Quadratic roots, wide range

Percentage Accurate: 18.3% → 97.5%

Time: 12.5s

Alternatives: 7

Speedup: 29.0×

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

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

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: 18.3% 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: 97.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. neg-sub0 18.3%

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

   2. associate-+l- 18.3%

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

   3. sub0-neg 18.3%

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

   4. neg-mul-1 18.3%

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

   5. associate-*l/ 18.3%

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

   6. *-commutative 18.3%

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

   7. associate-/r* 18.3%

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

   8. /-rgt-identity 18.3%

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

   9. metadata-eval 18.3%

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

3. Simplified 18.3%

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

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%

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

7. Taylor expanded in b around 0 97.0%

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

   1. associate-/l* 97.0%

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

   2. distribute-rgt-out 97.0%

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

   3. metadata-eval 97.0%

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

9. Simplified 97.0%

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

10. Final simplification 97.0%

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

Alternative 2: 97.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. /-rgt-identity 18.3%

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

   2. metadata-eval 18.3%

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

   3. associate-/l* 18.3%

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

   4. associate-*r/ 18.3%

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

   5. +-commutative 18.3%

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

   6. unsub-neg 18.3%

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

   7. fma-neg 18.3%

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

   8. associate-*l* 18.3%

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

   9. *-commutative 18.3%

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

   10. distribute-rgt-neg-in 18.3%

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

   11. metadata-eval 18.3%

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

   12. associate-/r* 18.3%

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

   13. metadata-eval 18.3%

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

   14. metadata-eval 18.3%

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

3. Simplified 18.3%

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

   1. fma-udef 18.3%

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

   2. *-commutative 18.3%

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

5. Applied egg-rr 18.3%

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

6. Taylor expanded in a around 0 96.4%

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

8. Simplified 96.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{{a}^{4}}{b} \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}, -4 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right) + -2 \cdot \left(a \cdot \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right)} \cdot \frac{0.5}{a} \]

9. Taylor expanded in a around 0 97.0%

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

    1. +-commutative 97.0%

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

    2. mul-1-neg 97.0%

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

    3. unsub-neg 97.0%

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

11. Simplified 97.0%

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

12. Final simplification 97.0%

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

Alternative 3: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-0.5, \frac{{a}^{4}}{b} \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}, -4 \cdot \frac{\left(a \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{5}}\right) + -2 \cdot \left(a \cdot \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right) \cdot \frac{0.5}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (+
   (fma
    -0.5
    (* (/ (pow a 4.0) b) (/ (* (pow c 4.0) 20.0) (pow b 6.0)))
    (* -4.0 (/ (* (* a c) (* (* a c) (* a c))) (pow b 5.0))))
   (* -2.0 (* a (+ (/ c b) (* a (/ (* c c) (pow b 3.0)))))))
  (/ 0.5 a)))

C

double code(double a, double b, double c) {
	return (fma(-0.5, ((pow(a, 4.0) / b) * ((pow(c, 4.0) * 20.0) / pow(b, 6.0))), (-4.0 * (((a * c) * ((a * c) * (a * c))) / pow(b, 5.0)))) + (-2.0 * (a * ((c / b) + (a * ((c * c) / pow(b, 3.0))))))) * (0.5 / a);
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(-0.5, Float64(Float64((a ^ 4.0) / b) * Float64(Float64((c ^ 4.0) * 20.0) / (b ^ 6.0))), Float64(-4.0 * Float64(Float64(Float64(a * c) * Float64(Float64(a * c) * Float64(a * c))) / (b ^ 5.0)))) + Float64(-2.0 * Float64(a * Float64(Float64(c / b) + Float64(a * Float64(Float64(c * c) / (b ^ 3.0))))))) * Float64(0.5 / a))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.5 * N[(N[(N[Power[a, 4.0], $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(N[(a * c), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(a * N[(N[(c / b), $MachinePrecision] + N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.3%

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

   1. /-rgt-identity 18.3%

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

   2. metadata-eval 18.3%

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

   3. associate-/l* 18.3%

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

   4. associate-*r/ 18.3%

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

   5. +-commutative 18.3%

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

   6. unsub-neg 18.3%

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

   7. fma-neg 18.3%

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

   8. associate-*l* 18.3%

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

   9. *-commutative 18.3%

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

   10. distribute-rgt-neg-in 18.3%

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

   11. metadata-eval 18.3%

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

   12. associate-/r* 18.3%

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

   13. metadata-eval 18.3%

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

   14. metadata-eval 18.3%

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

3. Simplified 18.3%

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

   1. fma-udef 18.3%

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

   2. *-commutative 18.3%

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

5. Applied egg-rr 18.3%

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

6. Taylor expanded in a around 0 96.4%

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

8. Simplified 96.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{{a}^{4}}{b} \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}, -4 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right) + -2 \cdot \left(a \cdot \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right)} \cdot \frac{0.5}{a} \]
9. Step-by-step derivation

