rsin B (should all be same)

Percentage Accurate: 76.1% → 99.4%

Time: 14.6s

Alternatives: 14

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: 76.1% 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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. associate-*r/ 79.6%

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

   2. associate-/l* 79.5%

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

   3. +-commutative 79.5%

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

3. Simplified 79.5%

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

   1. cos-sum 99.5%

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

   2. div-sub 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in b around 0 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. associate-*r/ 79.6%

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

   2. associate-/l* 79.5%

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

   3. +-commutative 79.5%

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

3. Simplified 79.5%

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

   1. cos-sum 99.5%

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

   2. div-sub 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in b around 0 99.5%

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

   1. sub-neg 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-/l* 99.5%

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

8. Applied egg-rr 99.5%

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

   1. unsub-neg 99.5%

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

   2. quot-tan 99.5%

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

10. Applied egg-rr 99.5%

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

11. Final simplification 99.5%

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

Alternative 3: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-6} \lor \neg \left(b \leq 16500\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -9e-6) (not (<= b 16500.0)))
   (* r (/ (sin b) (cos b)))
   (* b (/ r (cos (+ b a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -9e-6) || !(b <= 16500.0)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = b * (r / cos((b + 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 <= (-9d-6)) .or. (.not. (b <= 16500.0d0))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = b * (r / cos((b + a)))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -9e-6) || !(b <= 16500.0)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = b * (r / Math.cos((b + a)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -9e-6) or not (b <= 16500.0):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = b * (r / math.cos((b + a)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -9e-6) || !(b <= 16500.0))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(b * Float64(r / cos(Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -9e-6) || ~((b <= 16500.0)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = b * (r / cos((b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -9e-6], N[Not[LessEqual[b, 16500.0]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.00000000000000023e-6 or 16500 < b

   1. Initial program 59.8%

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

      1. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in a around 0 58.9%

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

   if -9.00000000000000023e-6 < b < 16500

   1. Initial program 98.6%

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

      1. associate-*r/ 98.7%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

   3. Simplified 98.6%

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

      1. associate-/r/ 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in b around 0 98.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-6} \lor \neg \left(b \leq 16500\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 4: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-5} \lor \neg \left(b \leq 16500\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.5e-5) (not (<= b 16500.0)))
   (* (sin b) (/ r (cos b)))
   (* b (/ r (cos (+ b a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.5e-5) || !(b <= 16500.0)) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = b * (r / cos((b + a)));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.5d-5)) .or. (.not. (b <= 16500.0d0))) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = b * (r / cos((b + a)))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.5e-5) || !(b <= 16500.0)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = b * (r / Math.cos((b + a)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.5e-5) or not (b <= 16500.0):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = b * (r / math.cos((b + a)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.5e-5) || !(b <= 16500.0))
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(b * Float64(r / cos(Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.5e-5) || ~((b <= 16500.0)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = b * (r / cos((b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.5e-5], N[Not[LessEqual[b, 16500.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.50000000000000004e-5 or 16500 < b

   1. Initial program 59.8%

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

      1. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in a around 0 59.0%

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

      1. associate-/l* 59.0%

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

      2. associate-/r/ 59.0%

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

   6. Simplified 59.0%

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

   if -1.50000000000000004e-5 < b < 16500

   1. Initial program 98.6%

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

      1. associate-*r/ 98.7%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

   3. Simplified 98.6%

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

      1. associate-/r/ 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in b around 0 98.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-5} \lor \neg \left(b \leq 16500\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 5: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.8e-5)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 16500.0) (* b (/ r (cos (+ b a)))) (* (sin b) (/ r (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.8e-5) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 16500.0) {
		tmp = b * (r / cos((b + a)));
	} 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 <= (-1.8d-5)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 16500.0d0) then
        tmp = b * (r / cos((b + a)))
    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 <= -1.8e-5) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 16500.0) {
		tmp = b * (r / Math.cos((b + a)));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.8e-5:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 16500.0:
		tmp = b * (r / math.cos((b + a)))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.8e-5)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 16500.0)
		tmp = Float64(b * Float64(r / cos(Float64(b + a))));
	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 <= -1.8e-5)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 16500.0)
		tmp = b * (r / cos((b + a)));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.8e-5], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 16500.0], N[(b * N[(r / N[Cos[N[(b + a), $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 < -1.80000000000000005e-5

   1. Initial program 60.0%

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

      1. associate-*r/ 60.0%

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

      2. associate-/l* 60.1%

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

      3. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in a around 0 57.9%

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

   if -1.80000000000000005e-5 < b < 16500

   1. Initial program 98.6%

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

      1. associate-*r/ 98.7%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

   3. Simplified 98.6%

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

      1. associate-/r/ 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in b around 0 98.8%

