rsin A (should all be same)

?

Percentage Accurate: 76.1% → 99.5%
Time: 19.8s
Precision: binary64
Cost: 39040

?

\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \cdot r \]
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (* (/ (sin b) (fma (cos b) (cos a) (* (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(b), cos(a), (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(b), cos(a), 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[b], $MachinePrecision] * N[Cos[a], $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 b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \cdot r

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Derivation?

  1. Initial program 77.2%

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

  2. Simplified77.2%

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

    [Start]77.2

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

    associate-*r/ [<=]77.2

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

    *-commutative [<=]77.2

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

    +-commutative [=>]77.2

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

  3. Applied egg-rr99.5%

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

    [Start]77.2

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

    cos-sum [=>]99.5

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

    cancel-sign-sub-inv [=>]99.5

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

    fma-def [=>]99.5

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

  4. Final simplification99.5%

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

Alternatives

Alternative 1
Accuracy99.5%
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 2
Accuracy99.5%
Cost32704
\[\frac{\sin b \cdot r}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 3
Accuracy99.4%
Cost32512
\[\frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, -\sin a\right)} \]
Alternative 4
Accuracy77.1%
Cost26048
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \]
Alternative 5
Accuracy75.4%
Cost13385
\[\begin{array}{l} \mathbf{if}\;a \leq -1150000000 \lor \neg \left(a \leq 8.5 \cdot 10^{-17}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
Alternative 6
Accuracy75.4%
Cost13385
\[\begin{array}{l} \mathbf{if}\;a \leq -1150000000 \lor \neg \left(a \leq 8.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
Alternative 7
Accuracy76.1%
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 8
Accuracy76.1%
Cost13248
\[\frac{\sin b \cdot r}{\cos \left(b + a\right)} \]
Alternative 9
Accuracy75.9%
Cost7113
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00074 \lor \neg \left(b \leq 1.55 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \end{array} \]
Alternative 10
Accuracy75.9%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00074 \lor \neg \left(b \leq 1.55 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 11
Accuracy75.9%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00015 \lor \neg \left(b \leq 1.55 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 12
Accuracy60.6%
Cost6592
\[r \cdot \tan b \]
Alternative 13
Accuracy34.1%
Cost192
\[b \cdot r \]

Error

Reproduce?

herbie shell --seed 2023160 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))