rsin A (should all be same)

Average Error: 14.5 → 0.3

Time: 14.5s

Precision: binary64

Cost: 51840

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (/
  (* r (sin b))
  (fma (cos a) (cos b) (- (log1p (expm1 (* (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), -log1p(expm1((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), Float64(-log1p(expm1(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[Log[1 + N[(Exp[N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.5

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

2. Applied egg-rr 0.3

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

3. Applied egg-rr 0.3

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	39040

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

Alternative 2
Error	0.3
Cost	32704

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

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	13.9
Cost	19648

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

Alternative 6
Error	14.7
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 0.018:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 7
Error	14.5
Cost	13248

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

Alternative 8
Error	14.6
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-5} \lor \neg \left(b \leq 0.018\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 9
Error	39.3
Cost	6592

\[r \cdot \sin b \]

Alternative 10
Error	25.1
Cost	6592

\[r \cdot \tan b \]

Alternative 11
Error	41.6
Cost	576

\[\frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]

Alternative 12
Error	42.1
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023066 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))