rsin B (should all be same)

Average Accuracy: 76.6% → 99.5%

Time: 16.0s

Precision: binary64

Cost: 51840

Math

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

↓

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

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) (- (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(b), cos(a), -expm1(log1p((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(-expm1(log1p(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[(Exp[N[Log[1 + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

2. Simplified 76.6%

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

   [Start] 76.6	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 76.6	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	39040

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

Alternative 2
Accuracy	99.5%
Cost	32704

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

Alternative 3
Accuracy	99.4%
Cost	32512

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

Alternative 4
Accuracy	77.6%
Cost	19648

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

Alternative 5
Accuracy	76.4%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-6} \lor \neg \left(b \leq 4.5 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \end{array} \]

Alternative 6
Accuracy	76.4%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-6} \lor \neg \left(b \leq 6.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \end{array} \]

Alternative 7
Accuracy	76.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 8
Accuracy	76.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 9
Accuracy	76.6%
Cost	13248

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

Alternative 10
Accuracy	54.4%
Cost	13120

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

Alternative 11
Accuracy	54.5%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+17} \lor \neg \left(b \leq 55\right):\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \end{array} \]

Alternative 12
Accuracy	54.5%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+17} \lor \neg \left(b \leq 55\right):\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 13
Accuracy	50.0%
Cost	6720

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

Alternative 14
Accuracy	33.8%
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023129 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))