rsin B (should all be same)

Average Error: 15.0 → 0.3

Time: 17.8s

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 15.0

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

2. Simplified 15.0

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

   [Start] 15.0	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 15.0	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.3

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

4. Simplified 0.3

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

   [Start] 0.3	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \left(\left(-\sin b\right) \cdot \sin a + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)} \]
   fma-def [=>] 0.3	\[ r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)}} \]
   fma-def [=>] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\mathsf{fma}\left(-\sin b, \sin a, \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)}\right)} \]
   *-commutative [=>] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin b, \sin a, \mathsf{fma}\left(-\sin a, \sin b, \color{blue}{\sin a \cdot \sin b}\right)\right)\right)} \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	45504

\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)} \]

Alternative 2
Error	0.4
Cost	39232

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

Alternative 3
Error	0.3
Cost	32704

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

Alternative 4
Error	0.4
Cost	32512

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

Alternative 5
Error	15.2
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.18:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.000145:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 6
Error	15.2
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.18:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.00035:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 7
Error	15.0
Cost	13248

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

Alternative 8
Error	15.1
Cost	13248

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

Alternative 9
Error	29.3
Cost	13120

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

Alternative 10
Error	29.3
Cost	13120

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

Alternative 11
Error	30.4
Cost	7364

\[\begin{array}{l} \mathbf{if}\;b \leq 0.6:\\ \;\;\;\;\frac{\frac{r}{\cos \left(b + a\right)}}{b \cdot 0.16666666666666666 + \frac{1}{b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]

Alternative 12
Error	30.5
Cost	7236

\[\begin{array}{l} \mathbf{if}\;b \leq 0.6:\\ \;\;\;\;\frac{\frac{r}{\cos a}}{b \cdot 0.16666666666666666 + \frac{1}{b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]

Alternative 13
Error	30.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;b \leq 0.55:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]

Alternative 14
Error	30.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;b \leq 0.6:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]

Alternative 15
Error	39.6
Cost	6592

\[r \cdot \sin b \]

Alternative 16
Error	42.5
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023045 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))