rsin B (should all be same)

Percentage Accurate: 76.4% → 99.4%

Time: 15.0s

Alternatives: 15

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: 76.4% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(N[Sin[b], $MachinePrecision] * N[(r / 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 70.1%

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

   1. *-commutative 70.1%

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

   2. associate-/r/ 70.1%

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

   3. +-commutative 70.1%

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

3. Simplified 70.1%

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

   1. div-inv 70.1%

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

   2. clear-num 70.2%

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

   3. *-commutative 70.2%

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

5. Applied egg-rr 70.2%

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

   1. cos-sum 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. +-commutative 70.1%

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

3. Simplified 70.1%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (cos b) (cos a) 0.0)))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / fma(cos(b), cos(a), 0.0);
}

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(cos(b), cos(a), 0.0))
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.1%

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

   1. associate-*r/ 70.2%

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

   2. +-commutative 70.2%

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

3. Simplified 70.2%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-def 99.4%

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

5. Applied egg-rr 99.4%

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

   1. add-sqr-sqrt 45.4%

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

   2. sqrt-unprod 83.3%

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

   3. sqr-neg 83.3%

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

   4. sqrt-unprod 37.8%

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

   5. add-sqr-sqrt 70.2%

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

   6. sin-mult 71.4%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\frac{\cos \left(b - a\right) - \cos \left(b + a\right)}{2}}\right)} \]

   7. div-sub 71.4%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\frac{\cos \left(b - a\right)}{2} - \frac{\cos \left(b + a\right)}{2}}\right)} \]

   8. cos-diff 70.6%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{\color{blue}{\cos b \cdot \cos a + \sin b \cdot \sin a}}{2} - \frac{\cos \left(b + a\right)}{2}\right)} \]

   9. add-sqr-sqrt 38.2%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{\cos b \cdot \cos a + \color{blue}{\left(\sqrt{\sin b} \cdot \sqrt{\sin b}\right)} \cdot \sin a}{2} - \frac{\cos \left(b + a\right)}{2}\right)} \]

   10. sqrt-unprod 71.5%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{\cos b \cdot \cos a + \color{blue}{\sqrt{\sin b \cdot \sin b}} \cdot \sin a}{2} - \frac{\cos \left(b + a\right)}{2}\right)} \]

   11. sqr-neg 71.5%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{\cos b \cdot \cos a + \sqrt{\color{blue}{\left(-\sin b\right) \cdot \left(-\sin b\right)}} \cdot \sin a}{2} - \frac{\cos \left(b + a\right)}{2}\right)} \]

   12. sqrt-unprod 33.3%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{\cos b \cdot \cos a + \color{blue}{\left(\sqrt{-\sin b} \cdot \sqrt{-\sin b}\right)} \cdot \sin a}{2} - \frac{\cos \left(b + a\right)}{2}\right)} \]

   13. add-sqr-sqrt 72.4%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{\cos b \cdot \cos a + \color{blue}{\left(-\sin b\right)} \cdot \sin a}{2} - \frac{\cos \left(b + a\right)}{2}\right)} \]

   14. cancel-sign-sub-inv 72.4%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}}{2} - \frac{\cos \left(b + a\right)}{2}\right)} \]

   15. cos-sum 71.6%

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \frac{\color{blue}{\cos \left(b + a\right)}}{2} - \frac{\cos \left(b + a\right)}{2}\right)} \]

7. Applied egg-rr 71.6%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\frac{\cos \left(b + a\right)}{2} - \frac{\cos \left(b + a\right)}{2}}\right)} \]
8. Step-by-step derivation

   1. +-inverses 71.6%

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

9. Simplified 71.6%

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

10. Final simplification 71.6%

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \]

Alternative 4: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0048 \lor \neg \left(b \leq 5.4 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0048) (not (<= b 5.4e-5)))
   (* r (/ (sin b) (cos b)))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0048) || !(b <= 5.4e-5)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = (r * b) / 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 <= (-0.0048d0)) .or. (.not. (b <= 5.4d-5))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0048) || !(b <= 5.4e-5)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0048) or not (b <= 5.4e-5):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0048) || !(b <= 5.4e-5))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.0048) || ~((b <= 5.4e-5)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0048], N[Not[LessEqual[b, 5.4e-5]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00479999999999999958 or 5.3999999999999998e-5 < b

   1. Initial program 47.4%

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

      1. +-commutative 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in a around 0 47.6%

