rsin B (should all be same)

Average Error: 15.0 → 0.3

Time: 16.8s

Precision: binary64

Cost: 32704

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (/ (* (sin b) r) (- (* (cos a) (cos 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 (sin(b) * r) / ((cos(a) * cos(b)) - (sin(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 = (sin(b) * r) / ((cos(a) * cos(b)) - (sin(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 (Math.sin(b) * r) / ((Math.cos(a) * Math.cos(b)) - (Math.sin(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 (math.sin(b) * r) / ((math.cos(a) * math.cos(b)) - (math.sin(b) * math.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(Float64(sin(b) * r) / Float64(Float64(cos(a) * cos(b)) - Float64(sin(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 = (sin(b) * r) / ((cos(a) * cos(b)) - (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[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.0

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

2. Simplified 15.0

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

   [Start] 15.0	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   *-commutative [=>] 15.0	\[ \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]
   associate-/r/ [<=] 15.0	\[ \color{blue}{\frac{\sin b}{\frac{\cos \left(a + b\right)}{r}}} \]
   +-commutative [=>] 15.0	\[ \frac{\sin b}{\frac{\cos \color{blue}{\left(b + a\right)}}{r}} \]

3. Applied egg-rr 0.4

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

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. Applied egg-rr 0.3

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

6. Final simplification 0.3

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

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	14.3
Cost	19776

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

Alternative 3
Error	14.3
Cost	19648

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

Alternative 4
Error	14.3
Cost	19648

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

Alternative 5
Error	15.2
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-7} \lor \neg \left(a \leq 2.6 \cdot 10^{-11}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 6
Error	15.2
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

Alternative 7
Error	15.1
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 8
Error	15.1
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 9
Error	15.0
Cost	13376

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

Alternative 10
Error	15.0
Cost	13248

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

Alternative 11
Error	15.0
Cost	13248

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

Alternative 12
Error	29.0
Cost	13120

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

Alternative 13
Error	30.5
Cost	7108

\[\begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{+42}:\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(b \cdot r\right)\\ \end{array} \]

Alternative 14
Error	30.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;b \leq -95:\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 15
Error	30.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 16
Error	39.0
Cost	6592

\[\sin b \cdot r \]

Alternative 17
Error	41.9
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023053 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))