rsin B (should all be same)

Percentage Accurate: 76.7% → 99.5%

Time: 12.6s

Alternatives: 7

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: 76.7% 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 76.9%

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

   1. +-commutative 76.9%

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

3. Simplified 76.9%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 2: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -29000 \lor \neg \left(b \leq 2.4 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -29000.0) (not (<= b 2.4e-8)))
   (* r (tan b))
   (* r (/ (sin b) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -29000.0) || !(b <= 2.4e-8)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (sin(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 <= (-29000.0d0)) .or. (.not. (b <= 2.4d-8))) then
        tmp = r * tan(b)
    else
        tmp = r * (sin(b) / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -29000.0) || !(b <= 2.4e-8)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -29000.0) or not (b <= 2.4e-8):
		tmp = r * math.tan(b)
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -29000.0) || !(b <= 2.4e-8))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -29000.0) || ~((b <= 2.4e-8)))
		tmp = r * tan(b);
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -29000.0], N[Not[LessEqual[b, 2.4e-8]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -29000 or 2.39999999999999998e-8 < b

   1. Initial program 54.7%

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

      1. associate-*r/ 54.6%

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

      2. +-commutative 54.6%

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

   3. Simplified 54.6%

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

      1. add-cube-cbrt 53.9%

         \[\leadsto \frac{r \cdot \color{blue}{\left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \sqrt[3]{\sin b}\right)}}{\cos \left(b + a\right)} \]

      2. pow3 54.0%

         \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

   6. Applied egg-rr 54.0%

      \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

   7. Taylor expanded in a around 0 54.5%

      \[\leadsto \frac{r \cdot {\left(\sqrt[3]{\sin b}\right)}^{3}}{\color{blue}{\cos b}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 55.1%

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

      2. associate-/l* 55.2%

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

      3. *-commutative 55.2%

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

      4. quot-tan 55.3%

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

   9. Applied egg-rr 55.3%

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

   if -29000 < b < 2.39999999999999998e-8

   1. Initial program 98.1%

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

      1. +-commutative 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in b around 0 98.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -29000 \lor \neg \left(b \leq 2.4 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% 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 76.9%

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

3. Final simplification 76.9%

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

Alternative 4: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-5} \lor \neg \left(b \leq 2.4 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -3.5e-5) (not (<= b 2.4e-8)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -3.5e-5) || !(b <= 2.4e-8)) {
		tmp = r * tan(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 <= (-3.5d-5)) .or. (.not. (b <= 2.4d-8))) then
        tmp = r * tan(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 <= -3.5e-5) || !(b <= 2.4e-8)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -3.5e-5) or not (b <= 2.4e-8):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -3.5e-5) || !(b <= 2.4e-8))
		tmp = Float64(r * tan(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 <= -3.5e-5) || ~((b <= 2.4e-8)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -3.5e-5], N[Not[LessEqual[b, 2.4e-8]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.4999999999999997e-5 or 2.39999999999999998e-8 < b

   1. Initial program 54.0%

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

      1. associate-*r/ 53.8%

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

      2. +-commutative 53.8%

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

   3. Simplified 53.8%

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

      1. add-cube-cbrt 53.2%

         \[\leadsto \frac{r \cdot \color{blue}{\left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \sqrt[3]{\sin b}\right)}}{\cos \left(b + a\right)} \]

      2. pow3 53.3%

         \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

   6. Applied egg-rr 53.3%

      \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

   7. Taylor expanded in a around 0 53.4%

      \[\leadsto \frac{r \cdot {\left(\sqrt[3]{\sin b}\right)}^{3}}{\color{blue}{\cos b}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 54.0%

