Average Error: 15.1 → 0.3
Time: 15.4s
Precision: binary64
Cost: 39040
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)} \cdot r \]
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (* (/ (sin b) (fma (sin b) (- (sin a)) (* (cos a) (cos b)))) r))
double code(double r, double a, double b) {
	return (r * sin(b)) / cos((a + b));
}
↓
double code(double r, double a, double b) {
	return (sin(b) / fma(sin(b), -sin(a), (cos(a) * cos(b)))) * r;
}

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(a + b)))
end
↓
function code(r, a, b)
	return Float64(Float64(sin(b) / fma(sin(b), Float64(-sin(a)), Float64(cos(a) * cos(b)))) * r)
end

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[r_, a_, b_] := N[(N[(N[Sin[b], $MachinePrecision] / N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision]) + N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]

\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)} \cdot r

Error

Derivation

  1. Initial program 15.1

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

  2. Simplified15.1

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

  3. Applied egg-rr0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

  4. Taylor expanded in r around 0 0.3

    \[\leadsto \color{blue}{\frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin a \cdot \sin b}} \]

  5. Simplified0.3

    \[\leadsto \color{blue}{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot r} \]
    Proof
    (*.f64 (/.f64 (sin.f64 b) (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 b) (sin.f64 a)))) r): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sin.f64 b) (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 a) (sin.f64 b))))) r): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sin.f64 b) r) (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 a) (sin.f64 b))))): 33 points increase in error, 30 points decrease in error

  6. Taylor expanded in b around inf 0.3

    \[\leadsto \color{blue}{\frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}} \cdot r \]

  7. Simplified0.3

    \[\leadsto \color{blue}{\frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)}} \cdot r \]
    Proof
    (/.f64 (sin.f64 b) (fma.f64 (sin.f64 b) (neg.f64 (sin.f64 a)) (*.f64 (cos.f64 a) (cos.f64 b)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (sin.f64 b) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 b) (neg.f64 (sin.f64 a))) (*.f64 (cos.f64 a) (cos.f64 b))))): 4 points increase in error, 6 points decrease in error
    (/.f64 (sin.f64 b) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (sin.f64 a)) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (sin.f64 b) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (neg.f64 (sin.f64 a)) (sin.f64 b))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (sin.f64 b) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 a) (sin.f64 b))))): 0 points increase in error, 0 points decrease in error

  8. Final simplification0.3

    \[\leadsto \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)} \cdot r \]

Alternatives

Alternative 1
Error0.3
Cost32704
\[\sin b \cdot \frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 2
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 3
Error15.6
Cost13384
\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -0.00062025530690066:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9.392951315321692 \cdot 10^{-21}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error15.4
Cost13384
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00062025530690066:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 1.2224342414590965 \cdot 10^{-14}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \]
Alternative 5
Error15.6
Cost13384
\[\begin{array}{l} t_0 := \frac{\sin b \cdot r}{\cos b}\\ \mathbf{if}\;b \leq -0.00062025530690066:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9.392951315321692 \cdot 10^{-21}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error15.2
Cost13248
\[\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}} \]
Alternative 7
Error15.2
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 8
Error15.1
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 9
Error28.8
Cost13120
\[r \cdot \frac{\sin b}{\cos a} \]
Alternative 10
Error31.5
Cost6848
\[r \cdot \left(b \cdot \frac{1}{\cos a}\right) \]
Alternative 11
Error31.5
Cost6720
\[\frac{r}{\frac{\cos a}{b}} \]
Alternative 12
Error31.5
Cost6720
\[r \cdot \frac{b}{\cos a} \]
Alternative 13
Error41.9
Cost192
\[b \cdot r \]

Error

Reproduce

herbie shell --seed 2022315 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))