rsin A (should all be same)

Percentage Accurate: 76.7% → 99.5%

Time: 16.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

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

Initial Program: 76.7% accurate, 1.0× speedup

Math

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

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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) / ((cos(a) * cos(b)) - (sin(b) * sin(a)))) * r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. +-commutative 74.7%

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

3. Simplified 74.7%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in r around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. mul-1-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. sub-neg 99.4%

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

8. Simplified 99.4%

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

   1. log1p-expm1-u 99.5%

      \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

10. Applied egg-rr 99.5%

    \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

11. Taylor expanded in b around inf 99.4%

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

    1. associate-/l* 99.4%

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

    2. associate-/r/ 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 77.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. +-commutative 74.7%

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

3. Simplified 74.7%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in r around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. mul-1-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. sub-neg 99.4%

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

8. Simplified 99.4%

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

   1. sin-mult 76.0%

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

   2. div-sub 76.0%

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

   3. cos-diff 75.8%

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

   4. *-commutative 75.8%

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

   5. add-sqr-sqrt 35.4%

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

   6. sqrt-unprod 74.9%

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

   7. sqr-neg 74.9%

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

   8. sqrt-unprod 39.5%

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

   9. add-sqr-sqrt 74.3%

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

   10. cancel-sign-sub-inv 74.3%

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

   11. *-commutative 74.3%

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

   12. cos-sum 76.0%

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

10. Applied egg-rr 76.0%

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

    1. +-inverses 76.0%

       \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{0}} \]

12. Simplified 76.0%

    \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{0}} \]

13. Final simplification 76.0%

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

Alternative 3: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -350000000 \lor \neg \left(b \leq 8.8 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -350000000.0) (not (<= b 8.8e-6)))
   (* r (/ (sin b) (cos b)))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -350000000.0) || !(b <= 8.8e-6)) {
		tmp = r * (sin(b) / cos(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 <= (-350000000.0d0)) .or. (.not. (b <= 8.8d-6))) then
        tmp = r * (sin(b) / cos(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 <= -350000000.0) || !(b <= 8.8e-6)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -350000000.0) or not (b <= 8.8e-6):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -350000000.0) || !(b <= 8.8e-6))
		tmp = Float64(r * Float64(sin(b) / cos(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 <= -350000000.0) || ~((b <= 8.8e-6)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -350000000.0], N[Not[LessEqual[b, 8.8e-6]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.5e8 or 8.8000000000000004e-6 < b

   1. Initial program 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-/l* 56.9%

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

      3. +-commutative 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in a around 0 56.7%

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

      1. associate-/l* 56.7%

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

      2. associate-/r/ 56.7%

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

   6. Simplified 56.7%

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

   if -3.5e8 < b < 8.8000000000000004e-6

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

   3. Simplified 97.6%

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

      1. cos-sum 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in r around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. sub-neg 99.8%

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

   8. Simplified 99.8%

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

      1. log1p-expm1-u 99.8%

         \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   11. Taylor expanded in b around 0 97.6%

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

       1. associate-/l* 97.5%

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

       2. associate-/r/ 97.7%

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

   13. Simplified 97.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -350000000 \lor \neg \left(b \leq 8.8 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 4: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -350000000:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 0.000136:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -350000000.0)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 0.000136) (* r (/ b (cos a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -350000000.0) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 0.000136) {
		tmp = r * (b / cos(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 <= (-350000000.0d0)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 0.000136d0) then
        tmp = r * (b / cos(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 <= -350000000.0) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 0.000136) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -350000000.0:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 0.000136:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -350000000.0)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 0.000136)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -350000000.0)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 0.000136)
		tmp = r * (b / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -350000000.0], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.000136], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e8

   1. Initial program 58.2%

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

      1. +-commutative 58.2%

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

   3. Simplified 58.2%

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

      1. cos-sum 99.3%

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

      2. cancel-sign-sub-inv 99.3%

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

      3. fma-def 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in r around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. mul-1-neg 99.3%

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

      3. *-commutative 99.3%

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

      4. sub-neg 99.3%

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

   8. Simplified 99.3%

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

      1. log1p-expm1-u 99.3%

         \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   10. Applied egg-rr 99.3%

       \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   11. Taylor expanded in a around 0 57.7%

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

       1. *-commutative 57.7%

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

       2. associate-/l* 57.6%

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

   13. Simplified 57.6%

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

   if -3.5e8 < b < 1.36e-4

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

   3. Simplified 97.6%

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

      1. cos-sum 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in r around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. sub-neg 99.8%

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

   8. Simplified 99.8%

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

      1. log1p-expm1-u 99.8%

         \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   11. Taylor expanded in b around 0 97.6%

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

       1. associate-/l* 97.5%

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

       2. associate-/r/ 97.7%

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

   13. Simplified 97.7%

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

   if 1.36e-4 < b

   1. Initial program 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-/l* 55.6%

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

      3. +-commutative 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in a around 0 55.7%

