rsin A (should all be same)

Average Error: 15.2 → 0.3

Time: 15.6s

Precision: binary64

Cost: 32704

Math

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

↓

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

FPCore

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

↓

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

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 (sin(b) / ((cos(a) * cos(b)) - (sin(b) * sin(a)))) * r;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (r * sin(b)) / cos((a + b))
end function

↓

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (sin(b) / ((cos(a) * cos(b)) - (sin(b) * sin(a)))) * r
end function

Java

public static double code(double r, double a, double b) {
	return (r * Math.sin(b)) / Math.cos((a + b));
}

↓

public static double code(double r, double a, double b) {
	return (Math.sin(b) / ((Math.cos(a) * Math.cos(b)) - (Math.sin(b) * Math.sin(a)))) * r;
}

Python

def code(r, a, b):
	return (r * math.sin(b)) / math.cos((a + b))

↓

def code(r, a, b):
	return (math.sin(b) / ((math.cos(a) * math.cos(b)) - (math.sin(b) * math.sin(a)))) * r

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(sin(b) / Float64(Float64(cos(a) * cos(b)) - Float64(sin(b) * sin(a)))) * r)
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / cos((a + b));
end

↓

function tmp = code(r, a, b)
	tmp = (sin(b) / ((cos(a) * cos(b)) - (sin(b) * sin(a)))) * r;
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[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]

Derivation

1. Initial program 15.2

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

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. Taylor expanded in r around 0 0.3

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

4. Simplified 0.4

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

   [Start] 0.3	\[ \frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin a \cdot \sin b} \]
   associate-/l* [=>] 0.4	\[ \color{blue}{\frac{\sin b}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{r}}} \]
   *-commutative [<=] 0.4	\[ \frac{\sin b}{\frac{\cos a \cdot \cos b - \color{blue}{\sin b \cdot \sin a}}{r}} \]
   fma-neg [=>] 0.4	\[ \frac{\sin b}{\frac{\color{blue}{\mathsf{fma}\left(\cos a, \cos b, -\sin b \cdot \sin a\right)}}{r}} \]
   distribute-rgt-neg-in [=>] 0.4	\[ \frac{\sin b}{\frac{\mathsf{fma}\left(\cos a, \cos b, \color{blue}{\sin b \cdot \left(-\sin a\right)}\right)}{r}} \]

5. Taylor expanded in b around inf 0.3

   \[\leadsto \color{blue}{\frac{\sin b \cdot r}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b}} \]

6. Simplified 0.3

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

   [Start] 0.3	\[ \frac{\sin b \cdot r}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \]
   associate-/l* [=>] 0.4	\[ \color{blue}{\frac{\sin b}{\frac{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b}{r}}} \]
   associate-/r/ [=>] 0.3	\[ \color{blue}{\frac{\sin b}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \cdot r} \]
   +-commutative [=>] 0.3	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b + -1 \cdot \left(\sin a \cdot \sin b\right)}} \cdot r \]
   *-commutative [<=] 0.3	\[ \frac{\sin b}{\cos a \cdot \cos b + -1 \cdot \color{blue}{\left(\sin b \cdot \sin a\right)}} \cdot r \]
   mul-1-neg [=>] 0.3	\[ \frac{\sin b}{\cos a \cdot \cos b + \color{blue}{\left(-\sin b \cdot \sin a\right)}} \cdot r \]
   sub-neg [<=] 0.3	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin b \cdot \sin a}} \cdot r \]
   *-commutative [=>] 0.3	\[ \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sin a \cdot \sin b}} \cdot r \]

7. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	32512

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

Alternative 2
Error	0.4
Cost	26176

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

Alternative 3
Error	14.7
Cost	13513

\[\begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{+16} \lor \neg \left(a \leq 0.00021\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 4
Error	14.4
Cost	13513

\[\begin{array}{l} \mathbf{if}\;a \leq -1750000 \lor \neg \left(a \leq 0.0095\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b} - a}\\ \end{array} \]

Alternative 5
Error	15.4
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -1750 \lor \neg \left(b \leq 2.75 \cdot 10^{-11}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 6
Error	15.4
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -1750 \lor \neg \left(b \leq 2.75 \cdot 10^{-11}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 7
Error	15.4
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -1750:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 8
Error	15.2
Cost	13248

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

Alternative 9
Error	15.2
Cost	13248

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

Alternative 10
Error	28.9
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1.15 \lor \neg \left(b \leq 22000000000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 11
Error	39.4
Cost	6592

\[\sin b \cdot r \]

Alternative 12
Error	42.3
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2022356 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))