rsin B (should all be same)

Percentage Accurate: 76.5% → 99.5%

Time: 20.3s

Alternatives: 9

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: 76.5% 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 81.3%

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

   1. +-commutative 81.3%

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

3. Simplified 81.3%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00013 \lor \neg \left(a \leq 0.0033\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00013) (not (<= a 0.0033)))
   (* r (/ (sin b) (cos a)))
   (* r (tan b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00013) || !(a <= 0.0033)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * tan(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 ((a <= (-0.00013d0)) .or. (.not. (a <= 0.0033d0))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00013) || !(a <= 0.0033)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00013) or not (a <= 0.0033):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.00013) || !(a <= 0.0033))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.00013) || ~((a <= 0.0033)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00013], N[Not[LessEqual[a, 0.0033]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.29999999999999989e-4 or 0.0033 < a

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in b around 0 62.5%

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

   if -1.29999999999999989e-4 < a < 0.0033

   1. Initial program 98.7%

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

      1. associate-*r/ 98.7%

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

      2. +-commutative 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.1%

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

      2. associate-/r/ 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in a around 0 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*r* 98.7%

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

      3. div-inv 98.7%

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

      4. clear-num 98.6%

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

      5. un-div-inv 98.6%

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

      6. clear-num 98.6%

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

      7. quot-tan 98.6%

         \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

   9. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]

   11. Simplified 98.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00013 \lor \neg \left(a \leq 0.0033\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% 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 81.3%

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

3. Final simplification 81.3%

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

Alternative 4: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-5} \lor \neg \left(b \leq 7.9 \cdot 10^{-16}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.2e-5) (not (<= b 7.9e-16)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.2e-5) || !(b <= 7.9e-16)) {
		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 <= (-4.2d-5)) .or. (.not. (b <= 7.9d-16))) 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 <= -4.2e-5) || !(b <= 7.9e-16)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.2e-5) or not (b <= 7.9e-16):
		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 <= -4.2e-5) || !(b <= 7.9e-16))
		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 <= -4.2e-5) || ~((b <= 7.9e-16)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.2e-5], N[Not[LessEqual[b, 7.9e-16]], $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 < -4.19999999999999977e-5 or 7.9000000000000002e-16 < b

   1. Initial program 61.8%

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

      1. associate-*r/ 61.8%

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

      2. +-commutative 61.8%

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

   3. Simplified 61.8%

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

      1. clear-num 61.7%

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

      2. associate-/r/ 61.7%

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

   6. Applied egg-rr 61.7%

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

   7. Taylor expanded in a around 0 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*r* 61.9%

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

      3. div-inv 62.0%

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

      4. clear-num 61.9%

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

      5. un-div-inv 62.0%

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

      6. clear-num 61.9%

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

      7. quot-tan 62.0%

         \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]

   11. Simplified 62.1%

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

   if -4.19999999999999977e-5 < b < 7.9000000000000002e-16

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in b around 0 99.5%

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

      1. associate-/l* 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-5} \lor \neg \left(b \leq 7.9 \cdot 10^{-16}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 7.9 \cdot 10^{-16}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00014) (not (<= b 7.9e-16)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00014) || !(b <= 7.9e-16)) {
		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 <= (-0.00014d0)) .or. (.not. (b <= 7.9d-16))) 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 <= -0.00014) || !(b <= 7.9e-16)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00014) or not (b <= 7.9e-16):
		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 <= -0.00014) || !(b <= 7.9e-16))
		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 <= -0.00014) || ~((b <= 7.9e-16)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00014], N[Not[LessEqual[b, 7.9e-16]], $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 < -1.3999999999999999e-4 or 7.9000000000000002e-16 < b

   1. Initial program 61.8%

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

      1. associate-*r/ 61.8%

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

      2. +-commutative 61.8%

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

   3. Simplified 61.8%

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

      1. clear-num 61.7%

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

      2. associate-/r/ 61.7%

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

   6. Applied egg-rr 61.7%

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

   7. Taylor expanded in a around 0 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*r* 61.9%

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

      3. div-inv 62.0%

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

      4. clear-num 61.9%

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

      5. un-div-inv 62.0%

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

      6. clear-num 61.9%

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

      7. quot-tan 62.0%

         \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]

