rsin B (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 14.9s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

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(r * Float64(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[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

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(r * Float64(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[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 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 80.4%

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

   1. +-commutative 80.4%

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

   2. cos-sum 99.5%

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

4. Applied egg-rr 99.5%

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

Alternative 2: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0058 \lor \neg \left(b \leq 8000\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0058) (not (<= b 8000.0)))
   (* (sin b) (/ r (cos b)))
   (*
    (/ 1.0 (cos (+ b a)))
    (* b (+ r (* -0.16666666666666666 (* r (* b b))))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0058) || !(b <= 8000.0)) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = (1.0 / cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * 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 <= (-0.0058d0)) .or. (.not. (b <= 8000.0d0))) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = (1.0d0 / cos((b + a))) * (b * (r + ((-0.16666666666666666d0) * (r * (b * b)))))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0058) || !(b <= 8000.0)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = (1.0 / Math.cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0058) or not (b <= 8000.0):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = (1.0 / math.cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0058) || !(b <= 8000.0))
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(Float64(1.0 / cos(Float64(b + a))) * Float64(b * Float64(r + Float64(-0.16666666666666666 * Float64(r * Float64(b * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.0058) || ~((b <= 8000.0)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = (1.0 / cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0058], N[Not[LessEqual[b, 8000.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(r + N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0058 or 8e3 < b

   1. Initial program 60.2%

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

      1. +-commutative 60.2%

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

      2. cos-sum 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in r around 0 99.3%

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

      1. associate-/l* 99.2%

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

      2. /-rgt-identity 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. sub0-neg 99.2%

         \[\leadsto \frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b + \color{blue}{\left(0 - \sin b \cdot \sin a\right)}} \]

      6. *-commutative 99.2%

         \[\leadsto \frac{r}{1} \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} + \left(0 - \sin b \cdot \sin a\right)} \]

      7. times-frac 99.3%

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{1 \cdot \left(\cos b \cdot \cos a + \left(0 - \sin b \cdot \sin a\right)\right)}} \]

      8. *-commutative 99.3%

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{1 \cdot \left(\cos b \cdot \cos a + \left(0 - \sin b \cdot \sin a\right)\right)} \]

      9. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{\sin b}{1} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b \cdot \sin a\right)}} \]

      10. /-rgt-identity 99.1%

          \[\leadsto \color{blue}{\sin b} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b \cdot \sin a\right)} \]

      11. associate-+r- 99.1%

          \[\leadsto \sin b \cdot \frac{r}{\color{blue}{\left(\cos b \cdot \cos a + 0\right) - \sin b \cdot \sin a}} \]

   7. Simplified 99.1%

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

   8. Taylor expanded in a around 0 60.3%

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

   if -0.0058 < b < 8e3

   1. Initial program 97.7%

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

      1. associate-*r/ 97.5%

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

      2. clear-num 97.4%

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

      3. associate-/r/ 97.6%

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

      4. *-commutative 97.6%

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in b around 0 97.6%

      \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \color{blue}{\left(b \cdot \left(r + -0.16666666666666666 \cdot \left({b}^{2} \cdot r\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \color{blue}{\left(r \cdot {b}^{2}\right)}\right)\right) \]

      2. unpow2 97.6%

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

   7. Simplified 97.6%

      \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \color{blue}{\left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0058 \lor \neg \left(b \leq 8000\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0023:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 8000:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.0023)
   (* r (/ (sin b) (cos b)))
   (if (<= b 8000.0)
     (*
      (/ 1.0 (cos (+ b a)))
      (* b (+ r (* -0.16666666666666666 (* r (* b b))))))
     (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0023) {
		tmp = r * (sin(b) / cos(b));
	} else if (b <= 8000.0) {
		tmp = (1.0 / cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	} 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 <= (-0.0023d0)) then
        tmp = r * (sin(b) / cos(b))
    else if (b <= 8000.0d0) then
        tmp = (1.0d0 / cos((b + a))) * (b * (r + ((-0.16666666666666666d0) * (r * (b * b)))))
    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 <= -0.0023) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else if (b <= 8000.0) {
		tmp = (1.0 / Math.cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.0023:
		tmp = r * (math.sin(b) / math.cos(b))
	elif b <= 8000.0:
		tmp = (1.0 / math.cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))))
	else:
		tmp = (r * math.sin(b)) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.0023)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (b <= 8000.0)
		tmp = Float64(Float64(1.0 / cos(Float64(b + a))) * Float64(b * Float64(r + Float64(-0.16666666666666666 * Float64(r * Float64(b * b))))));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.0023)
		tmp = r * (sin(b) / cos(b));
	elseif (b <= 8000.0)
		tmp = (1.0 / cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.0023], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8000.0], N[(N[(1.0 / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(r + N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0023

