rsin B (should all be same)

Average Error: 15.2 → 15.2

Time: 18.7s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 15.2

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

2. Simplified 15.2

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

   [Start] 15.2	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   rational.json-simplify-48 [=>] 15.2	\[ \color{blue}{\frac{\sin b \cdot r}{\cos \left(a + b\right)}} \]
   rational.json-simplify-1 [=>] 15.2	\[ \frac{\sin b \cdot r}{\cos \color{blue}{\left(b + a\right)}} \]

3. Final simplification 15.2

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

Alternatives

Alternative 1
Error	15.4
Cost	13384

\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -10:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1700:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	15.4
Cost	13384

\[\begin{array}{l} t_0 := \frac{r}{\cos a} \cdot \sin b\\ \mathbf{if}\;a \leq -10:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1700:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.4
Cost	13384

\[\begin{array}{l} t_0 := \frac{r}{\cos a} \cdot \sin b\\ \mathbf{if}\;a \leq -10:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1700:\\ \;\;\;\;\frac{r}{\cos b} \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.4
Cost	13384

\[\begin{array}{l} t_0 := \frac{r}{\cos a} \cdot \sin b\\ \mathbf{if}\;a \leq -10:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1700:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.2
Cost	13248

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

Alternative 6
Error	15.2
Cost	13248

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

Alternative 7
Error	28.2
Cost	13120

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

Alternative 8
Error	28.1
Cost	6984

\[\begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -1550:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.94:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	38.8
Cost	6592

\[\sin b \cdot r \]

Alternative 10
Error	41.6
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023073 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))