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

   if 16500 < b

   1. Initial program 59.5%

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

      1. +-commutative 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in a around 0 60.2%

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

      1. associate-/l* 60.0%

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

      2. associate-/r/ 60.1%

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

   6. Simplified 60.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 6: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.000102:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.000102)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 16500.0) (* b (/ r (cos (+ b a)))) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.000102) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 16500.0) {
		tmp = b * (r / cos((b + a)));
	} else {
		tmp = (r * sin(b)) / cos(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.000102d0)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 16500.0d0) then
        tmp = b * (r / cos((b + a)))
    else
        tmp = (r * sin(b)) / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.000102) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 16500.0) {
		tmp = b * (r / Math.cos((b + a)));
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.000102:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 16500.0:
		tmp = b * (r / math.cos((b + a)))
	else:
		tmp = (r * math.sin(b)) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.000102)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 16500.0)
		tmp = Float64(b * Float64(r / cos(Float64(b + a))));
	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 <= -0.000102)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 16500.0)
		tmp = b * (r / cos((b + a)));
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.000102], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 16500.0], N[(b * N[(r / N[Cos[N[(b + a), $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 < -1.01999999999999999e-4

   1. Initial program 60.0%

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

      1. associate-*r/ 60.0%

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

      2. associate-/l* 60.1%

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

      3. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in a around 0 57.9%

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

   if -1.01999999999999999e-4 < b < 16500

   1. Initial program 98.6%

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

      1. associate-*r/ 98.7%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

   3. Simplified 98.6%

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

      1. associate-/r/ 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in b around 0 98.8%

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

   if 16500 < b

   1. Initial program 59.5%

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

      1. +-commutative 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in a around 0 60.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000102:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]

Alternative 7: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

2. Final simplification 79.5%

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

Alternative 8: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. associate-*r/ 79.6%

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

   2. associate-/l* 79.5%

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

   3. +-commutative 79.5%

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

3. Simplified 79.5%

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

   1. associate-/r/ 79.6%

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

5. Applied egg-rr 79.6%

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

6. Final simplification 79.6%

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

Alternative 9: 54.5% 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 79.5%

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

   1. +-commutative 79.5%

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

3. Simplified 79.5%

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

4. Taylor expanded in b around 0 56.0%

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

5. Final simplification 56.0%

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

Alternative 10: 54.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.9:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 2.5:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -3.9)
   (* r (sin b))
   (if (<= b 2.5) (* b (/ r (cos (+ b a)))) (/ (sin b) (/ 1.0 r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.9) {
		tmp = r * sin(b);
	} else if (b <= 2.5) {
		tmp = b * (r / cos((b + a)));
	} else {
		tmp = sin(b) / (1.0 / r);
	}
	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 <= (-3.9d0)) then
        tmp = r * sin(b)
    else if (b <= 2.5d0) then
        tmp = b * (r / cos((b + a)))
    else
        tmp = sin(b) / (1.0d0 / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -3.9) {
		tmp = r * Math.sin(b);
	} else if (b <= 2.5) {
		tmp = b * (r / Math.cos((b + a)));
	} else {
		tmp = Math.sin(b) / (1.0 / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -3.9:
		tmp = r * math.sin(b)
	elif b <= 2.5:
		tmp = b * (r / math.cos((b + a)))
	else:
		tmp = math.sin(b) / (1.0 / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -3.9)
		tmp = Float64(r * sin(b));
	elseif (b <= 2.5)
		tmp = Float64(b * Float64(r / cos(Float64(b + a))));
	else
		tmp = Float64(sin(b) / Float64(1.0 / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -3.9)
		tmp = r * sin(b);
	elseif (b <= 2.5)
		tmp = b * (r / cos((b + a)));
	else
		tmp = sin(b) / (1.0 / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -3.9], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5], N[(b * N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.89999999999999991

   1. Initial program 60.0%

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

      1. associate-*r/ 60.0%

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

      2. associate-/l* 60.1%

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

      3. +-commutative 60.1%

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

   3. Simplified 60.1%

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

      1. associate-/r/ 60.1%

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

   5. Applied egg-rr 60.1%

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

   6. Taylor expanded in b around 0 12.0%

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

   7. Taylor expanded in a around 0 13.1%

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

   if -3.89999999999999991 < b < 2.5

   1. Initial program 99.4%

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

      1. associate-*r/ 99.5%

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

      2. associate-/l* 99.3%

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

      3. +-commutative 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in b around 0 99.5%

