Average Error: 15.1 → 0.3
Time: 17.8s
Precision: binary64
Cost: 39040
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{\sin b \cdot r}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \]
(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))
↓
(FPCore (r a b)
 :precision binary64
 (/ (* (sin b) r) (fma (cos b) (cos a) (* (sin b) (- (sin a))))))
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) * r) / fma(cos(b), cos(a), (sin(b) * -sin(a)));
}

function code(r, a, b)
	return Float64(r * Float64(sin(b) / cos(Float64(a + b))))
end
↓
function code(r, a, b)
	return Float64(Float64(sin(b) * r) / fma(cos(b), cos(a), Float64(sin(b) * Float64(-sin(a)))))
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[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 15.1

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

  2. Simplified15.1

    \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
    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. Taylor expanded in r around 0 0.3

    \[\leadsto \color{blue}{\frac{\sin b \cdot r}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b}} \]

  5. Simplified0.3

    \[\leadsto \color{blue}{\frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin a \cdot \sin b}} \]
    Proof
    (/.f64 (*.f64 (sin.f64 b) r) (-.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
    (/.f64 (*.f64 (sin.f64 b) r) (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (neg.f64 (*.f64 (sin.f64 a) (sin.f64 b)))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (*.f64 (sin.f64 b) r) (+.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b)))))): 0 points increase in error, 0 points decrease in error
    (/.f64 (*.f64 (sin.f64 b) r) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))))): 0 points increase in error, 0 points decrease in error

  6. Applied egg-rr0.3

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

  7. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 2
Error0.3
Cost32704
\[\frac{\sin b \cdot r}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 3
Error0.4
Cost26176
\[\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a} \]
Alternative 4
Error15.3
Cost13384
\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -6.853767436369812 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.6579676623039976:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error15.3
Cost13384
\[\begin{array}{l} t_0 := \frac{\sin b \cdot r}{\cos b}\\ \mathbf{if}\;b \leq -6.853767436369812 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.6579676623039976:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error15.3
Cost13384
\[\begin{array}{l} \mathbf{if}\;b \leq -6.853767436369812 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 0.6579676623039976:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
Alternative 7
Error15.4
Cost13384
\[\begin{array}{l} t_0 := \sin b \cdot r\\ t_1 := \frac{t_0}{\cos a}\\ \mathbf{if}\;a \leq -12071.683251395545:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.8730953776777996 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error15.4
Cost13384
\[\begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;a \leq -12071.683251395545:\\ \;\;\;\;\frac{t_0}{\cos a}\\ \mathbf{elif}\;a \leq 2.8730953776777996 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
Alternative 9
Error15.4
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -12071.683251395545:\\ \;\;\;\;r \cdot \frac{1}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 2.8730953776777996 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
Alternative 10
Error15.2
Cost13248
\[\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}} \]
Alternative 11
Error15.1
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 12
Error31.3
Cost6720
\[\frac{b}{\frac{\cos a}{r}} \]
Alternative 13
Error31.3
Cost6720
\[\frac{b \cdot r}{\cos a} \]
Alternative 14
Error31.3
Cost6720
\[b \cdot \frac{r}{\cos a} \]
Alternative 15
Error42.0
Cost192
\[b \cdot r \]

Error

Reproduce

herbie shell --seed 2022300 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))