Quadratic roots, medium range

Percentage Accurate: 31.9% → 95.2%

Time: 12.1s

Alternatives: 6

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.9% 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: 95.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, \mathsf{fma}\left(-1, \frac{c}{b}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -1.0
  (/ (* a (* c c)) (pow b 3.0))
  (fma
   -0.25
   (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0)))
   (fma -1.0 (/ c b) (* -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a))))))))

C

double code(double a, double b, double c) {
	return fma(-1.0, ((a * (c * c)) / pow(b, 3.0)), fma(-0.25, ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0))), fma(-1.0, (c / b), (-2.0 * (pow(c, 3.0) / (pow(b, 5.0) / (a * a)))))));
}

Julia

function code(a, b, c)
	return fma(-1.0, Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)), fma(-0.25, Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0))), fma(-1.0, Float64(c / b), Float64(-2.0 * Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a)))))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(c / b), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.6%

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

   1. /-rgt-identity 34.6%

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

   2. metadata-eval 34.6%

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

   3. associate-/l* 34.6%

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

   4. associate-*r/ 34.6%

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

   5. +-commutative 34.6%

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

   6. unsub-neg 34.6%

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

   7. fma-neg 34.5%

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

   8. associate-*l* 34.5%

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

   9. *-commutative 34.5%

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

   10. distribute-rgt-neg-in 34.5%

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

   11. metadata-eval 34.5%

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

   12. associate-/r* 34.5%

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

   13. metadata-eval 34.5%

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

   14. metadata-eval 34.5%

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

3. Simplified 34.5%

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

4. Taylor expanded in b around inf 95.3%

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

   1. fma-def 95.3%

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

   2. *-commutative 95.3%

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

   3. unpow2 95.3%

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

   4. fma-def 95.3%

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

6. Simplified 95.3%

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

7. Taylor expanded in c around 0 95.3%

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

   1. +-commutative 95.3%

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

   2. distribute-rgt-out 95.3%

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

   3. associate-*l* 95.3%

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

   4. times-frac 95.3%

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

9. Simplified 95.3%

   \[\leadsto \mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}}, \mathsf{fma}\left(-1, \frac{c}{b}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right)\right)\right) \]

10. Final simplification 95.3%

    \[\leadsto \mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, \mathsf{fma}\left(-1, \frac{c}{b}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right)\right)\right) \]

Alternative 2: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-0.5, \frac{\left({c}^{4} \cdot {a}^{3}\right) \cdot 10}{{b}^{7}}, \frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (-
   (fma
    -0.5
    (/ (* (* (pow c 4.0) (pow a 3.0)) 10.0) (pow b 7.0))
    (/ (* -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(-0.5, (((pow(c, 4.0) * pow(a, 3.0)) * 10.0) / pow(b, 7.0)), ((-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(Float64(fma(-0.5, Float64(Float64(Float64((c ^ 4.0) * (a ^ 3.0)) * 10.0) / (b ^ 7.0)), Float64(Float64(-2.0 * (c ^ 3.0)) / Float64((b ^ 5.0) / Float64(a * a)))) - Float64(c / b)) - Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.5 * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.6%

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

   1. /-rgt-identity 34.6%

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

   2. metadata-eval 34.6%

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

   3. associate-/l* 34.6%

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

   4. associate-*r/ 34.6%

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

   5. +-commutative 34.6%

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

   6. unsub-neg 34.6%

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

   7. fma-neg 34.5%

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

   8. associate-*l* 34.5%

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

   9. *-commutative 34.5%

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

   10. distribute-rgt-neg-in 34.5%

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

   11. metadata-eval 34.5%

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

   12. associate-/r* 34.5%

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

   13. metadata-eval 34.5%

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

   14. metadata-eval 34.5%

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

3. Simplified 34.5%

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

   1. fma-udef 34.6%

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

   2. *-commutative 34.6%

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

   3. metadata-eval 34.6%

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

   4. cancel-sign-sub-inv 34.6%

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

   5. associate-*l* 34.6%

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

   6. *-un-lft-identity 34.6%

      \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

   7. prod-diff 34.5%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

5. Applied egg-rr 34.5%

   \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
6. Step-by-step derivation