   1. Initial program 56.9%

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

   3. Taylor expanded in a around 0 57.2%

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

      1. associate-/l* 57.3%

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

      2. *-commutative 57.3%

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

   5. Simplified 57.3%

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

   if -0.0023 < b < 8e3

   1. Initial program 97.7%

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

      1. associate-*r/ 97.5%

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

      2. clear-num 97.4%

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

      3. associate-/r/ 97.6%

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

      4. *-commutative 97.6%

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in b around 0 97.6%

      \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \color{blue}{\left(b \cdot \left(r + -0.16666666666666666 \cdot \left({b}^{2} \cdot r\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \color{blue}{\left(r \cdot {b}^{2}\right)}\right)\right) \]

      2. unpow2 97.6%

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

   7. Simplified 97.6%

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

   if 8e3 < b

   1. Initial program 63.3%

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

   3. Taylor expanded in a around 0 63.4%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0023:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 8000:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0068:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 8000:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.0068)
   (* r (/ (sin b) (cos b)))
   (if (<= b 8000.0)
     (*
      (/ 1.0 (cos (+ b a)))
      (* b (+ r (* -0.16666666666666666 (* r (* b b))))))
     (* (sin b) (/ r (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0068) {
		tmp = r * (sin(b) / cos(b));
	} else if (b <= 8000.0) {
		tmp = (1.0 / cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	} else {
		tmp = sin(b) * (r / 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 <= (-0.0068d0)) then
        tmp = r * (sin(b) / cos(b))
    else if (b <= 8000.0d0) then
        tmp = (1.0d0 / cos((b + a))) * (b * (r + ((-0.16666666666666666d0) * (r * (b * b)))))
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0068) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else if (b <= 8000.0) {
		tmp = (1.0 / Math.cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.0068:
		tmp = r * (math.sin(b) / math.cos(b))
	elif b <= 8000.0:
		tmp = (1.0 / math.cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.0068)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (b <= 8000.0)
		tmp = Float64(Float64(1.0 / cos(Float64(b + a))) * Float64(b * Float64(r + Float64(-0.16666666666666666 * Float64(r * Float64(b * b))))));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.0068)
		tmp = r * (sin(b) / cos(b));
	elseif (b <= 8000.0)
		tmp = (1.0 / cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.0068], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8000.0], N[(N[(1.0 / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(r + N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.00679999999999999962

   1. Initial program 56.9%

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

   3. Taylor expanded in a around 0 57.2%

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

      1. associate-/l* 57.3%

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

      2. *-commutative 57.3%

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

   5. Simplified 57.3%

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

   if -0.00679999999999999962 < b < 8e3

   1. Initial program 97.7%

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

      1. associate-*r/ 97.5%

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

      2. clear-num 97.4%

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

      3. associate-/r/ 97.6%

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

      4. *-commutative 97.6%

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in b around 0 97.6%

      \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \color{blue}{\left(b \cdot \left(r + -0.16666666666666666 \cdot \left({b}^{2} \cdot r\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \color{blue}{\left(r \cdot {b}^{2}\right)}\right)\right) \]

      2. unpow2 97.6%

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

   7. Simplified 97.6%

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

   if 8e3 < b

   1. Initial program 63.3%

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

      1. +-commutative 63.3%

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

      2. cos-sum 99.1%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in r around 0 99.3%

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

      1. associate-/l* 99.1%

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

      2. /-rgt-identity 99.1%

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

      3. sub-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. sub0-neg 99.1%

         \[\leadsto \frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b + \color{blue}{\left(0 - \sin b \cdot \sin a\right)}} \]

      6. *-commutative 99.1%

         \[\leadsto \frac{r}{1} \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} + \left(0 - \sin b \cdot \sin a\right)} \]

      7. times-frac 99.3%

         \[\leadsto \color{blue}{\frac{r \cdot \sin b}{1 \cdot \left(\cos b \cdot \cos a + \left(0 - \sin b \cdot \sin a\right)\right)}} \]

      8. *-commutative 99.3%

         \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{1 \cdot \left(\cos b \cdot \cos a + \left(0 - \sin b \cdot \sin a\right)\right)} \]

      9. times-frac 99.2%

         \[\leadsto \color{blue}{\frac{\sin b}{1} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b \cdot \sin a\right)}} \]

      10. /-rgt-identity 99.2%

          \[\leadsto \color{blue}{\sin b} \cdot \frac{r}{\cos b \cdot \cos a + \left(0 - \sin b \cdot \sin a\right)} \]

      11. associate-+r- 99.2%

          \[\leadsto \sin b \cdot \frac{r}{\color{blue}{\left(\cos b \cdot \cos a + 0\right) - \sin b \cdot \sin a}} \]

   7. Simplified 99.2%

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

   8. Taylor expanded in a around 0 63.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0068:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 8000:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% 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 80.4%