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

   if 2.5 < b

   1. Initial program 58.6%

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

      1. associate-*r/ 58.7%

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

      2. associate-/l* 58.6%

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

      3. +-commutative 58.6%

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

   3. Simplified 58.6%

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

      1. associate-/r/ 58.6%

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

   5. Applied egg-rr 58.6%

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

   6. Taylor expanded in b around 0 11.7%

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

   7. Taylor expanded in r around 0 11.7%

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

      1. *-commutative 11.7%

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

      2. *-rgt-identity 11.7%

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

      3. times-frac 11.7%

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

      4. /-rgt-identity 11.7%

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

      5. associate-/r/ 11.7%

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

   9. Simplified 11.7%

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

   10. Taylor expanded in a around 0 13.2%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 2.5:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \]

Alternative 11: 54.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \lor \neg \left(b \leq 1.9 \cdot 10^{-18}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.45) (not (<= b 1.9e-18))) (* r (sin b)) (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.45) || !(b <= 1.9e-18)) {
		tmp = r * sin(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.45d0)) .or. (.not. (b <= 1.9d-18))) then
        tmp = r * sin(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.45) || !(b <= 1.9e-18)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.45) or not (b <= 1.9e-18):
		tmp = r * math.sin(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.45) || !(b <= 1.9e-18))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.45) || ~((b <= 1.9e-18)))
		tmp = r * sin(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.45], N[Not[LessEqual[b, 1.9e-18]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.44999999999999996 or 1.8999999999999999e-18 < b

   1. Initial program 59.9%

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

      1. associate-*r/ 60.0%

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

      2. associate-/l* 60.0%

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

      3. +-commutative 60.0%

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

   3. Simplified 60.0%

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

      1. associate-/r/ 60.0%

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

   5. Applied egg-rr 60.0%

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

   6. Taylor expanded in b around 0 13.2%

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

   7. Taylor expanded in a around 0 14.5%

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

   if -1.44999999999999996 < b < 1.8999999999999999e-18

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in b around 0 99.4%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \lor \neg \left(b \leq 1.9 \cdot 10^{-18}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 12: 54.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 1.1:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.25)
   (* r (sin b))
   (if (<= b 1.1) (/ (* r b) (cos a)) (/ (sin b) (/ 1.0 r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.25) {
		tmp = r * sin(b);
	} else if (b <= 1.1) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = sin(b) / (1.0 / r);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.25d0)) then
        tmp = r * sin(b)
    else if (b <= 1.1d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = sin(b) / (1.0d0 / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.25) {
		tmp = r * Math.sin(b);
	} else if (b <= 1.1) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = Math.sin(b) / (1.0 / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.25:
		tmp = r * math.sin(b)
	elif b <= 1.1:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = math.sin(b) / (1.0 / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.25)
		tmp = Float64(r * sin(b));
	elseif (b <= 1.1)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(sin(b) / Float64(1.0 / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1.25)
		tmp = r * sin(b);
	elseif (b <= 1.1)
		tmp = (r * b) / cos(a);
	else
		tmp = sin(b) / (1.0 / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.25], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25

   1. Initial program 60.0%

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

      1. associate-*r/ 60.0%

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

      2. associate-/l* 60.1%

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

      3. +-commutative 60.1%

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

   3. Simplified 60.1%

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

      1. associate-/r/ 60.1%

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

   5. Applied egg-rr 60.1%

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

   6. Taylor expanded in b around 0 12.0%

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

   7. Taylor expanded in a around 0 13.1%

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

   if -1.25 < b < 1.1000000000000001

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in b around 0 99.5%

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

   if 1.1000000000000001 < b

   1. Initial program 58.6%

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

      1. associate-*r/ 58.7%

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

      2. associate-/l* 58.6%

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

      3. +-commutative 58.6%

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

   3. Simplified 58.6%

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

      1. associate-/r/ 58.6%

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

   5. Applied egg-rr 58.6%

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

   6. Taylor expanded in b around 0 11.7%

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

   7. Taylor expanded in r around 0 11.7%

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

      1. *-commutative 11.7%

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

      2. *-rgt-identity 11.7%

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

      3. times-frac 11.7%

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

      4. /-rgt-identity 11.7%

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

      5. associate-/r/ 11.7%

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

   9. Simplified 11.7%

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

   10. Taylor expanded in a around 0 13.2%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 1.1:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \]

Alternative 13: 38.5% 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 79.5%

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

   1. associate-*r/ 79.6%

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

   2. associate-/l* 79.5%

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

   3. +-commutative 79.5%

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

3. Simplified 79.5%

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

   1. associate-/r/ 79.6%

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

5. Applied egg-rr 79.6%

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

6. Taylor expanded in b around 0 56.0%

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

7. Taylor expanded in a around 0 41.1%

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

8. Final simplification 41.1%

   \[\leadsto r \cdot \sin b \]

Alternative 14: 34.2% 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 79.5%

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

   1. +-commutative 79.5%

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

3. Simplified 79.5%

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

4. Taylor expanded in b around 0 52.2%

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

5. Taylor expanded in a around 0 36.4%

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

6. Final simplification 36.4%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023318 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))