rsin B (should all be same)

Percentage Accurate: 77.2% → 99.5%

Time: 18.2s

Alternatives: 13

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: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. associate-*r/ 72.0%

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

   2. +-commutative 72.0%

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

3. Simplified 72.0%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 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. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]
8. Add Preprocessing

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 71.9%

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

   1. +-commutative 71.9%

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

3. Simplified 71.9%

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

   1. cos-sum 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

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

Derivation

1. Initial program 71.9%

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

   1. associate-*r/ 72.0%

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

   2. +-commutative 72.0%

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

3. Simplified 72.0%

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

   1. cos-sum 99.4%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 78.0% accurate, 0.5× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. +-commutative 71.9%

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

3. Simplified 71.9%

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

   1. log1p-expm1-u 71.9%

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

6. Applied egg-rr 71.9%

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

   1. cos-sum 99.3%

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

   2. log1p-expm1-u 99.4%

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

   3. fma-neg 99.5%

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

   4. distribute-rgt-neg-in 99.5%

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

8. Applied egg-rr 99.5%

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

   1. add-sqr-sqrt 46.5%

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

   2. sqrt-unprod 84.3%

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

   3. sqr-neg 84.3%

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

   4. sqrt-unprod 37.7%

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

   5. add-sqr-sqrt 71.5%

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

   6. sin-mult 73.2%

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

   7. div-sub 73.2%

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

10. Applied egg-rr 73.3%

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

    1. +-inverses 73.3%

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

12. Simplified 73.3%

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

13. Final simplification 73.3%

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

Alternative 5: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-5} \lor \neg \left(a \leq 0.000205\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -1.9e-5) (not (<= a 0.000205)))
   (* r (/ (sin b) (cos a)))
   (* r (tan b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.9e-5) || !(a <= 0.000205)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.9d-5)) .or. (.not. (a <= 0.000205d0))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.9e-5) || !(a <= 0.000205)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -1.9e-5) or not (a <= 0.000205):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -1.9e-5) || !(a <= 0.000205))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -1.9e-5) || ~((a <= 0.000205)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -1.9e-5], N[Not[LessEqual[a, 0.000205]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9000000000000001e-5 or 2.05e-4 < a

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in b around 0 49.9%

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

   if -1.9000000000000001e-5 < a < 2.05e-4

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

   3. Simplified 98.7%

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

      1. add-cbrt-cube 36.9%

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

      2. pow3 36.9%

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

      3. associate-*r/ 37.0%

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

      4. *-commutative 37.0%

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

      5. associate-/l* 37.0%

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

   6. Applied egg-rr 37.0%

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

   7. Taylor expanded in a around 0 27.6%

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

      1. cube-prod 37.0%

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

   9. Simplified 37.0%

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

       1. cbrt-div 37.0%

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

       2. rem-cbrt-cube 98.7%

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

       3. rem-cbrt-cube 98.9%

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

       4. clear-num 98.2%

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

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{\cos b}{r \cdot \sin b}}} \]
   12. Step-by-step derivation

       1. clear-num 98.9%

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

       2. associate-*r/ 98.7%

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

       3. *-commutative 98.7%

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

       4. quot-tan 98.9%

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

   13. Applied egg-rr 98.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-5} \lor \neg \left(a \leq 0.000205\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% 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 71.9%

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

3. Final simplification 71.9%

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

Alternative 7: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. associate-*r/ 72.0%

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

   2. +-commutative 72.0%

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

3. Simplified 72.0%

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

   1. *-commutative 72.0%

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

   2. associate-/l* 72.0%

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

6. Applied egg-rr 72.0%

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

7. Final simplification 72.0%

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

Alternative 8: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00023 \lor \neg \left(b \leq 2.9 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00023) (not (<= b 2.9e-6)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00023) || !(b <= 2.9e-6)) {
		tmp = r * tan(b);
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.00023d0)) .or. (.not. (b <= 2.9d-6))) then
        tmp = r * tan(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00023) || !(b <= 2.9e-6)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00023) or not (b <= 2.9e-6):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.00023) || !(b <= 2.9e-6))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.00023) || ~((b <= 2.9e-6)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00023], N[Not[LessEqual[b, 2.9e-6]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.3000000000000001e-4 or 2.9000000000000002e-6 < b

