rsin B (should all be same)

Average Error: 14.8 → 0.3

Time: 15.5s

Precision: binary64

Cost: 32704

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 14.8

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

2. Simplified 14.8

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

   [Start] 14.8	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 14.8	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.3

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	14.3
Cost	19648

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

Alternative 2
Error	15.2
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -1700000000000 \lor \neg \left(a \leq 4.2 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 3
Error	15.2
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -1150000000000:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 4
Error	15.2
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -1150000000000:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 5
Error	14.8
Cost	13248

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

Alternative 6
Error	14.8
Cost	13248

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

Alternative 7
Error	28.7
Cost	13120

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

Alternative 8
Error	28.7
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -4.6 \lor \neg \left(b \leq 21500000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 9
Error	38.9
Cost	6592

\[r \cdot \sin b \]

Alternative 10
Error	41.8
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023034 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))