Average Error: 15.3 → 0.3
Time: 13.8s
Precision: binary64
Cost: 39040
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)} \]
(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(Float64(r * sin(b)) / cos(Float64(a + b)))
end
↓
function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(sin(a), Float64(-sin(b)), Float64(cos(a) * cos(b))))
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[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 15.3

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

  2. Simplified15.3

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
    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-rr15.4

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

  4. Applied egg-rr0.3

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

  5. Simplified0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)}} \]
    Proof
    (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
    (fma.f64 (sin.f64 a) (neg.f64 (sin.f64 b)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 b) (cos.f64 a)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 a) (neg.f64 (sin.f64 b))) (*.f64 (cos.f64 b) (cos.f64 a)))): 7 points increase in error, 6 points decrease in error
    (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (sin.f64 b)) (sin.f64 a))) (*.f64 (cos.f64 b) (cos.f64 a))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (sin.f64 b) (sin.f64 a)))) (*.f64 (cos.f64 b) (cos.f64 a))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (cos.f64 b) (cos.f64 a)) (neg.f64 (*.f64 (sin.f64 b) (sin.f64 a))))): 0 points increase in error, 0 points decrease in error

  6. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 2
Error0.3
Cost32704
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 3
Error15.3
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 4
Error15.3
Cost13248
\[\frac{r \cdot \sin b}{\cos \left(b + a\right)} \]
Alternative 5
Error15.5
Cost7112
\[\begin{array}{l} \mathbf{if}\;b \leq -0.0045:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 800:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot 3\right) \cdot \left(\tan b \cdot 0.3333333333333333\right)\\ \end{array} \]
Alternative 6
Error15.5
Cost7112
\[\begin{array}{l} \mathbf{if}\;b \leq -0.0045:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 800:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot 3}{\frac{3}{\tan b}}\\ \end{array} \]
Alternative 7
Error15.5
Cost6984
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.0045:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 800:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error15.5
Cost6984
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.0045:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 800:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error25.1
Cost6592
\[r \cdot \tan b \]
Alternative 10
Error41.4
Cost576
\[\frac{r \cdot 3}{\frac{3}{b} - b} \]
Alternative 11
Error42.0
Cost192
\[r \cdot b \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))