rsin A (should all be same)

Percentage Accurate: 77.1% → 99.5%

Time: 14.3s

Alternatives: 11

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sin b}{\cos b \cdot \cos a - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)} \cdot r \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* (/ (sin b) (- (* (cos b) (cos a)) (expm1 (log1p (* (sin b) (sin a)))))) r))

C

double code(double r, double a, double b) {
	return (sin(b) / ((cos(b) * cos(a)) - expm1(log1p((sin(b) * sin(a)))))) * r;
}

Java

public static double code(double r, double a, double b) {
	return (Math.sin(b) / ((Math.cos(b) * Math.cos(a)) - Math.expm1(Math.log1p((Math.sin(b) * Math.sin(a)))))) * r;
}

Python

def code(r, a, b):
	return (math.sin(b) / ((math.cos(b) * math.cos(a)) - math.expm1(math.log1p((math.sin(b) * math.sin(a)))))) * r

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) / Float64(Float64(cos(b) * cos(a)) - expm1(log1p(Float64(sin(b) * sin(a)))))) * r)
end

Wolfram

code[r_, a_, b_] := N[(N[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(Exp[N[Log[1 + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 72.9%

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

   2. *-commutative 72.9%

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

   3. +-commutative 72.9%

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

3. Simplified 72.9%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
4. Step-by-step derivation

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

   \[\leadsto \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \cdot r \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.4%

      \[\leadsto \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)}} \cdot r \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)}} \cdot r \]

8. Final simplification 99.4%

   \[\leadsto \frac{\sin b}{\cos b \cdot \cos a - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)} \cdot r \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \end{array} \]

FPCore

(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(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(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(b) * Math.cos(a)) - (Math.sin(b) * Math.sin(a))));
}

Python

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) / Float64(Float64(cos(b) * cos(a)) - Float64(sin(b) * sin(a)))))
end

MATLAB

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[(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 72.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 72.9%

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

   2. *-commutative 72.9%

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

   3. +-commutative 72.9%

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

3. Simplified 72.9%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
4. Step-by-step derivation

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \lor \neg \left(b \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.0) (not (<= b 5e-16)))
   (* r (/ (sin b) (cos b)))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.0) || !(b <= 5e-16)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = r * (b / cos(a));
	}
	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
    real(8) :: tmp
    if ((b <= (-1.0d0)) .or. (.not. (b <= 5d-16))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.0) || !(b <= 5e-16)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.0) or not (b <= 5e-16):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.0) || !(b <= 5e-16))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.0) || ~((b <= 5e-16)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.0], N[Not[LessEqual[b, 5e-16]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1 or 5.0000000000000004e-16 < b

   1. Initial program 51.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 51.2%

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

      2. *-commutative 51.2%

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

      3. +-commutative 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in a around 0 51.6%

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

   if -1 < b < 5.0000000000000004e-16

   1. Initial program 98.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 98.8%

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

      2. *-commutative 98.8%

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

      3. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in b around 0 98.7%

      \[\leadsto \color{blue}{\frac{b}{\cos a}} \cdot r \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \lor \neg \left(b \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 4: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.0)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 5e-16) (* r (/ b (cos a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.0) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 5e-16) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r * (sin(b) / cos(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
    real(8) :: tmp
    if (b <= (-1.0d0)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 5d-16) then
        tmp = r * (b / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.0) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 5e-16) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.0:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 5e-16:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.0)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 5e-16)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1.0)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 5e-16)
		tmp = r * (b / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.0], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-16], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1

   1. Initial program 53.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 53.4%

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

      2. +-commutative 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in a around 0 54.6%

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

   if -1 < b < 5.0000000000000004e-16

   1. Initial program 98.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 98.8%

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

      2. *-commutative 98.8%

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

      3. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in b around 0 98.7%

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

   if 5.0000000000000004e-16 < b

   1. Initial program 49.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 49.7%

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

      2. *-commutative 49.7%

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

      3. +-commutative 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in a around 0 49.6%

      \[\leadsto \frac{\sin b}{\color{blue}{\cos b}} \cdot r \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 5: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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 = sin(b) * (r / cos((b + a)))
end function

Java

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 math.sin(b) * (r / math.cos((b + a)))

Julia

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

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-/l* 72.8%

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

   2. +-commutative 72.8%

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

3. Simplified 72.8%

   \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}}} \]
4. Step-by-step derivation

   1. associate-/r/ 72.9%

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

5. Applied egg-rr 72.9%

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

6. Final simplification 72.9%

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

Alternative 6: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return r * (sin(b) / 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((b + a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 72.9%

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

   2. *-commutative 72.9%

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

   3. +-commutative 72.9%

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

3. Simplified 72.9%

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

4. Final simplification 72.9%

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

Alternative 7: 52.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 5e-16) (* r (/ b (cos a))) (/ r (- (/ 1.0 b) (sin a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 5e-16) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r / ((1.0 / b) - sin(a));
	}
	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
    real(8) :: tmp
    if (b <= 5d-16) then
        tmp = r * (b / cos(a))
    else
        tmp = r / ((1.0d0 / b) - sin(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 5e-16) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r / ((1.0 / b) - Math.sin(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 5e-16:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r / ((1.0 / b) - math.sin(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 5e-16)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r / Float64(Float64(1.0 / b) - sin(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 5e-16)
		tmp = r * (b / cos(a));
	else
		tmp = r / ((1.0 / b) - sin(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 5e-16], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(N[(1.0 / b), $MachinePrecision] - N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0000000000000004e-16