   1. +-commutative 34.5%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

   2. fma-udef 34.5%

      \[\leadsto \left(\sqrt{\color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   3. *-rgt-identity 34.5%

      \[\leadsto \left(\sqrt{\left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   4. *-rgt-identity 34.5%

      \[\leadsto \left(\sqrt{\left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   5. count-2 34.5%

      \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   6. *-commutative 34.5%

      \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(\left(c \cdot -4\right) \cdot a\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   7. *-commutative 34.5%

      \[\leadsto \left(\sqrt{2 \cdot \left(\color{blue}{\left(-4 \cdot c\right)} \cdot a\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   8. associate-*r* 34.5%

      \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(-4 \cdot \left(c \cdot a\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   9. *-rgt-identity 34.5%

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

   10. fma-neg 34.5%

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

   11. *-commutative 34.5%

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

   12. *-commutative 34.5%

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

   13. associate-*r* 34.5%

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

7. Simplified 34.5%

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

   1. add-log-exp 24.9%

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

   2. associate-*r* 24.9%

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

   3. metadata-eval 24.9%

      \[\leadsto \log \left(e^{\sqrt{\color{blue}{-8} \cdot \left(c \cdot a\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} - b}\right) \cdot \frac{0.5}{a} \]

   4. cancel-sign-sub-inv 24.9%

      \[\leadsto \log \left(e^{\sqrt{-8 \cdot \left(c \cdot a\right) + \color{blue}{\left(b \cdot b + \left(--4\right) \cdot \left(c \cdot a\right)\right)}} - b}\right) \cdot \frac{0.5}{a} \]

   5. metadata-eval 24.9%

      \[\leadsto \log \left(e^{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + \color{blue}{4} \cdot \left(c \cdot a\right)\right)} - b}\right) \cdot \frac{0.5}{a} \]

9. Applied egg-rr 24.9%

   \[\leadsto \color{blue}{\log \left(e^{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} - b}\right)} \cdot \frac{0.5}{a} \]
10. Step-by-step derivation

    1. flip-- 24.8%

       \[\leadsto \log \left(e^{\color{blue}{\frac{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} \cdot \sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}}}\right) \cdot \frac{0.5}{a} \]

    2. add-sqr-sqrt 25.4%

       \[\leadsto \log \left(e^{\frac{\color{blue}{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)\right)} - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    3. fma-def 25.4%

       \[\leadsto \log \left(e^{\frac{\color{blue}{\mathsf{fma}\left(-8, c \cdot a, b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    4. fma-def 25.3%

       \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-8, c \cdot a, \color{blue}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    5. associate-*r* 25.3%

       \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \color{blue}{\left(4 \cdot c\right) \cdot a}\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    6. fma-def 25.3%

       \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right) - b \cdot b}{\sqrt{\color{blue}{\mathsf{fma}\left(-8, c \cdot a, b \cdot b + 4 \cdot \left(c \cdot a\right)\right)}} + b}}\right) \cdot \frac{0.5}{a} \]

    7. fma-def 25.3%

       \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \color{blue}{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    8. associate-*r* 25.3%

       \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \color{blue}{\left(4 \cdot c\right) \cdot a}\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

11. Applied egg-rr 25.3%

    \[\leadsto \log \left(e^{\color{blue}{\frac{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}}\right) \cdot \frac{0.5}{a} \]
12. Step-by-step derivation

    1. fma-udef 25.3%

       \[\leadsto \log \left(e^{\frac{\color{blue}{\left(-8 \cdot \left(c \cdot a\right) + \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    2. fma-udef 25.4%

       \[\leadsto \log \left(e^{\frac{\left(-8 \cdot \left(c \cdot a\right) + \color{blue}{\left(b \cdot b + \left(4 \cdot c\right) \cdot a\right)}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    3. unpow2 25.4%

       \[\leadsto \log \left(e^{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left(\color{blue}{{b}^{2}} + \left(4 \cdot c\right) \cdot a\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    4. associate-*r* 25.4%

       \[\leadsto \log \left(e^{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left({b}^{2} + \color{blue}{4 \cdot \left(c \cdot a\right)}\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    5. *-commutative 25.4%