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

   if -0.00479999999999999958 < b < 5.3999999999999998e-5

   1. Initial program 98.5%

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

      1. associate-*r/ 98.5%

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

      2. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in b around 0 98.5%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0048 \lor \neg \left(b \leq 5.4 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 5: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0034 \lor \neg \left(b \leq 5.4 \cdot 10^{-5}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0034) (not (<= b 5.4e-5)))
   (* (sin b) (/ r (cos b)))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0034) || !(b <= 5.4e-5)) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = (r * b) / 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 <= (-0.0034d0)) .or. (.not. (b <= 5.4d-5))) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0034) || !(b <= 5.4e-5)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0034) or not (b <= 5.4e-5):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0034) || !(b <= 5.4e-5))
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.0034) || ~((b <= 5.4e-5)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0034], N[Not[LessEqual[b, 5.4e-5]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00339999999999999981 or 5.3999999999999998e-5 < b

   1. Initial program 47.4%

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

      1. *-commutative 47.4%

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

      2. associate-/r/ 47.4%

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

      3. +-commutative 47.4%

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

   3. Simplified 47.4%

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

      1. div-inv 47.4%

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

      2. clear-num 47.5%

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

      3. *-commutative 47.5%

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

   5. Applied egg-rr 47.5%

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

   6. Taylor expanded in a around 0 47.7%

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

   if -0.00339999999999999981 < b < 5.3999999999999998e-5

   1. Initial program 98.5%

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

      1. associate-*r/ 98.5%

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

      2. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in b around 0 98.5%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0034 \lor \neg \left(b \leq 5.4 \cdot 10^{-5}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 6: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0034:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 0.00022:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.0034)
   (* (sin b) (/ r (cos b)))
   (if (<= b 0.00022) (/ (* r b) (cos (+ b a))) (/ (sin b) (/ (cos b) r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0034) {
		tmp = sin(b) * (r / cos(b));
	} else if (b <= 0.00022) {
		tmp = (r * b) / cos((b + a));
	} else {
		tmp = sin(b) / (cos(b) / 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 <= (-0.0034d0)) then
        tmp = sin(b) * (r / cos(b))
    else if (b <= 0.00022d0) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = sin(b) / (cos(b) / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0034) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else if (b <= 0.00022) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = Math.sin(b) / (Math.cos(b) / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.0034:
		tmp = math.sin(b) * (r / math.cos(b))
	elif b <= 0.00022:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = math.sin(b) / (math.cos(b) / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.0034)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	elseif (b <= 0.00022)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = Float64(sin(b) / Float64(cos(b) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.0034)
		tmp = sin(b) * (r / cos(b));
	elseif (b <= 0.00022)
		tmp = (r * b) / cos((b + a));
	else
		tmp = sin(b) / (cos(b) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.0034], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.00022], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.00339999999999999981

   1. Initial program 50.1%

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

      1. *-commutative 50.1%

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

      2. associate-/r/ 50.1%

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

      3. +-commutative 50.1%

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

   3. Simplified 50.1%

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

      1. div-inv 50.1%

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

      2. clear-num 50.2%

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

      3. *-commutative 50.2%

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

   5. Applied egg-rr 50.2%

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

   6. Taylor expanded in a around 0 50.0%

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

   if -0.00339999999999999981 < b < 2.20000000000000008e-4

   1. Initial program 98.5%

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

      1. associate-*r/ 98.5%

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

      2. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in b around 0 98.5%

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

   if 2.20000000000000008e-4 < b

   1. Initial program 45.1%

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

      1. *-commutative 45.1%

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

      2. associate-/r/ 45.1%

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

      3. +-commutative 45.1%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in a around 0 45.8%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0034:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 0.00022:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \]

Alternative 7: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.004:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.004)
   (/ (* r (sin b)) (cos b))
   (if (<= b 6.5e-5) (/ (* r b) (cos (+ b a))) (/ (sin b) (/ (cos b) r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.004) {
		tmp = (r * sin(b)) / cos(b);
	} else if (b <= 6.5e-5) {
		tmp = (r * b) / cos((b + a));
	} else {
		tmp = sin(b) / (cos(b) / 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 <= (-0.004d0)) then
        tmp = (r * sin(b)) / cos(b)
    else if (b <= 6.5d-5) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = sin(b) / (cos(b) / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.004) {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	} else if (b <= 6.5e-5) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = Math.sin(b) / (Math.cos(b) / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.004:
		tmp = (r * math.sin(b)) / math.cos(b)
	elif b <= 6.5e-5:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = math.sin(b) / (math.cos(b) / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.004)
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	elseif (b <= 6.5e-5)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = Float64(sin(b) / Float64(cos(b) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.004)
		tmp = (r * sin(b)) / cos(b);
	elseif (b <= 6.5e-5)
		tmp = (r * b) / cos((b + a));
	else
		tmp = sin(b) / (cos(b) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.004], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-5], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0040000000000000001

