rsin A (should all be same)

Average Error: 15.2 → 0.3

Time: 16.1s

Precision: binary64

Cost: 90560

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (/
  (* r (sin b))
  (fma
   (cos a)
   (cos b)
   (fma
    (- (sin a))
    (sin b)
    (fma (- (sin b)) (sin a) (expm1 (log1p (* (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(a), cos(b), fma(-sin(a), sin(b), fma(-sin(b), sin(a), expm1(log1p((sin(b) * sin(a)))))));
}

Julia

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

↓

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(cos(a), cos(b), fma(Float64(-sin(a)), sin(b), fma(Float64(-sin(b)), sin(a), expm1(log1p(Float64(sin(b) * sin(a))))))))
end

Wolfram

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[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision] + N[((-N[Sin[a], $MachinePrecision]) * N[Sin[b], $MachinePrecision] + N[((-N[Sin[b], $MachinePrecision]) * N[Sin[a], $MachinePrecision] + N[(Exp[N[Log[1 + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.2

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

2. Applied egg-rr 0.3

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

3. Simplified 0.3

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

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

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	77760

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

Alternative 2
Error	0.3
Cost	39040

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

Alternative 3
Error	0.3
Cost	32704

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

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	0.4
Cost	26176

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

Alternative 6
Error	0.4
Cost	26176

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

Alternative 7
Error	15.3
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -1150 \lor \neg \left(a \leq 1700\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 8
Error	15.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -1150:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 1700:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 9
Error	15.2
Cost	13248

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

Alternative 10
Error	15.2
Cost	13248

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

Alternative 11
Error	15.4
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -0.00186 \lor \neg \left(b \leq 6.8 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 12
Error	15.4
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -0.00186 \lor \neg \left(b \leq 4.3 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 13
Error	26.0
Cost	6592

\[r \cdot \tan b \]

Alternative 14
Error	39.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;b \leq -1750:\\ \;\;\;\;r\\ \mathbf{elif}\;b \leq 0.72:\\ \;\;\;\;r \cdot b\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \]

Alternative 15
Error	39.4
Cost	448

\[\frac{r}{\frac{1}{b} + -1} \]

Alternative 16
Error	59.1
Cost	64

\[r \]

Reproduce

herbie shell --seed 2023073 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))