Quadratic roots, wide range

Average Error: 52.4 → 0.1

Time: 13.1s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

↓

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

↓

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

Derivation

1. Initial program 52.4

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

2. Simplified 52.4

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   Proof
   (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))): 2 points increase in error, 0 points decrease in error

3. Applied egg-rr 52.1

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

4. Simplified 52.0

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}}{a \cdot 2} \]
   Proof
   (/.f64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 c (*.f64 a -4) (*.f64 b b))) (neg.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 (*.f64 b b) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 c (*.f64 a -4)) (*.f64 b b)))) (neg.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))))) (*.f64 a 2)): 12 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 (*.f64 b b) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 c (*.f64 a -4))))) (neg.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))))) (*.f64 a 2)): 12 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 (*.f64 b b) (Rewrite=> fma-def_binary64 (fma.f64 b b (*.f64 c (*.f64 a -4))))) (neg.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))))) (*.f64 a 2)): 0 points increase in error, 12 points decrease in error
   (/.f64 (/.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 c (*.f64 a -4)))) 1)) (neg.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))))) (*.f64 a 2)): 0 points increase in error, 12 points decrease in error
   (/.f64 (/.f64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 c (*.f64 a -4)))) 1) (neg.f64 (+.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 c (*.f64 a -4)) (*.f64 b b))))))) (*.f64 a 2)): 12 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 c (*.f64 a -4)))) 1) (neg.f64 (+.f64 b (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 c (*.f64 a -4)))))))) (*.f64 a 2)): 12 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 c (*.f64 a -4)))) 1) (neg.f64 (+.f64 b (sqrt.f64 (Rewrite=> fma-def_binary64 (fma.f64 b b (*.f64 c (*.f64 a -4)))))))) (*.f64 a 2)): 0 points increase in error, 12 points decrease in error
   (/.f64 (/.f64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 c (*.f64 a -4)))) 1) (neg.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4)))))) (sqrt.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4)))))))))) (*.f64 a 2)): 0 points increase in error, 12 points decrease in error
   (/.f64 (/.f64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 c (*.f64 a -4)))) 1) (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (sqrt.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4))))))) (sqrt.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4))))))))) (*.f64 a 2)): 12 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite=> /-rgt-identity_binary64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 c (*.f64 a -4))))) (*.f64 (neg.f64 (sqrt.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4))))))) (sqrt.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4)))))))) (*.f64 a 2)): 12 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 c (*.f64 a -4)))) (sqrt.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4))))))) (neg.f64 (sqrt.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4))))))))) (*.f64 a 2)): 12 points increase in error, 0 points decrease in error

5. Taylor expanded in b around 0 0.4

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

6. Applied egg-rr 0.4

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

7. Simplified 0.2

   \[\leadsto \frac{\color{blue}{\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot a}}{a \cdot 2} \]
   Proof
   (/.f64 (*.f64 (/.f64 (*.f64 -4 c) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))) a) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 4 -1)) c) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))) a) (*.f64 a 2)): 0 points increase in error, 9 points decrease in error
   (/.f64 (*.f64 (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 4 (/.f64 -1 c))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))) a) (*.f64 a 2)): 5 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 4 c) -1)) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))) a) (*.f64 a 2)): 0 points increase in error, 5 points decrease in error
   (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (/.f64 (*.f64 4 c) -1) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))) a))) (*.f64 a 2)): 5 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 4 (/.f64 -1 c))) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))) a)) (*.f64 a 2)): 0 points increase in error, 5 points decrease in error
   (/.f64 (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 4 -1) c)) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))) a)) (*.f64 a 2)): 5 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (*.f64 (Rewrite=> metadata-eval -4) c) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))) a)) (*.f64 a 2)): 0 points increase in error, 5 points decrease in error
   (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 -4 (/.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))) a)) c)) (*.f64 a 2)): 8 points increase in error, 0 points decrease in error

8. Applied egg-rr 51.1

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(c \cdot \left(-4 \cdot a\right)\right) \cdot \frac{0.5}{a}}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right)} - 1} \]

9. Simplified 0.1

   \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]
   Proof
   (*.f64 -2 (/.f64 c (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 -2 (/.f64 c (+.f64 b (sqrt.f64 (fma.f64 c (Rewrite<= *-commutative_binary64 (*.f64 -4 a)) (*.f64 b b)))))): 5 points increase in error, 3 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -2 c) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b)))))): 0 points increase in error, 5 points decrease in error
   (/.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 -4 2)) c) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 1 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 -4 2) (Rewrite<= /-rgt-identity_binary64 (/.f64 c 1))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 0 points increase in error, 1 points decrease in error
   (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -4 c) (*.f64 2 1))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 7 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (*.f64 -4 c) (*.f64 2 (Rewrite<= *-inverses_binary64 (/.f64 a a)))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 0 points increase in error, 7 points decrease in error
   (/.f64 (/.f64 (*.f64 -4 c) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 a) a))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 1 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (*.f64 -4 c) (/.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 1 1/2)) a) a)) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 16 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (*.f64 -4 c) (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 1/2 a))) a)) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 0 points increase in error, 16 points decrease in error
   (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 -4 c) a) (/.f64 1 (/.f64 1/2 a)))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 0 points increase in error, 1 points decrease in error
   (/.f64 (/.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 c -4)) a) (/.f64 1 (/.f64 1/2 a))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 3 points increase in error, 1 points decrease in error
   (/.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 c (*.f64 -4 a))) (/.f64 1 (/.f64 1/2 a))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 1 points increase in error, 3 points decrease in error
   (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (*.f64 c (*.f64 -4 a)) 1) (/.f64 1/2 a))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 1 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (Rewrite=> /-rgt-identity_binary64 (*.f64 c (*.f64 -4 a))) (/.f64 1/2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))): 0 points increase in error, 1 points decrease in error
   (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (/.f64 (*.f64 (*.f64 c (*.f64 -4 a)) (/.f64 1/2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b)))))))): 6 points increase in error, 1 points decrease in error
   (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (/.f64 (*.f64 (*.f64 c (*.f64 -4 a)) (/.f64 1/2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 -4 a) (*.f64 b b))))))) 1)): 11 points increase in error, 0 points decrease in error

10. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	7744

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

Alternative 2
Error	3.1
Cost	7232

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

Alternative 3
Error	3.2
Cost	1344

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

Alternative 4
Error	6.3
Cost	256

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

Reproduce

herbie shell --seed 2022343 
(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)))