   1. Initial program 84.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

      3. +-commutative 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in b around 0 68.3%

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

   if 5.0000000000000004e-16 < b

   1. Initial program 49.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 49.7%

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

      2. +-commutative 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in b around 0 12.5%

      \[\leadsto \frac{r}{\color{blue}{-1 \cdot \sin a + \frac{\cos a}{b}}} \]
   5. Step-by-step derivation

      1. neg-mul-1 12.5%

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

      2. +-commutative 12.5%

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

      3. unsub-neg 12.5%

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

   6. Simplified 12.5%

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

   7. Taylor expanded in a around 0 12.5%

      \[\leadsto \frac{r}{\color{blue}{\frac{1}{b}} - \sin a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \end{array} \]

Alternative 8: 52.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{\sin a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 1.1) (* r (/ b (cos a))) (/ (- r) (sin a))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 1.1) {
		tmp = r * (b / cos(a));
	} else {
		tmp = -r / sin(a);
	}
	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
    real(8) :: tmp
    if (b <= 1.1d0) then
        tmp = r * (b / cos(a))
    else
        tmp = -r / sin(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 1.1) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = -r / Math.sin(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 1.1:
		tmp = r * (b / math.cos(a))
	else:
		tmp = -r / math.sin(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(Float64(-r) / sin(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 1.1)
		tmp = r * (b / cos(a));
	else
		tmp = -r / sin(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 1.1], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   1. Initial program 84.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 84.5%

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

      2. *-commutative 84.5%

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

      3. +-commutative 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in b around 0 68.4%

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

   if 1.1000000000000001 < b

   1. Initial program 48.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 48.5%

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

      2. +-commutative 48.5%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in b around 0 11.0%

      \[\leadsto \frac{r}{\color{blue}{-1 \cdot \sin a + \frac{\cos a}{b}}} \]
   5. Step-by-step derivation

      1. neg-mul-1 11.0%

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

      2. +-commutative 11.0%

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

      3. unsub-neg 11.0%

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

   6. Simplified 11.0%

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

   7. Taylor expanded in b around inf 10.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{r}{\sin a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 10.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot r}{\sin a}} \]

      2. neg-mul-1 10.8%

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

   9. Simplified 10.8%

      \[\leadsto \color{blue}{\frac{-r}{\sin a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{\sin a}\\ \end{array} \]

Alternative 9: 36.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{r}{\frac{1}{b} + b \cdot -0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{\sin a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 5.5e+16)
   (/ r (+ (/ 1.0 b) (* b -0.3333333333333333)))
   (/ (- r) (sin a))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 5.5e+16) {
		tmp = r / ((1.0 / b) + (b * -0.3333333333333333));
	} else {
		tmp = -r / sin(a);
	}
	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
    real(8) :: tmp
    if (b <= 5.5d+16) then
        tmp = r / ((1.0d0 / b) + (b * (-0.3333333333333333d0)))
    else
        tmp = -r / sin(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 5.5e+16) {
		tmp = r / ((1.0 / b) + (b * -0.3333333333333333));
	} else {
		tmp = -r / Math.sin(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 5.5e+16:
		tmp = r / ((1.0 / b) + (b * -0.3333333333333333))
	else:
		tmp = -r / math.sin(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 5.5e+16)
		tmp = Float64(r / Float64(Float64(1.0 / b) + Float64(b * -0.3333333333333333)));
	else
		tmp = Float64(Float64(-r) / sin(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 5.5e+16)
		tmp = r / ((1.0 / b) + (b * -0.3333333333333333));
	else
		tmp = -r / sin(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 5.5e+16], N[(r / N[(N[(1.0 / b), $MachinePrecision] + N[(b * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5e16

   1. Initial program 83.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.5%

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

      2. +-commutative 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in b around 0 66.6%

      \[\leadsto \frac{r}{\color{blue}{b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \left(-1 \cdot \sin a + \frac{\cos a}{b}\right)}} \]
   5. Step-by-step derivation

      1. fma-def 66.6%

         \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(b, -0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a, -1 \cdot \sin a + \frac{\cos a}{b}\right)}} \]

      2. distribute-rgt-out-- 66.6%

         \[\leadsto \frac{r}{\mathsf{fma}\left(b, \color{blue}{\cos a \cdot \left(-0.5 - -0.16666666666666666\right)}, -1 \cdot \sin a + \frac{\cos a}{b}\right)} \]

      3. metadata-eval 66.6%

         \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot \color{blue}{-0.3333333333333333}, -1 \cdot \sin a + \frac{\cos a}{b}\right)} \]

