Average Error: 14.9 → 0.3
Time: 19.5s
Precision: binary64
Cost: 39040
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \left(-\sin a\right)\right)} \cdot r \]
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (* (/ (sin b) (fma (cos a) (cos b) (* (sin b) (- (sin a))))) 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(cos(a), cos(b), (sin(b) * -sin(a)))) * 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(cos(a), cos(b), Float64(sin(b) * Float64(-sin(a))))) * 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[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]

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

Error

Derivation

  1. Initial program 14.9

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

  2. Simplified14.9

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

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

  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

    \[\leadsto \color{blue}{\frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \left(-\sin a\right)\right)} \cdot r} \]
    Proof
    (*.f64 (/.f64 (sin.f64 b) (fma.f64 (cos.f64 a) (cos.f64 b) (*.f64 (sin.f64 b) (neg.f64 (sin.f64 a))))) r): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sin.f64 b) (fma.f64 (cos.f64 a) (cos.f64 b) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 b) (sin.f64 a)))))) r): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sin.f64 b) (fma.f64 (cos.f64 a) (cos.f64 b) (neg.f64 (Rewrite=> *-commutative_binary64 (*.f64 (sin.f64 a) (sin.f64 b)))))) r): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sin.f64 b) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 a) (sin.f64 b))))) r): 9 points increase in error, 8 points decrease in error
    (Rewrite<= associate-/r/_binary64 (/.f64 (sin.f64 b) (/.f64 (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 a) (sin.f64 b))) r))): 40 points increase in error, 27 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))))): 36 points increase in error, 37 points decrease in error

  6. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost32704
\[\frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 2
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 3
Error14.3
Cost19648
\[\frac{\sin b \cdot r}{\cos a \cdot \cos b} \]
Alternative 4
Error14.3
Cost19648
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b} \]
Alternative 5
Error14.9
Cost13384
\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -0.0073:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.00052:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error14.9
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -0.0073:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 0.00035:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
Alternative 7
Error14.9
Cost13248
\[\frac{\sin b}{\frac{\cos \left(b + a\right)}{r}} \]
Alternative 8
Error15.0
Cost13248
\[\frac{r}{\frac{\cos \left(a - b\right)}{\sin b}} \]
Alternative 9
Error14.9
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 10
Error14.9
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 11
Error15.0
Cost6984
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.00017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.041:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 12
Error15.0
Cost6984
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.041:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 13
Error25.8
Cost6592
\[r \cdot \tan b \]
Alternative 14
Error42.0
Cost192
\[b \cdot r \]

Error

Reproduce

herbie shell --seed 2022296 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))