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

   3. Simplified 51.6%

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

      1. add-cbrt-cube 16.0%

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

      2. pow3 16.0%

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

      3. associate-*r/ 16.1%

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

      4. *-commutative 16.1%

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

      5. associate-/l* 16.1%

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

   6. Applied egg-rr 16.1%

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

   7. Taylor expanded in a around 0 15.6%

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

      1. cube-prod 15.7%

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

   9. Simplified 15.7%

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

       1. cbrt-div 15.6%

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

       2. rem-cbrt-cube 50.9%

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

       3. rem-cbrt-cube 51.1%

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

       4. clear-num 51.0%

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

   11. Applied egg-rr 51.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\cos b}{r \cdot \sin b}}} \]
   12. Step-by-step derivation

       1. clear-num 51.1%

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

       2. associate-*r/ 51.0%

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

       3. *-commutative 51.0%

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

       4. quot-tan 51.1%

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

   13. Applied egg-rr 51.1%

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

   if -2.3000000000000001e-4 < b < 2.9000000000000002e-6

   1. Initial program 99.3%

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

      1. associate-*r/ 99.3%

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

      2. +-commutative 99.3%

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

   3. Simplified 99.3%

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

      1. cos-sum 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 99.3%

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

      1. associate-/l* 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00023 \lor \neg \left(b \leq 2.9 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-6} \lor \neg \left(b \leq 1.95 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -2.7e-6) (not (<= b 1.95e-6)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.7e-6) || !(b <= 1.95e-6)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-2.7d-6)) .or. (.not. (b <= 1.95d-6))) then
        tmp = r * tan(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.7e-6) || !(b <= 1.95e-6)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -2.7e-6) or not (b <= 1.95e-6):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -2.7e-6) || !(b <= 1.95e-6))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -2.7e-6) || ~((b <= 1.95e-6)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -2.7e-6], N[Not[LessEqual[b, 1.95e-6]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.69999999999999998e-6 or 1.95e-6 < b

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

   3. Simplified 51.6%

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

      1. add-cbrt-cube 16.0%

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

      2. pow3 16.0%

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

      3. associate-*r/ 16.1%

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

      4. *-commutative 16.1%

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

      5. associate-/l* 16.1%

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

   6. Applied egg-rr 16.1%

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

   7. Taylor expanded in a around 0 15.6%

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

      1. cube-prod 15.7%

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

   9. Simplified 15.7%

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

       1. cbrt-div 15.6%

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

       2. rem-cbrt-cube 50.9%

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

       3. rem-cbrt-cube 51.1%

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

       4. clear-num 51.0%

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

   11. Applied egg-rr 51.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\cos b}{r \cdot \sin b}}} \]
   12. Step-by-step derivation

       1. clear-num 51.1%

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

       2. associate-*r/ 51.0%

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

       3. *-commutative 51.0%

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

       4. quot-tan 51.1%

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

   13. Applied egg-rr 51.1%

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

   if -2.69999999999999998e-6 < b < 1.95e-6

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in b around 0 99.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-6} \lor \neg \left(b \leq 1.95 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -7.2e-5)
   (/ r (/ 1.0 (tan b)))
   (if (<= b 3.9e-6) (* r (/ b (cos a))) (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -7.2e-5) {
		tmp = r / (1.0 / tan(b));
	} else if (b <= 3.9e-6) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-7.2d-5)) then
        tmp = r / (1.0d0 / tan(b))
    else if (b <= 3.9d-6) then
        tmp = r * (b / cos(a))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -7.2e-5) {
		tmp = r / (1.0 / Math.tan(b));
	} else if (b <= 3.9e-6) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -7.2e-5:
		tmp = r / (1.0 / math.tan(b))
	elif b <= 3.9e-6:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -7.2e-5)
		tmp = Float64(r / Float64(1.0 / tan(b)));
	elseif (b <= 3.9e-6)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -7.2e-5)
		tmp = r / (1.0 / tan(b));
	elseif (b <= 3.9e-6)
		tmp = r * (b / cos(a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -7.2e-5], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-6], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.20000000000000018e-5

