rsin B (should all be same)

Average Accuracy: 76.8% → 99.5%

Time: 17.7s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 76.8%

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

2. Simplified 76.8%

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

   [Start] 76.8	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 76.8	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   Proof

   [Start] 76.8	\[ r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
   cos-sum [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

4. Applied egg-rr 99.5%

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

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

5. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	32704

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

Alternative 2
Accuracy	98.8%
Cost	26432

\[\frac{1}{\frac{\cos b}{r} \cdot \frac{\cos a}{\sin b} - \frac{\sin a}{r}} \]

Alternative 3
Accuracy	98.9%
Cost	26432

\[\frac{1}{\cos a \cdot \frac{\cos b}{r \cdot \sin b} - \frac{\sin a}{r}} \]

Alternative 4
Accuracy	76.4%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 28000:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{\cos a}{\sin b}}\\ \end{array} \]

Alternative 5
Accuracy	76.4%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -3.95 \cdot 10^{-7} \lor \neg \left(a \leq 28000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 6
Accuracy	76.8%
Cost	13248

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

Alternative 7
Accuracy	54.9%
Cost	13120

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

Alternative 8
Accuracy	55.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -40000 \lor \neg \left(b \leq 330\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 9
Accuracy	55.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -68 \lor \neg \left(b \leq 0.45\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 10
Accuracy	38.9%
Cost	6592

\[r \cdot \sin b \]

Alternative 11
Accuracy	34.3%
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023146 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))