rsin B (should all be same)

Average Error: 15.3 → 15.3

Time: 26.2s

Precision: binary64

Cost: 13384

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0007:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.00155:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.0007)
   (* r (/ (sin b) (cos a)))
   (if (<= a 0.00155) (/ (* (sin b) r) (cos b)) (/ r (/ (cos a) (sin b))))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / cos((a + b)));
}

↓

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0007) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 0.00155) {
		tmp = (sin(b) * r) / cos(b);
	} else {
		tmp = r / (cos(a) / sin(b));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= (-0.0007d0)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 0.00155d0) then
        tmp = (sin(b) * r) / cos(b)
    else
        tmp = r / (cos(a) / sin(b))
    end if
    code = tmp
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) {
	double tmp;
	if (a <= -0.0007) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 0.00155) {
		tmp = (Math.sin(b) * r) / Math.cos(b);
	} else {
		tmp = r / (Math.cos(a) / Math.sin(b));
	}
	return tmp;
}

Python

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

↓

def code(r, a, b):
	tmp = 0
	if a <= -0.0007:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 0.00155:
		tmp = (math.sin(b) * r) / math.cos(b)
	else:
		tmp = r / (math.cos(a) / math.sin(b))
	return tmp

Julia

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

↓

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.0007)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 0.00155)
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	else
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.0007)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 0.00155)
		tmp = (sin(b) * r) / cos(b);
	else
		tmp = r / (cos(a) / sin(b));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[a, -0.0007], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00155], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.99999999999999993e-4

   1. Initial program 29.2

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

   2. Simplified 29.2

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

      [Start] 29.2	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
      rational.json-simplify-1 [=>] 29.2	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Taylor expanded in b around 0 29.3

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

   if -6.99999999999999993e-4 < a < 0.00154999999999999995

   1. Initial program 0.6

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

   2. Simplified 0.6

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

      [Start] 0.6	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
      rational.json-simplify-1 [=>] 0.6	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Taylor expanded in a around 0 0.7

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

   if 0.00154999999999999995 < a

   1. Initial program 30.0

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

   2. Simplified 30.0

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

      [Start] 30.0	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
      rational.json-simplify-1 [=>] 30.0	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Applied egg-rr 30.0

      \[\leadsto \color{blue}{\frac{r \cdot \frac{0.5}{\cos \left(b + a\right)}}{\frac{0.5}{\sin b}}} \]

   4. Taylor expanded in b around 0 30.1

      \[\leadsto \frac{r \cdot \color{blue}{\frac{0.5}{\cos a}}}{\frac{0.5}{\sin b}} \]

   5. Applied egg-rr 41.3

      \[\leadsto \color{blue}{\left(\sin b \cdot \sin b\right) \cdot \frac{4}{\frac{\cos a}{r} \cdot \left(\sin b \cdot 4\right)}} \]

   6. Simplified 41.3

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

      [Start] 41.3	\[ \left(\sin b \cdot \sin b\right) \cdot \frac{4}{\frac{\cos a}{r} \cdot \left(\sin b \cdot 4\right)} \]
      rational.json-simplify-46 [=>] 41.3	\[ \left(\sin b \cdot \sin b\right) \cdot \color{blue}{\frac{\frac{4}{\frac{\cos a}{r}}}{\sin b \cdot 4}} \]
      rational.json-simplify-44 [=>] 41.3	\[ \left(\sin b \cdot \sin b\right) \cdot \color{blue}{\frac{\frac{4}{\sin b \cdot 4}}{\frac{\cos a}{r}}} \]
      rational.json-simplify-2 [=>] 41.3	\[ \left(\sin b \cdot \sin b\right) \cdot \frac{\frac{4}{\color{blue}{4 \cdot \sin b}}}{\frac{\cos a}{r}} \]
      rational.json-simplify-46 [=>] 41.3	\[ \left(\sin b \cdot \sin b\right) \cdot \frac{\color{blue}{\frac{\frac{4}{4}}{\sin b}}}{\frac{\cos a}{r}} \]
      metadata-eval [=>] 41.3	\[ \left(\sin b \cdot \sin b\right) \cdot \frac{\frac{\color{blue}{1}}{\sin b}}{\frac{\cos a}{r}} \]

   7. Applied egg-rr 30.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos a}{r} \cdot \frac{1}{\sin b}} + 0} \]

   8. Simplified 30.0

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

      [Start] 30.4	\[ \frac{1}{\frac{\cos a}{r} \cdot \frac{1}{\sin b}} + 0 \]
      rational.json-simplify-4 [=>] 30.4	\[ \color{blue}{\frac{1}{\frac{\cos a}{r} \cdot \frac{1}{\sin b}}} \]
      rational.json-simplify-46 [=>] 30.1	\[ \color{blue}{\frac{\frac{1}{\frac{\cos a}{r}}}{\frac{1}{\sin b}}} \]
      rational.json-simplify-44 [=>] 30.1	\[ \color{blue}{\frac{\frac{1}{\frac{1}{\sin b}}}{\frac{\cos a}{r}}} \]
      rational.json-simplify-61 [=>] 30.0	\[ \frac{\color{blue}{\frac{\sin b}{\frac{1}{1}}}}{\frac{\cos a}{r}} \]
      metadata-eval [=>] 30.0	\[ \frac{\frac{\sin b}{\color{blue}{1}}}{\frac{\cos a}{r}} \]
      rational.json-simplify-7 [=>] 30.0	\[ \frac{\color{blue}{\sin b}}{\frac{\cos a}{r}} \]
      rational.json-simplify-61 [=>] 30.0	\[ \color{blue}{\frac{r}{\frac{\cos a}{\sin b}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0007:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.00155:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.3
Cost	13384

\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -0.000175:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.00055:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	15.3
Cost	13384

\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -0.00072:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.005:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.000116:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.00025:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \end{array} \]

Alternative 4
Error	15.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.000175:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

Alternative 5
Error	15.3
Cost	13248

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

Alternative 6
Error	15.3
Cost	13248

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

Alternative 7
Error	29.1
Cost	13120

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

Alternative 8
Error	30.6
Cost	7624

\[\begin{array}{l} t_0 := \frac{r \cdot \frac{0.5}{\cos a}}{b \cdot 0.08333333333333333 + 0.5 \cdot \frac{1}{b}}\\ \mathbf{if}\;a \leq -220000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.00014:\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	29.3
Cost	6984

\[\begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.82:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	39.5
Cost	6592

\[\sin b \cdot r \]

Alternative 11
Error	42.4
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023075 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))