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

      1. associate-/l* 55.7%

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

      2. associate-/r/ 55.8%

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

   6. Simplified 55.8%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -350000000:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 0.000136:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 5: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -350000000:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -350000000.0)
   (/ (* (sin b) r) (cos b))
   (if (<= b 2e-5) (* r (/ b (cos a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -350000000.0) {
		tmp = (sin(b) * r) / cos(b);
	} else if (b <= 2e-5) {
		tmp = r * (b / cos(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 <= (-350000000.0d0)) then
        tmp = (sin(b) * r) / cos(b)
    else if (b <= 2d-5) then
        tmp = r * (b / cos(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 <= -350000000.0) {
		tmp = (Math.sin(b) * r) / Math.cos(b);
	} else if (b <= 2e-5) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -350000000.0:
		tmp = (math.sin(b) * r) / math.cos(b)
	elif b <= 2e-5:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -350000000.0)
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	elseif (b <= 2e-5)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -350000000.0)
		tmp = (sin(b) * r) / cos(b);
	elseif (b <= 2e-5)
		tmp = r * (b / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -350000000.0], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-5], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e8

   1. Initial program 58.2%

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

      1. *-commutative 58.2%

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

      2. associate-/l* 58.2%

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

      3. +-commutative 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in a around 0 57.7%

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

   if -3.5e8 < b < 2.00000000000000016e-5

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

   3. Simplified 97.6%

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

      1. cos-sum 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in r around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. sub-neg 99.8%

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

   8. Simplified 99.8%

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

      1. log1p-expm1-u 99.8%

         \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   11. Taylor expanded in b around 0 97.6%

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

       1. associate-/l* 97.5%

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

       2. associate-/r/ 97.7%

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

   13. Simplified 97.7%

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

   if 2.00000000000000016e-5 < b

   1. Initial program 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-/l* 55.6%

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

      3. +-commutative 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in a around 0 55.7%

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

      1. associate-/l* 55.7%

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

      2. associate-/r/ 55.8%

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

   6. Simplified 55.8%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -350000000:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 6: 76.7% 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 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-/l* 74.6%

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

   3. +-commutative 74.6%

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

3. Simplified 74.6%

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

   1. clear-num 73.6%

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

   2. associate-/r/ 74.6%

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

   3. clear-num 74.7%

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

5. Applied egg-rr 74.7%

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

6. Final simplification 74.7%

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

Alternative 7: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-/l* 74.6%

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

   3. +-commutative 74.6%

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

3. Simplified 74.6%

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

   1. associate-/r/ 74.7%

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

5. Applied egg-rr 74.7%

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

6. Final simplification 74.7%

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

Alternative 8: 76.5% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 74.7%

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

   1. +-commutative 74.7%

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

3. Simplified 74.7%

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

   1. log1p-expm1-u 74.6%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(b + a\right)\right)\right)}} \]

5. Applied egg-rr 74.6%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(b + a\right)\right)\right)}} \]
6. Step-by-step derivation

   1. log1p-expm1-u 74.7%

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

   2. cos-sum 99.4%

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

   3. *-commutative 99.4%

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

   4. cancel-sign-sub-inv 99.4%

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

   5. add-sqr-sqrt 52.4%

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

   6. sqrt-unprod 87.7%

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

   7. sqr-neg 87.7%

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

   8. sqrt-unprod 35.3%

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

   9. add-sqr-sqrt 75.0%

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

   10. *-commutative 75.0%

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

   11. cos-diff 74.9%

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

7. Applied egg-rr 74.9%

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

8. Final simplification 74.9%

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

Alternative 9: 54.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -580000000 \lor \neg \left(b \leq 1.56\right):\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -580000000.0) (not (<= b 1.56)))
   (* r (- (sin b)))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -580000000.0) || !(b <= 1.56)) {
		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 <= (-580000000.0d0)) .or. (.not. (b <= 1.56d0))) 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 <= -580000000.0) || !(b <= 1.56)) {
		tmp = r * -Math.sin(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -580000000.0) or not (b <= 1.56):
		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 <= -580000000.0) || !(b <= 1.56))
		tmp = Float64(r * Float64(-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 <= -580000000.0) || ~((b <= 1.56)))
		tmp = r * -sin(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -580000000.0], N[Not[LessEqual[b, 1.56]], $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 < -5.8e8 or 1.5600000000000001 < b

   1. Initial program 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-/l* 56.9%

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

      3. +-commutative 56.9%

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

   3. Simplified 56.9%

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

      1. frac-2neg 56.9%

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

      2. div-inv 56.8%

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

      3. distribute-neg-frac 56.8%

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

   5. Applied egg-rr 56.8%

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

      1. expm1-log1p-u 42.0%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\cos \left(b + a\right)}{r}\right)\right)}} \]

      2. expm1-udef 11.4%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{-\cos \left(b + a\right)}{r}\right)} - 1}} \]

      3. add-sqr-sqrt 7.4%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\cos \left(b + a\right)} \cdot \sqrt{-\cos \left(b + a\right)}}}{r}\right)} - 1} \]

      4. sqrt-unprod 8.8%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\cos \left(b + a\right)\right) \cdot \left(-\cos \left(b + a\right)\right)}}}{r}\right)} - 1} \]

