rsin A (should all be same)

Average Error: 15.0 → 0.3

Time: 13.9s

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 15.0

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

2. Applied egg-rr 0.3

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

3. Simplified 0.3

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

   [Start] 0.3	\[ \frac{r \cdot \sin b}{\cos a \cdot \cos b + \left(-\sin a\right) \cdot \sin b} \]
   +-commutative [=>] 0.3	\[ \frac{r \cdot \sin b}{\color{blue}{\left(-\sin a\right) \cdot \sin b + \cos a \cdot \cos b}} \]
   *-commutative [=>] 0.3	\[ \frac{r \cdot \sin b}{\color{blue}{\sin b \cdot \left(-\sin a\right)} + \cos a \cdot \cos b} \]
   fma-def [=>] 0.3	\[ \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)}} \]
   *-commutative [=>] 0.3	\[ \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin b, -\sin a, \color{blue}{\cos b \cdot \cos a}\right)} \]

4. Taylor expanded in r around 0 0.3

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

5. Simplified 0.3

   \[\leadsto \color{blue}{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot r} \]
   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	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} + -1 \cdot \left(\sin a \cdot \sin b\right)} \cdot r \]
   *-commutative [<=] 0.3	\[ \frac{\sin b}{\cos b \cdot \cos a + -1 \cdot \color{blue}{\left(\sin b \cdot \sin a\right)}} \cdot r \]
   mul-1-neg [=>] 0.3	\[ \frac{\sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b \cdot \sin a\right)}} \cdot r \]
   unsub-neg [=>] 0.3	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \cdot r \]
   *-commutative [=>] 0.3	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b} - \sin b \cdot \sin a} \cdot r \]

6. 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	26176

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

Alternative 2
Error	15.2
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -22 \lor \neg \left(a \leq 9.5 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]

Alternative 3
Error	15.2
Cost	13376

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

Alternative 4
Error	15.0
Cost	13248

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

Alternative 5
Error	15.2
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-6} \lor \neg \left(b \leq 0.003\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 6
Error	15.2
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-6} \lor \neg \left(b \leq 0.003\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{b}}\\ \end{array} \]

Alternative 7
Error	15.2
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -7.3 \cdot 10^{-6} \lor \neg \left(b \leq 0.003\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 8
Error	39.8
Cost	6592

\[\sin b \cdot r \]

Alternative 9
Error	26.0
Cost	6592

\[r \cdot \tan b \]

Alternative 10
Error	42.1
Cost	576

\[\frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]

Alternative 11
Error	42.7
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023041 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))