Average Error: 15.4 → 0.3
Time: 14.2s
Precision: binary64
Cost: 39040
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)} \]
(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))
↓
(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (fma (sin a) (- (sin b)) (* (cos a) (cos b))))))
double code(double r, double a, double b) {
	return r * (sin(b) / cos((a + b)));
}
↓
double code(double r, double a, double b) {
	return r * (sin(b) / fma(sin(a), -sin(b), (cos(a) * cos(b))));
}

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

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

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

Error

Derivation

  1. Initial program 15.4

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

  2. Simplified15.4

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

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

  4. Applied egg-rr0.3

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

  5. Applied egg-rr0.3

    \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)}} \]

  6. Taylor expanded in b around inf 0.3

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

  7. Simplified0.3

    \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)}} \]
    Proof
    (/.f64 (sin.f64 b) (fma.f64 (sin.f64 a) (neg.f64 (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<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 a) (neg.f64 (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))))): 3 points increase in error, 4 points decrease in error
    (/.f64 (sin.f64 b) (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.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) (+.f64 (Rewrite<= distribute-lft-neg-out_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

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

Alternatives

Alternative 1
Error0.3
Cost32704
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 2
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 3
Error15.7
Cost13384
\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -1.4401327662638936 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.234609204574017 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error15.7
Cost13384
\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -1.4401327662638936 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.234609204574017 \cdot 10^{-7}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error15.4
Cost13248
\[\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}} \]
Alternative 6
Error15.4
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 7
Error29.2
Cost13120
\[r \cdot \frac{\sin b}{\cos a} \]
Alternative 8
Error29.2
Cost6984
\[\begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -7.594559937069208 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 844069690.1636496:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error39.5
Cost6592
\[r \cdot \sin b \]
Alternative 10
Error41.8
Cost576
\[\frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]
Alternative 11
Error42.4
Cost192
\[r \cdot b \]

Error

Reproduce

herbie shell --seed 2022297 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))