rsin B (should all be same)

Average Accuracy: 76.6% → 99.5%

Time: 22.9s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))

↓

(FPCore (r a b)
 :precision binary64
 (* (/ (sin b) (fma (sin b) (- (sin a)) (* (cos a) (cos b)))) r))

C

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(sin(b), -sin(a), (cos(a) * cos(b)))) * r;
}

Julia

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) / fma(sin(b), Float64(-sin(a)), Float64(cos(a) * cos(b)))) * r)
end

Wolfram

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] / N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision]) + N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]

Derivation

1. Initial program 74.6%

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

2. Simplified 74.6%

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

   [Start] 74.6	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 74.6	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 99.5%

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

   [Start] 74.6	\[ r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
   cos-sum [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   cancel-sign-sub-inv [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]
   fma-def [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

4. Taylor expanded in r around 0 99.4%

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

5. Simplified 99.5%

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

   [Start] 99.4	\[ \frac{\sin b \cdot r}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \]
   associate-/l* [=>] 99.4	\[ \color{blue}{\frac{\sin b}{\frac{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b}{r}}} \]
   associate-/r/ [=>] 99.5	\[ \color{blue}{\frac{\sin b}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \cdot r} \]
   mul-1-neg [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\left(-\sin a \cdot \sin b\right)} + \cos a \cdot \cos b} \cdot r \]
   distribute-lft-neg-out [<=] 99.5	\[ \frac{\sin b}{\color{blue}{\left(-\sin a\right) \cdot \sin b} + \cos a \cdot \cos b} \cdot r \]
   +-commutative [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b + \left(-\sin a\right) \cdot \sin b}} \cdot r \]
   *-commutative [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} + \left(-\sin a\right) \cdot \sin b} \cdot r \]
   cancel-sign-sub-inv [<=] 99.5	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin a \cdot \sin b}} \cdot r \]
   *-commutative [<=] 99.5	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b} - \sin a \cdot \sin b} \cdot r \]

6. Applied egg-rr 99.5%

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

   [Start] 99.5	\[ \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b} \cdot r \]
   sub-neg [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b + \left(-\sin a \cdot \sin b\right)}} \cdot r \]
   +-commutative [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\left(-\sin a \cdot \sin b\right) + \cos a \cdot \cos b}} \cdot r \]
   *-commutative [=>] 99.5	\[ \frac{\sin b}{\left(-\color{blue}{\sin b \cdot \sin a}\right) + \cos a \cdot \cos b} \cdot r \]
   distribute-rgt-neg-in [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\sin b \cdot \left(-\sin a\right)} + \cos a \cdot \cos b} \cdot r \]
   fma-def [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)}} \cdot r \]

7. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	39040

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

Alternative 2
Accuracy	99.5%
Cost	32704

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

Alternative 3
Accuracy	98.7%
Cost	26432

\[\frac{1}{\frac{\cos b}{r} \cdot \frac{\cos a}{\sin b} - \frac{\sin a}{r}} \]

Alternative 4
Accuracy	98.8%
Cost	26432

\[\frac{1}{\cos a \cdot \frac{\cos b}{\sin b \cdot r} - \frac{\sin a}{r}} \]

Alternative 5
Accuracy	77.5%
Cost	26048

\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \]

Alternative 6
Accuracy	76.4%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.000125 \lor \neg \left(a \leq 0.0132\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 7
Accuracy	76.4%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.00186 \lor \neg \left(a \leq 0.0132\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 8
Accuracy	76.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.0132:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 9
Accuracy	76.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.0132:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 10
Accuracy	76.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0018:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 0.0132:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \end{array} \]

Alternative 11
Accuracy	76.6%
Cost	13248

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

Alternative 12
Accuracy	76.6%
Cost	13248

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

Alternative 13
Accuracy	54.9%
Cost	13120

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

Alternative 14
Accuracy	55.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -23 \lor \neg \left(b \leq 550\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 15
Accuracy	55.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -23 \lor \neg \left(b \leq 1150\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 16
Accuracy	55.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -23 \lor \neg \left(b \leq 580\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 17
Accuracy	38.7%
Cost	6592

\[\sin b \cdot r \]

Alternative 18
Accuracy	34.3%
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023157 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))