      4. neg-mul-1 66.6%

         \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \color{blue}{\left(-\sin a\right)} + \frac{\cos a}{b}\right)} \]

      5. +-commutative 66.6%

         \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \color{blue}{\frac{\cos a}{b} + \left(-\sin a\right)}\right)} \]

      6. unsub-neg 66.6%

         \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \color{blue}{\frac{\cos a}{b} - \sin a}\right)} \]

   6. Simplified 66.6%

      \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \frac{\cos a}{b} - \sin a\right)}} \]

   7. Taylor expanded in a around 0 42.8%

      \[\leadsto \color{blue}{\frac{r}{-0.3333333333333333 \cdot b + \frac{1}{b}}} \]

   if 5.5e16 < b

   1. Initial program 47.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 47.2%

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

      2. +-commutative 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in b around 0 11.1%

      \[\leadsto \frac{r}{\color{blue}{-1 \cdot \sin a + \frac{\cos a}{b}}} \]
   5. Step-by-step derivation

      1. neg-mul-1 11.1%

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

      2. +-commutative 11.1%

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

      3. unsub-neg 11.1%

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

   6. Simplified 11.1%

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

   7. Taylor expanded in b around inf 11.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{r}{\sin a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 11.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot r}{\sin a}} \]

      2. neg-mul-1 11.1%

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

   9. Simplified 11.1%

      \[\leadsto \color{blue}{\frac{-r}{\sin a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{r}{\frac{1}{b} + b \cdot -0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{\sin a}\\ \end{array} \]

Alternative 10: 34.9% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\frac{1}{b} + b \cdot -0.3333333333333333} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ r (+ (/ 1.0 b) (* b -0.3333333333333333))))

C

double code(double r, double a, double b) {
	return r / ((1.0 / b) + (b * -0.3333333333333333));
}

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 / ((1.0d0 / b) + (b * (-0.3333333333333333d0)))
end function

Java

public static double code(double r, double a, double b) {
	return r / ((1.0 / b) + (b * -0.3333333333333333));
}

Python

def code(r, a, b):
	return r / ((1.0 / b) + (b * -0.3333333333333333))

Julia

function code(r, a, b)
	return Float64(r / Float64(Float64(1.0 / b) + Float64(b * -0.3333333333333333)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r / ((1.0 / b) + (b * -0.3333333333333333));
end

Wolfram

code[r_, a_, b_] := N[(r / N[(N[(1.0 / b), $MachinePrecision] + N[(b * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-/l* 72.8%

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

   2. +-commutative 72.8%

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

3. Simplified 72.8%

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

4. Taylor expanded in b around 0 48.5%

   \[\leadsto \frac{r}{\color{blue}{b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \left(-1 \cdot \sin a + \frac{\cos a}{b}\right)}} \]
5. Step-by-step derivation

   1. fma-def 48.5%

      \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(b, -0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a, -1 \cdot \sin a + \frac{\cos a}{b}\right)}} \]

   2. distribute-rgt-out-- 48.5%

      \[\leadsto \frac{r}{\mathsf{fma}\left(b, \color{blue}{\cos a \cdot \left(-0.5 - -0.16666666666666666\right)}, -1 \cdot \sin a + \frac{\cos a}{b}\right)} \]

   3. metadata-eval 48.5%

      \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot \color{blue}{-0.3333333333333333}, -1 \cdot \sin a + \frac{\cos a}{b}\right)} \]

   4. neg-mul-1 48.5%

      \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \color{blue}{\left(-\sin a\right)} + \frac{\cos a}{b}\right)} \]

   5. +-commutative 48.5%

      \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \color{blue}{\frac{\cos a}{b} + \left(-\sin a\right)}\right)} \]

   6. unsub-neg 48.5%

      \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \color{blue}{\frac{\cos a}{b} - \sin a}\right)} \]

6. Simplified 48.5%

   \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \frac{\cos a}{b} - \sin a\right)}} \]

7. Taylor expanded in a around 0 31.6%

   \[\leadsto \color{blue}{\frac{r}{-0.3333333333333333 \cdot b + \frac{1}{b}}} \]

8. Final simplification 31.6%

   \[\leadsto \frac{r}{\frac{1}{b} + b \cdot -0.3333333333333333} \]

Alternative 11: 34.5% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ b \cdot r \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* b r))

C

double code(double r, double a, double b) {
	return b * r;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * r
end function

Java

public static double code(double r, double a, double b) {
	return b * r;
}

Python

def code(r, a, b):
	return b * r

Julia

function code(r, a, b)
	return Float64(b * r)
end

MATLAB

function tmp = code(r, a, b)
	tmp = b * r;
end

Wolfram

code[r_, a_, b_] := N[(b * r), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 72.9%

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

   2. *-commutative 72.9%

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

   3. +-commutative 72.9%

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

3. Simplified 72.9%

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

4. Taylor expanded in b around 0 47.6%

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

5. Taylor expanded in a around 0 30.9%

   \[\leadsto \color{blue}{b} \cdot r \]

6. Final simplification 30.9%

   \[\leadsto b \cdot r \]

Reproduce

herbie shell --seed 2023200 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))