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in a around 0 50.0%

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

   if -0.0040000000000000001 < b < 6.49999999999999943e-5

   1. Initial program 98.5%

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

      1. associate-*r/ 98.5%

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

      2. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in b around 0 98.5%

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

   if 6.49999999999999943e-5 < b

   1. Initial program 45.1%

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

      1. *-commutative 45.1%

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

      2. associate-/r/ 45.1%

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

      3. +-commutative 45.1%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in a around 0 45.8%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.004:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \]

Alternative 8: 76.4% 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 70.1%

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

2. Final simplification 70.1%

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

Alternative 9: 76.4% 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 70.1%

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

   1. *-commutative 70.1%

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

   2. associate-/r/ 70.1%

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

   3. +-commutative 70.1%

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

3. Simplified 70.1%

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

   1. div-inv 70.1%

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

   2. clear-num 70.2%

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

   3. *-commutative 70.2%

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

5. Applied egg-rr 70.2%

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

6. Final simplification 70.2%

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

Alternative 10: 54.6% 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 70.1%

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

   1. +-commutative 70.1%

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

3. Simplified 70.1%

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

4. Taylor expanded in b around 0 50.3%

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

5. Final simplification 50.3%

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

Alternative 11: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \lor \neg \left(b \leq 3.7\right):\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.6) (not (<= b 3.7)))
   (/ (sin b) (/ 1.0 r))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.6) || !(b <= 3.7)) {
		tmp = sin(b) / (1.0 / r);
	} else {
		tmp = (r * b) / 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 <= (-4.6d0)) .or. (.not. (b <= 3.7d0))) then
        tmp = sin(b) / (1.0d0 / r)
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.6) || !(b <= 3.7)) {
		tmp = Math.sin(b) / (1.0 / r);
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.6) or not (b <= 3.7):
		tmp = math.sin(b) / (1.0 / r)
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.6) || !(b <= 3.7))
		tmp = Float64(sin(b) / Float64(1.0 / r));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.6) || ~((b <= 3.7)))
		tmp = sin(b) / (1.0 / r);
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.6], N[Not[LessEqual[b, 3.7]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5999999999999996 or 3.7000000000000002 < b

   1. Initial program 46.8%

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

      1. *-commutative 46.8%

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

      2. associate-/r/ 46.8%

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

      3. +-commutative 46.8%

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

   3. Simplified 46.8%

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

      1. clear-num 46.8%

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

      2. inv-pow 46.8%

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

   5. Applied egg-rr 46.8%

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

      1. unpow-1 46.8%

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

      2. +-commutative 46.8%

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

   7. Simplified 46.8%

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

      1. associate-/r/ 46.8%

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

      2. +-commutative 46.8%

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

      3. cos-sum 99.1%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. fma-neg 99.1%

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

      7. cos-sum 46.8%

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

      8. distribute-lft-in 46.8%

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

      9. +-commutative 46.8%

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

      10. associate-/r/ 46.8%

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

      11. +-commutative 46.8%

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

      12. clear-num 46.8%

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

   9. Applied egg-rr 44.0%

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

       1. *-commutative 44.0%

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

       2. associate-*r/ 44.0%

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

       3. *-rgt-identity 44.0%

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

       4. count-2 44.0%

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

   11. Simplified 44.0%

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

   12. Taylor expanded in a around 0 44.7%

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

   13. Taylor expanded in b around 0 12.0%

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

   if -4.5999999999999996 < b < 3.7000000000000002

   1. Initial program 97.9%

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

      1. associate-*r/ 97.9%

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

      2. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 96.6%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \lor \neg \left(b \leq 3.7\right):\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 12: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \lor \neg \left(b \leq 4800\right):\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.0) (not (<= b 4800.0)))
   (/ (sin b) (/ 1.0 r))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.0) || !(b <= 4800.0)) {
		tmp = sin(b) / (1.0 / r);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.0d0)) .or. (.not. (b <= 4800.0d0))) then
        tmp = sin(b) / (1.0d0 / r)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.0) || !(b <= 4800.0)) {
		tmp = Math.sin(b) / (1.0 / r);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.0) or not (b <= 4800.0):
		tmp = math.sin(b) / (1.0 / r)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.0) || !(b <= 4800.0))
		tmp = Float64(sin(b) / Float64(1.0 / r));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.0) || ~((b <= 4800.0)))
		tmp = sin(b) / (1.0 / r);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.0], N[Not[LessEqual[b, 4800.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1 or 4800 < b

