rsin B (should all be same)

Average Error: 23.31% → 0.51%

Time: 17.4s

Precision: binary64

Cost: 77760

Math

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

↓

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

FPCore

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

↓

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

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

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / cos(Float64(a + b))))
end

↓

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

Derivation

1. Initial program 23.31

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

2. Simplified 23.31

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

   [Start] 23.31	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 23.31	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.51

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

4. Simplified 0.51

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

   [Start] 0.51	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \left(\left(-\sin b\right) \cdot \sin a + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)} \]
   fma-def [=>] 0.5	\[ r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)}} \]
   fma-def [=>] 0.51	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\mathsf{fma}\left(-\sin b, \sin a, \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)}\right)} \]
   *-commutative [=>] 0.51	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin b, \sin a, \mathsf{fma}\left(-\sin a, \sin b, \color{blue}{\sin a \cdot \sin b}\right)\right)\right)} \]

5. Final simplification 0.51

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

Alternatives

Alternative 1
Error	0.5%
Cost	39040

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

Alternative 2
Error	0.51%
Cost	32704

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

Alternative 3
Error	0.59%
Cost	32512

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

Alternative 4
Error	23.67%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -0.000105 \lor \neg \left(b \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 5
Error	23.65%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 0.0007:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 6
Error	23.66%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 0.0028:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 7
Error	23.31%
Cost	13248

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

Alternative 8
Error	44.84%
Cost	13120

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

Alternative 9
Error	44.78%
Cost	7500

\[\begin{array}{l} t_0 := \frac{r}{\frac{1}{\sin b}}\\ \mathbf{if}\;b \leq -29000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.8:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{+302}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{r}{\cos b}}{b \cdot 0.16666666666666666 + \frac{1}{b}}\\ \end{array} \]

Alternative 10
Error	44.73%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -29000000000 \lor \neg \left(b \leq 1.45\right):\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11
Error	49.18%
Cost	6720

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

Alternative 12
Error	65.52%
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023089 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))