       \[\leadsto \log \left(e^{\frac{\left(\color{blue}{\left(c \cdot a\right) \cdot -8} + \left({b}^{2} + 4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    6. +-commutative 25.4%

       \[\leadsto \log \left(e^{\frac{\left(\left(c \cdot a\right) \cdot -8 + \color{blue}{\left(4 \cdot \left(c \cdot a\right) + {b}^{2}\right)}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    7. associate-+r+ 25.5%

       \[\leadsto \log \left(e^{\frac{\color{blue}{\left(\left(\left(c \cdot a\right) \cdot -8 + 4 \cdot \left(c \cdot a\right)\right) + {b}^{2}\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    8. *-commutative 25.5%

       \[\leadsto \log \left(e^{\frac{\left(\left(\color{blue}{-8 \cdot \left(c \cdot a\right)} + 4 \cdot \left(c \cdot a\right)\right) + {b}^{2}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    9. associate-*r* 25.5%

       \[\leadsto \log \left(e^{\frac{\left(\left(\color{blue}{\left(-8 \cdot c\right) \cdot a} + 4 \cdot \left(c \cdot a\right)\right) + {b}^{2}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    10. associate-*r* 25.5%

        \[\leadsto \log \left(e^{\frac{\left(\left(\left(-8 \cdot c\right) \cdot a + \color{blue}{\left(4 \cdot c\right) \cdot a}\right) + {b}^{2}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    11. distribute-rgt-in 25.5%

        \[\leadsto \log \left(e^{\frac{\left(\color{blue}{a \cdot \left(-8 \cdot c + 4 \cdot c\right)} + {b}^{2}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    12. fma-def 25.5%

        \[\leadsto \log \left(e^{\frac{\color{blue}{\mathsf{fma}\left(a, -8 \cdot c + 4 \cdot c, {b}^{2}\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    13. distribute-rgt-out 25.5%

        \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(a, \color{blue}{c \cdot \left(-8 + 4\right)}, {b}^{2}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    14. metadata-eval 25.5%

        \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(a, c \cdot \color{blue}{-4}, {b}^{2}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

    15. unpow2 25.5%

        \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(a, c \cdot -4, \color{blue}{b \cdot b}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8, c \cdot a, \mathsf{fma}\left(b, b, \left(4 \cdot c\right) \cdot a\right)\right)} + b}}\right) \cdot \frac{0.5}{a} \]

13. Simplified 25.5%

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

14. Taylor expanded in b around inf 95.3%

    \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.5 \cdot \frac{2 \cdot \left({c}^{4} \cdot {a}^{3}\right) + 8 \cdot \left({c}^{4} \cdot {a}^{3}\right)}{{b}^{7}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]
15. Step-by-step derivation

    1. +-commutative 95.3%

       \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.5 \cdot \frac{2 \cdot \left({c}^{4} \cdot {a}^{3}\right) + 8 \cdot \left({c}^{4} \cdot {a}^{3}\right)}{{b}^{7}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

    2. mul-1-neg 95.3%

       \[\leadsto \left(-1 \cdot \frac{c}{b} + \left(-0.5 \cdot \frac{2 \cdot \left({c}^{4} \cdot {a}^{3}\right) + 8 \cdot \left({c}^{4} \cdot {a}^{3}\right)}{{b}^{7}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

    3. unsub-neg 95.3%

       \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.5 \cdot \frac{2 \cdot \left({c}^{4} \cdot {a}^{3}\right) + 8 \cdot \left({c}^{4} \cdot {a}^{3}\right)}{{b}^{7}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

16. Simplified 95.3%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{\left({c}^{4} \cdot {a}^{3}\right) \cdot 10}{{b}^{7}}, \frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}} \]

17. Final simplification 95.3%

    \[\leadsto \left(\mathsf{fma}\left(-0.5, \frac{\left({c}^{4} \cdot {a}^{3}\right) \cdot 10}{{b}^{7}}, \frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \]