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

3. Final simplification 80.4%

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

Alternative 6: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

3. Taylor expanded in b around 0 58.0%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a}} \]
4. Add Preprocessing

Alternative 7: 55.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -49 \lor \neg \left(b \leq 1.25 \cdot 10^{+19}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -49.0) (not (<= b 1.25e+19)))
   (* r (sin b))
   (*
    (/ 1.0 (cos (+ b a)))
    (* b (+ r (* -0.16666666666666666 (* r (* b b))))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -49.0) || !(b <= 1.25e+19)) {
		tmp = r * sin(b);
	} else {
		tmp = (1.0 / cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * 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 <= (-49.0d0)) .or. (.not. (b <= 1.25d+19))) then
        tmp = r * sin(b)
    else
        tmp = (1.0d0 / cos((b + a))) * (b * (r + ((-0.16666666666666666d0) * (r * (b * b)))))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -49.0) || !(b <= 1.25e+19)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = (1.0 / Math.cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -49.0) or not (b <= 1.25e+19):
		tmp = r * math.sin(b)
	else:
		tmp = (1.0 / math.cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -49.0) || !(b <= 1.25e+19))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(Float64(1.0 / cos(Float64(b + a))) * Float64(b * Float64(r + Float64(-0.16666666666666666 * Float64(r * Float64(b * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -49.0) || ~((b <= 1.25e+19)))
		tmp = r * sin(b);
	else
		tmp = (1.0 / cos((b + a))) * (b * (r + (-0.16666666666666666 * (r * (b * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -49.0], N[Not[LessEqual[b, 1.25e+19]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(r + N[(-0.16666666666666666 * N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -49 or 1.25e19 < b

   1. Initial program 59.8%

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

      1. associate-*r/ 59.8%

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

      2. clear-num 59.7%

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

      3. associate-/r/ 59.8%

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

      4. *-commutative 59.8%

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

   4. Applied egg-rr 59.8%

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

   5. Taylor expanded in b around 0 12.0%

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

   6. Taylor expanded in a around 0 12.3%

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

   if -49 < b < 1.25e19

   1. Initial program 96.4%

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

      1. associate-*r/ 96.3%

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

      2. clear-num 96.2%

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

      3. associate-/r/ 96.3%

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

      4. *-commutative 96.3%

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

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in b around 0 94.1%

      \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \color{blue}{\left(b \cdot \left(r + -0.16666666666666666 \cdot \left({b}^{2} \cdot r\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 94.1%

         \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \color{blue}{\left(r \cdot {b}^{2}\right)}\right)\right) \]

      2. unpow2 94.1%

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

   7. Simplified 94.1%

      \[\leadsto \frac{1}{\cos \left(a + b\right)} \cdot \color{blue}{\left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -49 \lor \neg \left(b \leq 1.25 \cdot 10^{+19}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(r + -0.16666666666666666 \cdot \left(r \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -54 \lor \neg \left(b \leq 56000000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -54.0) (not (<= b 56000000000000.0)))
   (* r (sin b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -54.0) || !(b <= 56000000000000.0)) {
		tmp = r * sin(b);
	} else {
		tmp = b * (r / 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 <= (-54.0d0)) .or. (.not. (b <= 56000000000000.0d0))) then
        tmp = r * sin(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -54.0) || !(b <= 56000000000000.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -54.0) or not (b <= 56000000000000.0):
		tmp = r * math.sin(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -54.0) || !(b <= 56000000000000.0))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -54.0) || ~((b <= 56000000000000.0)))
		tmp = r * sin(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -54.0], N[Not[LessEqual[b, 56000000000000.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -54 or 5.6e13 < b

   1. Initial program 59.3%

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

      1. associate-*r/ 59.3%

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

      2. clear-num 59.2%

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

      3. associate-/r/ 59.3%

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

      4. *-commutative 59.3%

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

   4. Applied egg-rr 59.3%

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

   5. Taylor expanded in b around 0 11.9%

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

   6. Taylor expanded in a around 0 12.2%

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

   if -54 < b < 5.6e13

   1. Initial program 97.0%

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

   3. Taylor expanded in b around 0 94.5%

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

      1. associate-/l* 94.7%

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

   5. Simplified 94.7%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -54 \lor \neg \left(b \leq 56000000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. associate-*r/ 80.3%

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

   2. clear-num 80.2%

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

   3. associate-/r/ 80.3%

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

   4. *-commutative 80.3%

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

4. Applied egg-rr 80.3%

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

5. Taylor expanded in b around 0 58.0%

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

6. Taylor expanded in a around 0 38.4%

   \[\leadsto \color{blue}{r \cdot \sin b} \]
7. Add Preprocessing

Alternative 10: 34.6% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

3. Taylor expanded in b around 0 54.3%

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

   1. associate-/l* 54.4%

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

5. Simplified 54.4%

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

6. Taylor expanded in a around 0 34.5%

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

7. Final simplification 34.5%

   \[\leadsto r \cdot b \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024097 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))