      5. sqr-neg 8.8%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\cos \left(b + a\right) \cdot \cos \left(b + a\right)}}}{r}\right)} - 1} \]

      6. sqrt-unprod 1.4%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\cos \left(b + a\right)} \cdot \sqrt{\cos \left(b + a\right)}}}{r}\right)} - 1} \]

      7. add-sqr-sqrt 3.6%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\cos \left(b + a\right)}}{r}\right)} - 1} \]

   7. Applied egg-rr 3.6%

      \[\leadsto \left(-\sin b\right) \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\cos \left(b + a\right)}{r}\right)} - 1}} \]

   8. Taylor expanded in a around 0 6.6%

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

   9. Taylor expanded in b around 0 12.9%

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

   if -5.8e8 < b < 1.5600000000000001

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

   3. Simplified 97.6%

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

      1. cos-sum 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in r around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. sub-neg 99.8%

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

   8. Simplified 99.8%

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

      1. log1p-expm1-u 99.8%

         \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \frac{\sin b \cdot r}{\cos a \cdot \cos b - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

   11. Taylor expanded in b around 0 97.6%

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

       1. associate-/l* 97.5%

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

       2. associate-/r/ 97.7%

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

   13. Simplified 97.7%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -580000000 \lor \neg \left(b \leq 1.56\right):\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 10: 37.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-11} \lor \neg \left(b \leq 2200\right):\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -2.9e-11) (not (<= b 2200.0))) (* r (- (sin b))) (* b r)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.9e-11) || !(b <= 2200.0)) {
		tmp = r * -sin(b);
	} else {
		tmp = 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 <= (-2.9d-11)) .or. (.not. (b <= 2200.0d0))) then
        tmp = r * -sin(b)
    else
        tmp = b * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.9e-11) || !(b <= 2200.0)) {
		tmp = r * -Math.sin(b);
	} else {
		tmp = b * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -2.9e-11) or not (b <= 2200.0):
		tmp = r * -math.sin(b)
	else:
		tmp = b * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -2.9e-11) || !(b <= 2200.0))
		tmp = Float64(r * Float64(-sin(b)));
	else
		tmp = Float64(b * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -2.9e-11) || ~((b <= 2200.0)))
		tmp = r * -sin(b);
	else
		tmp = b * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -2.9e-11], N[Not[LessEqual[b, 2200.0]], $MachinePrecision]], N[(r * (-N[Sin[b], $MachinePrecision])), $MachinePrecision], N[(b * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.9e-11 or 2200 < b

   1. Initial program 56.7%

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

      1. *-commutative 56.7%

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

      2. associate-/l* 56.7%

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

      3. +-commutative 56.7%

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

   3. Simplified 56.7%

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

      1. frac-2neg 56.7%

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

      2. div-inv 56.6%

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

      3. distribute-neg-frac 56.6%

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

   5. Applied egg-rr 56.6%

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

      1. expm1-log1p-u 42.0%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\cos \left(b + a\right)}{r}\right)\right)}} \]

      2. expm1-udef 11.8%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{-\cos \left(b + a\right)}{r}\right)} - 1}} \]

      3. add-sqr-sqrt 7.7%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\cos \left(b + a\right)} \cdot \sqrt{-\cos \left(b + a\right)}}}{r}\right)} - 1} \]

      4. sqrt-unprod 9.2%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\cos \left(b + a\right)\right) \cdot \left(-\cos \left(b + a\right)\right)}}}{r}\right)} - 1} \]

      5. sqr-neg 9.2%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\cos \left(b + a\right) \cdot \cos \left(b + a\right)}}}{r}\right)} - 1} \]

      6. sqrt-unprod 1.5%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\cos \left(b + a\right)} \cdot \sqrt{\cos \left(b + a\right)}}}{r}\right)} - 1} \]

      7. add-sqr-sqrt 3.5%

         \[\leadsto \left(-\sin b\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\cos \left(b + a\right)}}{r}\right)} - 1} \]

   7. Applied egg-rr 3.5%

      \[\leadsto \left(-\sin b\right) \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\cos \left(b + a\right)}{r}\right)} - 1}} \]

   8. Taylor expanded in a around 0 6.8%

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

   9. Taylor expanded in b around 0 13.0%

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

   if -2.9e-11 < b < 2200

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-/l* 98.4%

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

      3. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in b around 0 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-/l* 98.4%

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

      3. associate-/r/ 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in a around 0 69.1%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-11} \lor \neg \left(b \leq 2200\right):\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot r\\ \end{array} \]

Alternative 11: 33.7% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-/l* 74.6%

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

   3. +-commutative 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in b around 0 44.8%

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

   1. *-commutative 44.8%

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

   2. associate-/l* 44.8%

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

   3. associate-/r/ 44.8%

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

6. Simplified 44.8%

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

7. Taylor expanded in a around 0 32.2%

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

8. Final simplification 32.2%

   \[\leadsto b \cdot r \]

Reproduce

herbie shell --seed 2023264 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))