Alternative 3: 93.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.25, \frac{c}{\frac{b}{-4}}, \mathsf{fma}\left(0.03125, \left(a \cdot a\right) \cdot \frac{{c}^{3} \cdot -64}{{b}^{5}}, \frac{-0.0625}{\frac{{b}^{3}}{a \cdot \left(\left(c \cdot c\right) \cdot 16\right)}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  0.25
  (/ c (/ b -4.0))
  (fma
   0.03125
   (* (* a a) (/ (* (pow c 3.0) -64.0) (pow b 5.0)))
   (/ -0.0625 (/ (pow b 3.0) (* a (* (* c c) 16.0)))))))

C

double code(double a, double b, double c) {
	return fma(0.25, (c / (b / -4.0)), fma(0.03125, ((a * a) * ((pow(c, 3.0) * -64.0) / pow(b, 5.0))), (-0.0625 / (pow(b, 3.0) / (a * ((c * c) * 16.0))))));
}

Julia

function code(a, b, c)
	return fma(0.25, Float64(c / Float64(b / -4.0)), fma(0.03125, Float64(Float64(a * a) * Float64(Float64((c ^ 3.0) * -64.0) / (b ^ 5.0))), Float64(-0.0625 / Float64((b ^ 3.0) / Float64(a * Float64(Float64(c * c) * 16.0))))))
end

Wolfram

code[a_, b_, c_] := N[(0.25 * N[(c / N[(b / -4.0), $MachinePrecision]), $MachinePrecision] + N[(0.03125 * N[(N[(a * a), $MachinePrecision] * N[(N[(N[Power[c, 3.0], $MachinePrecision] * -64.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 / N[(N[Power[b, 3.0], $MachinePrecision] / N[(a * N[(N[(c * c), $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.6%

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

   1. /-rgt-identity 34.6%

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

   2. metadata-eval 34.6%

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

   3. associate-/l* 34.6%

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

   4. associate-*r/ 34.6%

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

   5. +-commutative 34.6%

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

   6. unsub-neg 34.6%

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

   7. fma-neg 34.5%

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

   8. associate-*l* 34.5%

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

   9. *-commutative 34.5%

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

   10. distribute-rgt-neg-in 34.5%

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

   11. metadata-eval 34.5%

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

   12. associate-/r* 34.5%

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

   13. metadata-eval 34.5%

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

   14. metadata-eval 34.5%

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

3. Simplified 34.5%

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

   1. fma-udef 34.6%

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

   2. *-commutative 34.6%

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

   3. metadata-eval 34.6%

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

   4. cancel-sign-sub-inv 34.6%

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

   5. associate-*l* 34.6%

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

   6. *-un-lft-identity 34.6%

      \[\leadsto \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]

   7. prod-diff 34.5%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

5. Applied egg-rr 34.5%

   \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
6. Step-by-step derivation

   1. +-commutative 34.5%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

   2. fma-udef 34.5%

      \[\leadsto \left(\sqrt{\color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   3. *-rgt-identity 34.5%

      \[\leadsto \left(\sqrt{\left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   4. *-rgt-identity 34.5%

      \[\leadsto \left(\sqrt{\left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   5. count-2 34.5%

      \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   6. *-commutative 34.5%

      \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(\left(c \cdot -4\right) \cdot a\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   7. *-commutative 34.5%

      \[\leadsto \left(\sqrt{2 \cdot \left(\color{blue}{\left(-4 \cdot c\right)} \cdot a\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   8. associate-*r* 34.5%

      \[\leadsto \left(\sqrt{2 \cdot \color{blue}{\left(-4 \cdot \left(c \cdot a\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   9. *-rgt-identity 34.5%

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

   10. fma-neg 34.5%

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

   11. *-commutative 34.5%

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

   12. *-commutative 34.5%

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

   13. associate-*r* 34.5%

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

7. Simplified 34.5%

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

8. Taylor expanded in a around 0 93.7%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-8 \cdot c - -4 \cdot c}{b} + \left(-0.0625 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{2} \cdot a}{{b}^{3}} + 0.03125 \cdot \frac{{\left(-8 \cdot c - -4 \cdot c\right)}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]

10. Simplified 93.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{c}{\frac{b}{-4}}, \mathsf{fma}\left(0.03125, \frac{{c}^{3} \cdot -64}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{-0.0625}{\frac{{b}^{3}}{a \cdot \left(\left(c \cdot c\right) \cdot 16\right)}}\right)\right)} \]