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

      2. associate-/l* 54.1%

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

      3. *-commutative 54.1%

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

      4. quot-tan 54.2%

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

   9. Applied egg-rr 54.2%

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

   if -3.4999999999999997e-5 < b < 2.39999999999999998e-8

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 99.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-5} \lor \neg \left(b \leq 2.4 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-7} \lor \neg \left(b \leq 2.4 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -8.5e-7) (not (<= b 2.4e-8)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -8.5e-7) || !(b <= 2.4e-8)) {
		tmp = r * tan(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 <= (-8.5d-7)) .or. (.not. (b <= 2.4d-8))) then
        tmp = r * tan(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 <= -8.5e-7) || !(b <= 2.4e-8)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -8.5e-7) or not (b <= 2.4e-8):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -8.5e-7) || !(b <= 2.4e-8))
		tmp = Float64(r * tan(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 <= -8.5e-7) || ~((b <= 2.4e-8)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -8.5e-7], N[Not[LessEqual[b, 2.4e-8]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.50000000000000014e-7 or 2.39999999999999998e-8 < b

   1. Initial program 54.0%

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

      1. associate-*r/ 53.8%

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

      2. +-commutative 53.8%

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

   3. Simplified 53.8%

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

      1. add-cube-cbrt 53.2%

         \[\leadsto \frac{r \cdot \color{blue}{\left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \sqrt[3]{\sin b}\right)}}{\cos \left(b + a\right)} \]

      2. pow3 53.3%

         \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

   6. Applied egg-rr 53.3%

      \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

   7. Taylor expanded in a around 0 53.4%

      \[\leadsto \frac{r \cdot {\left(\sqrt[3]{\sin b}\right)}^{3}}{\color{blue}{\cos b}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 54.0%

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

      2. associate-/l* 54.1%

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

      3. *-commutative 54.1%

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

      4. quot-tan 54.2%

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

   9. Applied egg-rr 54.2%

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

   if -8.50000000000000014e-7 < b < 2.39999999999999998e-8

   1. Initial program 99.9%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r/ 99.9%

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

      2. add-cbrt-cube 59.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

      3. pow1/3 40.4%

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

      4. pow3 40.4%

         \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      5. associate-*r/ 40.4%

         \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{{\left({\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in b around 0 99.8%

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

      1. associate-/l* 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-7} \lor \neg \left(b \leq 2.4 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. associate-*r/ 76.8%

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

   2. +-commutative 76.8%

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

3. Simplified 76.8%

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

   1. add-cube-cbrt 75.7%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \sqrt[3]{\sin b}\right)}}{\cos \left(b + a\right)} \]

   2. pow3 75.7%

      \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

6. Applied egg-rr 75.7%

   \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

7. Taylor expanded in a around 0 60.0%

   \[\leadsto \frac{r \cdot {\left(\sqrt[3]{\sin b}\right)}^{3}}{\color{blue}{\cos b}} \]
8. Step-by-step derivation

   1. rem-cube-cbrt 60.8%

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

   2. associate-/l* 60.8%

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

   3. *-commutative 60.8%

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

   4. quot-tan 60.9%

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

9. Applied egg-rr 60.9%

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

10. Final simplification 60.9%

    \[\leadsto r \cdot \tan b \]
11. Add Preprocessing

Alternative 7: 34.1% 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 76.9%

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

   1. associate-*r/ 76.8%

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

   2. +-commutative 76.8%

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

3. Simplified 76.8%

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

   1. add-cube-cbrt 75.7%

      \[\leadsto \frac{r \cdot \color{blue}{\left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \sqrt[3]{\sin b}\right)}}{\cos \left(b + a\right)} \]

   2. pow3 75.7%

      \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

6. Applied egg-rr 75.7%

   \[\leadsto \frac{r \cdot \color{blue}{{\left(\sqrt[3]{\sin b}\right)}^{3}}}{\cos \left(b + a\right)} \]

7. Taylor expanded in a around 0 60.0%

   \[\leadsto \frac{r \cdot {\left(\sqrt[3]{\sin b}\right)}^{3}}{\color{blue}{\cos b}} \]

8. Taylor expanded in b around 0 35.9%

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

9. Final simplification 35.9%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))