rsin A (should all be same)

Average Accuracy: 76.1% → 99.4%

Time: 18.0s

Precision: binary64

Cost: 19648

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b) :precision binary64 (/ r (- (/ (cos a) (tan 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 / ((cos(a) / tan(b)) - sin(a));
}

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 = r / ((cos(a) / tan(b)) - sin(a))
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 r / ((Math.cos(a) / Math.tan(b)) - Math.sin(a));
}

Python

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

↓

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

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(a + b)))
end

↓

function code(r, a, b)
	return Float64(r / Float64(Float64(cos(a) / tan(b)) - sin(a)))
end

MATLAB

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

↓

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

Derivation

1. Initial program 76.1%

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

2. Simplified 76.0%

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

   [Start] 76.1	\[ \frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   associate-/l* [=>] 76.0	\[ \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]
   +-commutative [=>] 76.0	\[ \frac{r}{\frac{\cos \color{blue}{\left(b + a\right)}}{\sin b}} \]

3. Applied egg-rr 99.4%

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

   [Start] 76.0	\[ \frac{r}{\frac{\cos \left(b + a\right)}{\sin b}} \]
   cos-sum [=>] 99.4	\[ \frac{r}{\frac{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}}{\sin b}} \]
   div-sub [=>] 99.4	\[ \frac{r}{\color{blue}{\frac{\cos b \cdot \cos a}{\sin b} - \frac{\sin b \cdot \sin a}{\sin b}}} \]

4. Taylor expanded in b around 0 99.4%

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

5. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{0 + \frac{r}{\frac{\cos a}{\tan b} - \sin a}} \]
   Proof

   [Start] 99.4	\[ \frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a} \]
   add-log-exp [=>] 16.8	\[ \color{blue}{\log \left(e^{\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a}}\right)} \]
   *-un-lft-identity [=>] 16.8	\[ \log \color{blue}{\left(1 \cdot e^{\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a}}\right)} \]
   log-prod [=>] 16.8	\[ \color{blue}{\log 1 + \log \left(e^{\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a}}\right)} \]
   metadata-eval [=>] 16.8	\[ \color{blue}{0} + \log \left(e^{\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a}}\right) \]
   add-log-exp [<=] 99.4	\[ 0 + \color{blue}{\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a}} \]
   *-commutative [=>] 99.4	\[ 0 + \frac{r}{\frac{\color{blue}{\cos a \cdot \cos b}}{\sin b} - \sin a} \]
   associate-/l* [=>] 99.4	\[ 0 + \frac{r}{\color{blue}{\frac{\cos a}{\frac{\sin b}{\cos b}}} - \sin a} \]
   quot-tan [=>] 99.4	\[ 0 + \frac{r}{\frac{\cos a}{\color{blue}{\tan b}} - \sin a} \]

6. Simplified 99.4%

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

   [Start] 99.4	\[ 0 + \frac{r}{\frac{\cos a}{\tan b} - \sin a} \]
   +-lft-identity [=>] 99.4	\[ \color{blue}{\frac{r}{\frac{\cos a}{\tan b} - \sin a}} \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	75.8%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -0.031 \lor \neg \left(b \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 2
Accuracy	75.8%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0075 \lor \neg \left(a \leq 13500000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 3
Accuracy	75.8%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.014 \lor \neg \left(a \leq 13500000\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 4
Accuracy	76.1%
Cost	13248

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

Alternative 5
Accuracy	76.1%
Cost	13248

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

Alternative 6
Accuracy	76.0%
Cost	13248

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

Alternative 7
Accuracy	74.4%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;b \leq -0.031 \lor \neg \left(b \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\tan b} - a}{r}}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 8
Accuracy	74.4%
Cost	7240

\[\begin{array}{l} t_0 := \frac{1}{\tan b} - a\\ \mathbf{if}\;b \leq -0.032:\\ \;\;\;\;\frac{r}{t_0}\\ \mathbf{elif}\;b \leq 0.00043:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{t_0}\\ \end{array} \]

Alternative 9
Accuracy	74.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -0.031 \lor \neg \left(b \leq 0.000215\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 10
Accuracy	54.9%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -0.45 \lor \neg \left(b \leq 5.2 \cdot 10^{+19}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11
Accuracy	54.9%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -0.45 \lor \neg \left(b \leq 5.2 \cdot 10^{+19}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{b}}\\ \end{array} \]

Alternative 12
Accuracy	54.9%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -0.45 \lor \neg \left(b \leq 4.5 \cdot 10^{+19}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 13
Accuracy	38.4%
Cost	6592

\[r \cdot \sin b \]

Alternative 14
Accuracy	33.9%
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023151 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))