rsin B (should all be same)

Average Error: 15.3 → 0.3

Time: 18.0s

Precision: binary64

Cost: 51840

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 15.3

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

2. Simplified 15.3

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

   [Start] 15.3	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 15.3	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.3

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

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
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 2
Error	0.3
Cost	32704

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

Alternative 3
Error	14.6
Cost	19648

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

Alternative 4
Error	15.3
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-5} \lor \neg \left(a \leq 5.9 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]

Alternative 5
Error	15.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.00017:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 6
Error	15.3
Cost	13248

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

Alternative 7
Error	15.5
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -0.0155 \lor \neg \left(b \leq 2.7 \cdot 10^{-10}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 8
Error	15.5
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -0.0155 \lor \neg \left(b \leq 2.7 \cdot 10^{-10}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 9
Error	39.4
Cost	6592

\[r \cdot \sin b \]

Alternative 10
Error	25.6
Cost	6592

\[r \cdot \tan b \]

Alternative 11
Error	42.2
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023030 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))