rsin A (should all be same)

?

Percentage Accurate: 76.4% → 99.4%
Time: 14.8s
Precision: binary64
Cost: 39040

?

\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{\sin b}{\frac{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)}{r}} \]
(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(sin(b) / Float64(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[Sin[b], $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]

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

Local Percentage Accuracy vs ?

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.

Herbie found 20 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Initial program 78.8%

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

  2. Simplified78.8%

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

    [Start]78.8%

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

    associate-*r/ [<=]78.8%

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

    *-commutative [<=]78.8%

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

    +-commutative [=>]78.8%

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

  3. Applied egg-rr78.8%

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

    [Start]78.8%

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

    associate-*l/ [=>]78.8%

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

    associate-/l* [=>]78.8%

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

  4. Applied egg-rr99.4%

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

    [Start]78.8%

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

    cos-sum [=>]99.4%

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

    cancel-sign-sub-inv [=>]99.4%

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

    fma-def [=>]99.4%

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

  5. Final simplification99.4%

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

Alternatives

Alternative 1
Accuracy99.4%
Cost39040
\[\frac{\sin b}{\frac{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)}{r}} \]
Alternative 2
Accuracy99.4%
Cost39040
\[\frac{r}{\frac{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)}{\sin b}} \]
Alternative 3
Accuracy99.4%
Cost32832
\[\frac{\sin b}{\frac{\cos b \cdot \cos a}{r} - \frac{\sin b}{\frac{r}{\sin a}}} \]
Alternative 4
Accuracy99.4%
Cost26176
\[\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a} \]
Alternative 5
Accuracy99.4%
Cost26112
\[\frac{r}{\mathsf{fma}\left(\frac{1}{\tan b}, \cos a, -\sin a\right)} \]
Alternative 6
Accuracy99.5%
Cost19648
\[\frac{r}{\frac{\cos a}{\tan b} - \sin a} \]
Alternative 7
Accuracy75.1%
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -0.036 \lor \neg \left(b \leq 8.8 \cdot 10^{+17}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \end{array} \]
Alternative 8
Accuracy75.1%
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -0.036 \lor \neg \left(b \leq 8.8 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \end{array} \]
Alternative 9
Accuracy76.4%
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 10
Accuracy76.3%
Cost13248
\[\frac{\sin b}{\frac{\cos \left(b + a\right)}{r}} \]
Alternative 11
Accuracy76.4%
Cost13248
\[\frac{\sin b \cdot r}{\cos \left(b + a\right)} \]
Alternative 12
Accuracy54.7%
Cost13120
\[r \cdot \frac{\sin b}{\cos a} \]
Alternative 13
Accuracy55.0%
Cost7113
\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \lor \neg \left(b \leq 29500000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \end{array} \]
Alternative 14
Accuracy55.0%
Cost7113
\[\begin{array}{l} \mathbf{if}\;b \leq -7800000 \lor \neg \left(b \leq 1500000000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \]
Alternative 15
Accuracy55.0%
Cost7113
\[\begin{array}{l} \mathbf{if}\;b \leq -7500000 \lor \neg \left(b \leq 29000000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]
Alternative 16
Accuracy55.0%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \lor \neg \left(b \leq 29500000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 17
Accuracy55.0%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \lor \neg \left(b \leq 170000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]
Alternative 18
Accuracy38.9%
Cost6592
\[\sin b \cdot r \]
Alternative 19
Accuracy35.2%
Cost576
\[\frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]
Alternative 20
Accuracy34.8%
Cost192
\[b \cdot r \]

Reproduce?

herbie shell --seed 2023165 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))