   11. Simplified 62.1%

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

   if -1.3999999999999999e-4 < b < 7.9000000000000002e-16

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in b around 0 99.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 7.9 \cdot 10^{-16}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 7.9 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2.55e-5)
   (* r (tan b))
   (if (<= b 7.9e-16) (* r (/ b (cos a))) (/ r (/ 1.0 (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.55e-5) {
		tmp = r * tan(b);
	} else if (b <= 7.9e-16) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r / (1.0 / tan(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 <= (-2.55d-5)) then
        tmp = r * tan(b)
    else if (b <= 7.9d-16) then
        tmp = r * (b / cos(a))
    else
        tmp = r / (1.0d0 / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -2.55e-5) {
		tmp = r * Math.tan(b);
	} else if (b <= 7.9e-16) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r / (1.0 / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2.55e-5:
		tmp = r * math.tan(b)
	elif b <= 7.9e-16:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r / (1.0 / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2.55e-5)
		tmp = Float64(r * tan(b));
	elseif (b <= 7.9e-16)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r / Float64(1.0 / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2.55e-5)
		tmp = r * tan(b);
	elseif (b <= 7.9e-16)
		tmp = r * (b / cos(a));
	else
		tmp = r / (1.0 / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2.55e-5], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.9e-16], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.54999999999999998e-5

   1. Initial program 60.9%

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

      1. associate-*r/ 60.8%

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

      2. +-commutative 60.8%

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

   3. Simplified 60.8%

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

      1. clear-num 60.7%

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

      2. associate-/r/ 60.7%

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

   6. Applied egg-rr 60.7%

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

   7. Taylor expanded in a around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*r* 61.6%

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

      3. div-inv 61.6%

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

      4. clear-num 61.6%

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

      5. un-div-inv 61.6%

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

      6. clear-num 61.5%

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

      7. quot-tan 61.5%

         \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]

   11. Simplified 61.7%

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

   if -2.54999999999999998e-5 < b < 7.9000000000000002e-16

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in b around 0 99.6%

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

   if 7.9000000000000002e-16 < b

   1. Initial program 62.8%

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

      1. associate-*r/ 62.9%

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

      2. +-commutative 62.9%

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

   3. Simplified 62.9%

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

      1. clear-num 62.8%

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

      2. associate-/r/ 62.8%

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

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in a around 0 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*r* 62.3%

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

      3. div-inv 62.3%

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

      4. clear-num 62.2%

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

      5. un-div-inv 62.4%

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

      6. clear-num 62.4%

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

      7. quot-tan 62.5%

         \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

   9. Applied egg-rr 62.5%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 7.9 \cdot 10^{-16}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.8% 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 81.3%

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

   1. +-commutative 81.3%

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

3. Simplified 81.3%

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

   1. add-cbrt-cube 59.4%

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

   2. add-cbrt-cube 59.3%

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

   3. cbrt-undiv 59.4%

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

   4. pow3 59.4%

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

   5. pow3 59.4%

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

6. Applied egg-rr 59.4%

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

7. Taylor expanded in b around 0 36.6%

   \[\leadsto r \cdot \sqrt[3]{\frac{{\sin b}^{3}}{\color{blue}{{\cos a}^{3}}}} \]

8. Taylor expanded in a around 0 41.3%

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

9. Final simplification 41.3%

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

Alternative 8: 60.0% 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 81.3%

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

   1. associate-*r/ 81.3%

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

   2. +-commutative 81.3%

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

3. Simplified 81.3%

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

   1. clear-num 80.7%

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

   2. associate-/r/ 81.2%

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

6. Applied egg-rr 81.2%

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

7. Taylor expanded in a around 0 63.9%

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

   1. *-commutative 63.9%

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

   2. associate-*r* 63.9%

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

   3. div-inv 64.0%

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

   4. clear-num 63.9%

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

   5. un-div-inv 63.9%

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

   6. clear-num 63.9%

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

   7. quot-tan 63.9%

      \[\leadsto \frac{r}{\frac{1}{\color{blue}{\tan b}}} \]

9. Applied egg-rr 63.9%

   \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]

11. Simplified 64.0%

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

12. Final simplification 64.0%

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

Alternative 9: 34.8% 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 81.3%

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

   1. +-commutative 81.3%

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

3. Simplified 81.3%

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

5. Taylor expanded in b around 0 54.3%

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

   1. associate-/l* 54.3%

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

7. Simplified 54.3%

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

8. Taylor expanded in a around 0 36.9%

   \[\leadsto \color{blue}{b \cdot r} \]
9. Step-by-step derivation

   1. *-commutative 36.9%

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

10. Simplified 36.9%

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

11. Final simplification 36.9%

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

Reproduce

herbie shell --seed 2024074 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))