   1. Initial program 46.8%

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

      1. *-commutative 46.8%

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

      2. associate-/r/ 46.8%

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

      3. +-commutative 46.8%

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

   3. Simplified 46.8%

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

      1. clear-num 46.8%

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

      2. inv-pow 46.8%

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

   5. Applied egg-rr 46.8%

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

      1. unpow-1 46.8%

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

      2. +-commutative 46.8%

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

   7. Simplified 46.8%

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

      1. associate-/r/ 46.8%

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

      2. +-commutative 46.8%

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

      3. cos-sum 99.1%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. fma-neg 99.1%

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

      7. cos-sum 46.8%

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

      8. distribute-lft-in 46.8%

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

      9. +-commutative 46.8%

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

      10. associate-/r/ 46.8%

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

      11. +-commutative 46.8%

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

      12. clear-num 46.8%

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

   9. Applied egg-rr 44.0%

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

       1. *-commutative 44.0%

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

       2. associate-*r/ 44.0%

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

       3. *-rgt-identity 44.0%

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

       4. count-2 44.0%

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

   11. Simplified 44.0%

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

   12. Taylor expanded in a around 0 44.7%

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

   13. Taylor expanded in b around 0 12.0%

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

   if -1 < b < 4800

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 96.2%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \lor \neg \left(b \leq 4800\right):\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 13: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -11 \lor \neg \left(b \leq 4800\right):\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -11.0) (not (<= b 4800.0)))
   (/ (sin b) (/ 1.0 r))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -11.0) || !(b <= 4800.0)) {
		tmp = sin(b) / (1.0 / r);
	} 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 <= (-11.0d0)) .or. (.not. (b <= 4800.0d0))) then
        tmp = sin(b) / (1.0d0 / r)
    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 <= -11.0) || !(b <= 4800.0)) {
		tmp = Math.sin(b) / (1.0 / r);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -11.0) or not (b <= 4800.0):
		tmp = math.sin(b) / (1.0 / r)
	else:
		tmp = (r * b) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -11.0) || !(b <= 4800.0))
		tmp = Float64(sin(b) / Float64(1.0 / r));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -11.0) || ~((b <= 4800.0)))
		tmp = sin(b) / (1.0 / r);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -11.0], N[Not[LessEqual[b, 4800.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -11 or 4800 < b

   1. Initial program 46.8%

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

      1. *-commutative 46.8%

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

      2. associate-/r/ 46.8%

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

      3. +-commutative 46.8%

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

   3. Simplified 46.8%

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

      1. clear-num 46.8%

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

      2. inv-pow 46.8%

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

   5. Applied egg-rr 46.8%

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

      1. unpow-1 46.8%

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

      2. +-commutative 46.8%

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

   7. Simplified 46.8%

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

      1. associate-/r/ 46.8%

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

      2. +-commutative 46.8%

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

      3. cos-sum 99.1%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. fma-neg 99.1%

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

      7. cos-sum 46.8%

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

      8. distribute-lft-in 46.8%

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

      9. +-commutative 46.8%

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

      10. associate-/r/ 46.8%

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

      11. +-commutative 46.8%

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

      12. clear-num 46.8%

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

   9. Applied egg-rr 44.0%

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

       1. *-commutative 44.0%

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

       2. associate-*r/ 44.0%

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

       3. *-rgt-identity 44.0%

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

       4. count-2 44.0%

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

   11. Simplified 44.0%

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

   12. Taylor expanded in a around 0 44.7%

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

   13. Taylor expanded in b around 0 12.0%

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

   if -11 < b < 4800

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 96.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -11 \lor \neg \left(b \leq 4800\right):\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 14: 50.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. +-commutative 70.1%

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

3. Simplified 70.1%

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

4. Taylor expanded in b around 0 46.0%

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

5. Final simplification 46.0%

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

Alternative 15: 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 70.1%

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

   1. +-commutative 70.1%

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

3. Simplified 70.1%

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

4. Taylor expanded in b around 0 46.0%

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

5. Taylor expanded in a around 0 29.3%

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

6. Final simplification 29.3%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023199 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))