rsin B (should all be same)

Average Error: 76.9% → 99.5%

Time: 16.4s

Precision: binary64

Cost: 77760.00

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 76.9

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

2. Simplified 76.9

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

   [Start] 76.9	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 76.9	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 99.5

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

4. Simplified 99.5

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

   [Start] 99.5	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \left(\left(-\sin b\right) \cdot \sin a + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)} \]
   fma-def [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)}} \]
   fma-def [=>] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\mathsf{fma}\left(-\sin b, \sin a, \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)}\right)} \]
   *-commutative [=>] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin b, \sin a, \mathsf{fma}\left(-\sin a, \sin b, \color{blue}{\sin a \cdot \sin b}\right)\right)\right)} \]

5. Final simplification 99.5

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

Alternatives

Alternative 1
Error	99.5%
Cost	39040.00

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

Alternative 2
Error	99.5%
Cost	32704.00

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

Alternative 3
Error	76.6%
Cost	13385.00

\[\begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-5} \lor \neg \left(a \leq 6.1 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 4
Error	76.6%
Cost	13385.00

\[\begin{array}{l} \mathbf{if}\;a \leq -0.000122 \lor \neg \left(a \leq 6.1 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 5
Error	76.9%
Cost	13248.00

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

Alternative 6
Error	54.9%
Cost	13120.00

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

Alternative 7
Error	51.5%
Cost	7232.00

\[\frac{\frac{r}{b \cdot 0.16666666666666666 + \frac{1}{b}}}{\cos \left(b + a\right)} \]

Alternative 8
Error	50.7%
Cost	6848.00

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

Alternative 9
Error	50.7%
Cost	6720.00

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

Alternative 10
Error	50.7%
Cost	6720.00

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

Alternative 11
Error	34.4%
Cost	192.00

\[r \cdot b \]

Reproduce

herbie shell --seed 2023093 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))