) + (a * ((-0.375 * ((c * c) / (b ^ 3.0))) + (a * ((-0.5625 * ((c ^ 3.0) / (b ^ 5.0))) + (((a * -1.0546875) * (c ^ 4.0)) / (b ^ 7.0))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * -1.0546875), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 97.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{a \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

4. Taylor expanded in c around 0 97.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
5. Step-by-step derivation

   1. associate-*r/ 97.0%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{\frac{-1.0546875 \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right) \]

   2. associate-*r* 97.0%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{\color{blue}{\left(-1.0546875 \cdot a\right) \cdot {c}^{4}}}{{b}^{7}}\right)\right) \]

6. Simplified 97.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{\frac{\left(-1.0546875 \cdot a\right) \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
7. Step-by-step derivation

   1. unpow2 97.0%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{\left(-1.0546875 \cdot a\right) \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

8. Applied egg-rr 97.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{\left(-1.0546875 \cdot a\right) \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

9. Final simplification 97.0%

   \[\leadsto \frac{c}{b} \cdot -0.5 + a \cdot \left(-0.375 \cdot \frac{c \cdot c}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{\left(a \cdot -1.0546875\right) \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
10. Add Preprocessing

Alternative 3: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* (/ c b) -0.5)
  (*
   a
   (+
    (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	return ((c / b) * -0.5) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * (pow(c, 2.0) / 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) * (-0.5d0)) + (a * (((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) + ((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c / b) * -0.5) + (a * ((-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0)))));
}

Python

def code(a, b, c):
	return ((c / b) * -0.5) + (a * ((-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0)))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c / b) * -0.5) + Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c / b) * -0.5) + (a * ((-0.5625 * ((a * (c ^ 3.0)) / (b ^ 5.0))) + (-0.375 * ((c ^ 2.0) / (b ^ 3.0)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 96.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]

4. Final simplification 96.2%

   \[\leadsto \frac{c}{b} \cdot -0.5 + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right) \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))
     (* -0.375 (/ a (pow b 3.0)))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / 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 * (((-0.5625d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) + ((-0.375d0) * (a / (b ** 3.0d0))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) + (-0.375 * (a / Math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.5625 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) + (-0.375 * (a / math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.5625 * ((c * (a ^ 2.0)) / (b ^ 5.0))) + (-0.375 * (a / (b ^ 3.0))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 95.9%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]

4. Final simplification 95.9%

   \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right) \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* (/ c b) -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return ((c / b) * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / 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) * (-0.5d0)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return ((c / b) * -0.5) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return ((c / b) * -0.5) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c / b) * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c / b) * -0.5) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 94.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Final simplification 94.8%

   \[\leadsto \frac{c}{b} \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 94.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Taylor expanded in b around inf 94.8%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
5. Step-by-step derivation

   1. +-commutative 94.8%

      \[\leadsto \frac{\color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.5 \cdot c}}{b} \]

   2. *-commutative 94.8%

      \[\leadsto \frac{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{c \cdot -0.5}}{b} \]

   3. fma-define 94.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, c \cdot -0.5\right)}}{b} \]

   4. associate-/l* 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}, c \cdot -0.5\right)}{b} \]

   5. unpow2 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, c \cdot -0.5\right)}{b} \]

   6. unpow2 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}, c \cdot -0.5\right)}{b} \]

   7. times-frac 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}, c \cdot -0.5\right)}{b} \]

   8. unpow1 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right), c \cdot -0.5\right)}{b} \]

   9. pow-plus 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}, c \cdot -0.5\right)}{b} \]

   10. metadata-eval 94.8%

       \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}, c \cdot -0.5\right)}{b} \]

6. Simplified 94.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]
7. Add Preprocessing

Alternative 7: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / 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 * (((-0.375d0) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 94.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Taylor expanded in c around 0 94.5%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
5. Step-by-step derivation

   1. associate-/l* 94.5%

      \[\leadsto c \cdot \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{3}}\right)} - 0.5 \cdot \frac{1}{b}\right) \]

   2. associate-*r/ 94.5%

      \[\leadsto c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   3. metadata-eval 94.5%

      \[\leadsto c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{\color{blue}{0.5}}{b}\right) \]

6. Simplified 94.5%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)} \]
7. Add Preprocessing

Alternative 8: 90.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* c -0.5) b))

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 * (-0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * -0.5) / b;
}

Python

def code(a, b, c):
	return (c * -0.5) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * -0.5) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * -0.5) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 90.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 90.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   2. *-commutative 90.2%

      \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

5. Simplified 90.2%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
6. Add Preprocessing

Alternative 9: 90.0% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* c (/ -0.5 b)))

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 * ((-0.5d0) / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * (-0.5 / b);
}

Python

def code(a, b, c):
	return c * (-0.5 / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(-0.5 / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (-0.5 / b);
end

Wolfram

code[a_, b_, c_] := N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 90.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 90.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   2. *-commutative 90.2%

      \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

5. Simplified 90.2%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]

6. Taylor expanded in c around 0 90.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
7. Step-by-step derivation

   1. *-commutative 90.2%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   2. associate-*l/ 90.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]

   3. associate-*r/ 90.0%

      \[\leadsto \color{blue}{c \cdot \frac{-0.5}{b}} \]

8. Simplified 90.0%

   \[\leadsto \color{blue}{c \cdot \frac{-0.5}{b}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))