11. Final simplification 93.7%

    \[\leadsto \mathsf{fma}\left(0.25, \frac{c}{\frac{b}{-4}}, \mathsf{fma}\left(0.03125, \left(a \cdot a\right) \cdot \frac{{c}^{3} \cdot -64}{{b}^{5}}, \frac{-0.0625}{\frac{{b}^{3}}{a \cdot \left(\left(c \cdot c\right) \cdot 16\right)}}\right)\right) \]

Alternative 4: 93.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.6%

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

   1. /-rgt-identity 34.6%

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

   2. metadata-eval 34.6%

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

   3. associate-/l* 34.6%

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

   4. associate-*r/ 34.6%

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

   5. +-commutative 34.6%

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

   6. unsub-neg 34.6%

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

   7. fma-neg 34.5%

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

   8. associate-*l* 34.5%

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

   9. *-commutative 34.5%

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

   10. distribute-rgt-neg-in 34.5%

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

   11. metadata-eval 34.5%

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

   12. associate-/r* 34.5%

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

   13. metadata-eval 34.5%

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

   14. metadata-eval 34.5%

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

3. Simplified 34.5%

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

4. Taylor expanded in b around inf 93.3%

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

   1. fma-def 93.3%

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

   2. associate-/l* 93.3%

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

   3. unpow2 93.3%

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

   4. unpow2 93.3%

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

   5. fma-def 93.3%

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

   6. cube-prod 93.3%

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

   7. associate-*r/ 93.3%

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

6. Simplified 93.3%

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

7. Taylor expanded in c around 0 93.7%

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

   1. +-commutative 93.7%

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

   2. mul-1-neg 93.7%

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

   3. unsub-neg 93.7%

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

   4. +-commutative 93.7%

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

   5. mul-1-neg 93.7%

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

   6. unsub-neg 93.7%

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

   7. unpow2 93.7%

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

   8. associate-/l* 93.7%

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

   9. associate-*r/ 93.7%

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

   10. associate-*l/ 93.7%

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

   11. unpow2 93.7%

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

9. Simplified 93.7%

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

10. Final simplification 93.7%

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

Alternative 5: 90.5% 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 34.6%

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

   1. /-rgt-identity 34.6%

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

   2. metadata-eval 34.6%

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

   3. associate-/l* 34.6%

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

   4. associate-*r/ 34.6%

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

   5. +-commutative 34.6%

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

   6. unsub-neg 34.6%

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

   7. fma-neg 34.5%

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

   8. associate-*l* 34.5%

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

   9. *-commutative 34.5%

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

   10. distribute-rgt-neg-in 34.5%

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

   11. metadata-eval 34.5%

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

   12. associate-/r* 34.5%

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

   13. metadata-eval 34.5%

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

   14. metadata-eval 34.5%

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

3. Simplified 34.5%

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

4. Taylor expanded in b around inf 90.2%

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

   1. +-commutative 90.2%

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

   2. mul-1-neg 90.2%

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

   3. unsub-neg 90.2%

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

   4. mul-1-neg 90.2%

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

   5. distribute-neg-frac 90.2%

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

   6. associate-/l* 90.2%

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

   7. unpow2 90.2%

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

6. Simplified 90.2%

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

7. Final simplification 90.2%

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

Alternative 6: 81.0% 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 34.6%

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

   1. /-rgt-identity 34.6%

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

   2. metadata-eval 34.6%

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

   3. associate-/l* 34.6%

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

   4. associate-*r/ 34.6%

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

   5. +-commutative 34.6%

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

   6. unsub-neg 34.6%

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

   7. fma-neg 34.5%

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

   8. associate-*l* 34.5%

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

   9. *-commutative 34.5%

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

   10. distribute-rgt-neg-in 34.5%

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

   11. metadata-eval 34.5%

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

   12. associate-/r* 34.5%

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

   13. metadata-eval 34.5%

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

   14. metadata-eval 34.5%

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

3. Simplified 34.5%

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

4. Taylor expanded in b around inf 79.4%

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

   1. mul-1-neg 79.4%

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

   2. distribute-neg-frac 79.4%

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

6. Simplified 79.4%

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

7. Final simplification 79.4%

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

Reproduce

herbie shell --seed 2023199 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))