   1. Initial program 55.7%

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

      1. +-commutative 55.7%

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

   3. Simplified 55.7%

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

      1. add-cbrt-cube 18.0%

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

      2. pow3 18.0%

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

      3. associate-*r/ 18.0%

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

      4. *-commutative 18.0%

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

      5. associate-/l* 18.1%

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

   6. Applied egg-rr 18.1%

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

   7. Taylor expanded in a around 0 17.7%

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

      1. cube-prod 17.7%

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

   9. Simplified 17.7%

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

       1. add-cube-cbrt 17.6%

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

       2. pow3 17.5%

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

       3. cbrt-div 17.5%

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

       4. rem-cbrt-cube 54.4%

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

       5. rem-cbrt-cube 54.4%

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

       6. associate-/l* 54.3%

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

   11. Applied egg-rr 54.3%

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

       1. rem-cube-cbrt 55.0%

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

       2. clear-num 55.0%

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

       3. un-div-inv 55.2%

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

       4. clear-num 55.1%

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

       5. quot-tan 55.2%

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

   13. Applied egg-rr 55.2%

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

   if -7.20000000000000018e-5 < b < 3.8999999999999999e-6

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in b around 0 99.3%

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

   if 3.8999999999999999e-6 < b

   1. Initial program 46.1%

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

      1. +-commutative 46.1%

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

   3. Simplified 46.1%

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

      1. add-cbrt-cube 13.4%

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

      2. pow3 13.4%

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

      3. associate-*r/ 13.4%

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

      4. *-commutative 13.4%

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

      5. associate-/l* 13.4%

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

   6. Applied egg-rr 13.4%

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

   7. Taylor expanded in a around 0 12.9%

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

      1. cube-prod 12.9%

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

   9. Simplified 12.9%

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

       1. cbrt-div 12.9%

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

       2. rem-cbrt-cube 45.4%

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

       3. rem-cbrt-cube 45.7%

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

       4. clear-num 45.6%

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

   11. Applied egg-rr 45.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{\cos b}{r \cdot \sin b}}} \]
   12. Step-by-step derivation

       1. clear-num 45.7%

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

       2. associate-*r/ 45.6%

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

       3. *-commutative 45.6%

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

       4. quot-tan 45.7%

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

   13. Applied egg-rr 45.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. +-commutative 71.9%

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

3. Simplified 71.9%

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

5. Taylor expanded in b around 0 47.0%

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

   1. mul-1-neg 47.0%

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

   2. unsub-neg 47.0%

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

7. Simplified 47.0%

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

8. Taylor expanded in a around 0 34.4%

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

9. Final simplification 34.4%

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

Alternative 12: 60.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. +-commutative 71.9%

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

3. Simplified 71.9%

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

   1. add-cbrt-cube 32.1%

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

   2. pow3 32.1%

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

   3. associate-*r/ 32.2%

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

   4. *-commutative 32.2%

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

   5. associate-/l* 32.2%

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

6. Applied egg-rr 32.2%

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

7. Taylor expanded in a around 0 21.4%

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

   1. cube-prod 26.3%

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

9. Simplified 26.3%

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

    1. cbrt-div 26.3%

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

    2. rem-cbrt-cube 56.2%

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

    3. rem-cbrt-cube 56.3%

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

    4. clear-num 56.0%

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

11. Applied egg-rr 56.0%

    \[\leadsto \color{blue}{\frac{1}{\frac{\cos b}{r \cdot \sin b}}} \]
12. Step-by-step derivation

    1. clear-num 56.3%

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

    2. associate-*r/ 56.2%

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

    3. *-commutative 56.2%

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

    4. quot-tan 56.3%

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

13. Applied egg-rr 56.3%

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

14. Final simplification 56.3%

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

Alternative 13: 35.1% 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 71.9%

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

   1. +-commutative 71.9%

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

3. Simplified 71.9%

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

5. Taylor expanded in b around 0 45.2%

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

6. Taylor expanded in a around 0 29.8%

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

7. Final simplification 29.8%

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

Reproduce

herbie shell --seed 2024075 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))