   1. unpow3 96.6%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, \frac{{a}^{4}}{b} \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}, -4 \cdot \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}}{{b}^{5}}\right) + -2 \cdot \left(a \cdot \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right) \cdot \frac{0.5}{a} \]

10. Applied egg-rr 96.6%

    \[\leadsto \left(\mathsf{fma}\left(-0.5, \frac{{a}^{4}}{b} \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}, -4 \cdot \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}}{{b}^{5}}\right) + -2 \cdot \left(a \cdot \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right) \cdot \frac{0.5}{a} \]

11. Final simplification 96.6%

    \[\leadsto \left(\mathsf{fma}\left(-0.5, \frac{{a}^{4}}{b} \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}, -4 \cdot \frac{\left(a \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{5}}\right) + -2 \cdot \left(a \cdot \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right) \cdot \frac{0.5}{a} \]

Alternative 4: 96.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-2.0d0) / ((b ** 5.0d0) / (a * (a * (c ** 3.0d0))))) - (c / b)) - (a / ((b ** 3.0d0) / (c * c)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. neg-sub0 18.3%

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

   2. associate-+l- 18.3%

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

   3. sub0-neg 18.3%

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

   4. neg-mul-1 18.3%

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

   5. associate-*l/ 18.3%

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

   6. *-commutative 18.3%

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

   7. associate-/r* 18.3%

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

   8. /-rgt-identity 18.3%

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

   9. metadata-eval 18.3%

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

3. Simplified 18.3%

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

4. Taylor expanded in b around inf 96.0%

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

   1. +-commutative 96.0%

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

   2. mul-1-neg 96.0%

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

   3. unsub-neg 96.0%

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

   4. +-commutative 96.0%

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

   5. mul-1-neg 96.0%

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

   6. unsub-neg 96.0%

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

   7. associate-*r/ 96.0%

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

   8. associate-/l* 96.0%

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

   9. *-commutative 96.0%

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

   10. unpow2 96.0%

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

   11. associate-*l* 96.0%

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

   12. *-commutative 96.0%

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

6. Simplified 96.0%

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

7. Final simplification 96.0%

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

Alternative 5: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. neg-sub0 18.3%

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

   2. associate-+l- 18.3%

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

   3. sub0-neg 18.3%

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

   4. neg-mul-1 18.3%

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

   5. associate-*l/ 18.3%

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

   6. *-commutative 18.3%

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

   7. associate-/r* 18.3%

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

   8. /-rgt-identity 18.3%

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

   9. metadata-eval 18.3%

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

3. Simplified 18.3%

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

4. Taylor expanded in b around inf 94.5%

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

   1. +-commutative 94.5%

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

   2. mul-1-neg 94.5%

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

   3. unsub-neg 94.5%

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

   4. associate-*r/ 94.5%

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

   5. neg-mul-1 94.5%

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

   6. *-commutative 94.5%

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

   7. associate-/l* 94.5%

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

   8. unpow2 94.5%

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

6. Simplified 94.5%

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

7. Final simplification 94.5%

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

Alternative 6: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. neg-sub0 18.3%

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

   2. associate-+l- 18.3%

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

   3. sub0-neg 18.3%

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

   4. neg-mul-1 18.3%

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

   5. associate-*l/ 18.3%

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

   6. *-commutative 18.3%

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

   7. associate-/r* 18.3%

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

   8. /-rgt-identity 18.3%

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

   9. metadata-eval 18.3%

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

3. Simplified 18.3%

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

4. Taylor expanded in b around inf 94.0%

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

   1. distribute-lft-out 94.0%

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

   2. associate-/l* 94.1%

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

   3. associate-/l* 94.1%

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

   4. unpow2 94.1%

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

   5. unpow2 94.1%

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

6. Simplified 94.1%

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

7. Taylor expanded in c around 0 94.5%

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

   1. +-commutative 94.5%

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

   2. mul-1-neg 94.5%

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

   3. unsub-neg 94.5%

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

   4. associate-*r/ 94.5%

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

   5. neg-mul-1 94.5%

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

   6. associate-/l* 94.5%

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

   7. unpow2 94.5%

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

9. Simplified 94.5%

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

10. Final simplification 94.5%

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

Alternative 7: 90.0% 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 18.3%

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

   1. neg-sub0 18.3%

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

   2. associate-+l- 18.3%

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

   3. sub0-neg 18.3%

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

   4. neg-mul-1 18.3%

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

   5. associate-*l/ 18.3%

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

   6. *-commutative 18.3%

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

   7. associate-/r* 18.3%

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

   8. /-rgt-identity 18.3%

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

   9. metadata-eval 18.3%

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

3. Simplified 18.3%

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

4. Taylor expanded in b around inf 90.1%

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

   1. associate-*r/ 90.1%

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

   2. neg-mul-1 90.1%

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

6. Simplified 90.1%

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

7. Final simplification 90.1%

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

Reproduce

herbie shell --seed 2023185 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))