rsin A (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 18.4s

Alternatives: 14

Speedup: 0.7×

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: 76.3% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-/l* 77.0%

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

   2. remove-double-neg 77.0%

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

   3. sin-neg 77.0%

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

   4. neg-mul-1 77.0%

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

   5. associate-/r* 77.0%

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

   6. associate-/l* 77.1%

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

   7. *-commutative 77.1%

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

   8. associate-*l/ 77.2%

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

   9. associate-/r/ 77.2%

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

   10. *-commutative 77.2%

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

   11. associate-*r/ 77.2%

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

   12. neg-mul-1 77.2%

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

   13. sin-neg 77.2%

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

   14. remove-double-neg 77.2%

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

   15. +-commutative 77.2%

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

3. Simplified 77.2%

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

   1. cos-sum 99.5%

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

   2. fma-neg 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-/l* 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. associate-/r/ 77.1%

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

6. Applied egg-rr 77.1%

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

   1. cos-sum 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 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 77.1%

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

   1. associate-/l* 77.0%

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

   2. remove-double-neg 77.0%

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

   3. sin-neg 77.0%

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

   4. neg-mul-1 77.0%

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

   5. associate-/r* 77.0%

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

   6. associate-/l* 77.1%

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

   7. *-commutative 77.1%

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

   8. associate-*l/ 77.2%

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

   9. associate-/r/ 77.2%

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

   10. *-commutative 77.2%

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

   11. associate-*r/ 77.2%

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

   12. neg-mul-1 77.2%

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

   13. sin-neg 77.2%

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

   14. remove-double-neg 77.2%

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

   15. +-commutative 77.2%

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

3. Simplified 77.2%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 77.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-/l* 77.0%

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

   2. remove-double-neg 77.0%

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

   3. sin-neg 77.0%

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

   4. neg-mul-1 77.0%

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

   5. associate-/r* 77.0%

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

   6. associate-/l* 77.1%

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

   7. *-commutative 77.1%

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

   8. associate-*l/ 77.2%

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

   9. associate-/r/ 77.2%

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

   10. *-commutative 77.2%

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

   11. associate-*r/ 77.2%

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

   12. neg-mul-1 77.2%

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

   13. sin-neg 77.2%

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

   14. remove-double-neg 77.2%

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

   15. +-commutative 77.2%

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

3. Simplified 77.2%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

   1. sin-mult 78.3%

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

   2. cos-sum 78.7%

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

   3. div-sub 78.7%

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

   4. sub-neg 78.7%

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

   5. add-sqr-sqrt 49.4%

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

   6. sqrt-unprod 78.2%

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

   7. sqr-neg 78.2%

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

   8. sqrt-unprod 39.0%

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

   9. add-sqr-sqrt 77.7%

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

   10. cos-diff 78.2%

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

8. Applied egg-rr 78.2%

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

   1. +-inverses 78.2%

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

10. Simplified 78.2%

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

11. Final simplification 78.2%

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

Alternative 5: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-6} \lor \neg \left(b \leq 54\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -6.5e-6) (not (<= b 54.0)))
   (* r (/ (sin b) (cos b)))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -6.5e-6) || !(b <= 54.0)) {
		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 <= (-6.5d-6)) .or. (.not. (b <= 54.0d0))) 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 <= -6.5e-6) || !(b <= 54.0)) {
		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 <= -6.5e-6) or not (b <= 54.0):
		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 <= -6.5e-6) || !(b <= 54.0))
		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 <= -6.5e-6) || ~((b <= 54.0)))
		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, -6.5e-6], N[Not[LessEqual[b, 54.0]], $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 < -6.4999999999999996e-6 or 54 < b

   1. Initial program 54.0%

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

      1. associate-/l* 54.0%

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

      2. remove-double-neg 54.0%

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

      3. sin-neg 54.0%

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

      4. neg-mul-1 54.0%

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

      5. associate-/r* 54.0%

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

      6. associate-/l* 54.0%

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

      7. *-commutative 54.0%

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

      8. associate-*l/ 54.1%

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

      9. associate-/r/ 54.1%

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

      10. *-commutative 54.1%

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

      11. associate-*r/ 54.1%

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

      12. neg-mul-1 54.1%

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

      13. sin-neg 54.1%

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

      14. remove-double-neg 54.1%

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

      15. +-commutative 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in a around 0 54.4%

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

   if -6.4999999999999996e-6 < b < 54

   1. Initial program 98.9%

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

      1. associate-/l* 98.7%

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

      2. remove-double-neg 98.7%

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

      3. sin-neg 98.7%

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

      4. neg-mul-1 98.7%

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

      5. associate-/r* 98.7%

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

      6. associate-/l* 98.9%

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

      7. *-commutative 98.9%

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

      8. associate-*l/ 98.9%

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

      9. associate-/r/ 98.9%

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

      10. *-commutative 98.9%

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

      11. associate-*r/ 98.9%

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

      12. neg-mul-1 98.9%

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

      13. sin-neg 98.9%

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

      14. remove-double-neg 98.9%

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

      15. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in b around 0 99.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-6} \lor \neg \left(b \leq 54\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 54:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -6.8e-6)
   (* r (/ (sin b) (cos b)))
   (if (<= b 54.0) (* r (/ b (cos a))) (/ (sin b) (/ (cos b) r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -6.8e-6) {
		tmp = r * (sin(b) / cos(b));
	} else if (b <= 54.0) {
		tmp = r * (b / cos(a));
	} else {
		tmp = sin(b) / (cos(b) / r);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.8d-6)) then
        tmp = r * (sin(b) / cos(b))
    else if (b <= 54.0d0) then
        tmp = r * (b / cos(a))
    else
        tmp = sin(b) / (cos(b) / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -6.8e-6) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else if (b <= 54.0) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = Math.sin(b) / (Math.cos(b) / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -6.8e-6:
		tmp = r * (math.sin(b) / math.cos(b))
	elif b <= 54.0:
		tmp = r * (b / math.cos(a))
	else:
		tmp = math.sin(b) / (math.cos(b) / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -6.8e-6)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (b <= 54.0)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(sin(b) / Float64(cos(b) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -6.8e-6)
		tmp = r * (sin(b) / cos(b));
	elseif (b <= 54.0)
		tmp = r * (b / cos(a));
	else
		tmp = sin(b) / (cos(b) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -6.8e-6], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 54.0], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.80000000000000012e-6

   1. Initial program 59.6%

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

      1. associate-/l* 59.6%

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

      2. remove-double-neg 59.6%

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

      3. sin-neg 59.6%

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

      4. neg-mul-1 59.6%

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

      5. associate-/r* 59.6%

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

      6. associate-/l* 59.6%

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

      7. *-commutative 59.6%

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

      8. associate-*l/ 59.8%

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

      9. associate-/r/ 59.8%

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

      10. *-commutative 59.8%

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

      11. associate-*r/ 59.8%

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

      12. neg-mul-1 59.8%

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

      13. sin-neg 59.8%

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

      14. remove-double-neg 59.8%

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

      15. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in a around 0 59.7%

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

   if -6.80000000000000012e-6 < b < 54

   1. Initial program 98.9%

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

      1. associate-/l* 98.7%

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

      2. remove-double-neg 98.7%

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

      3. sin-neg 98.7%

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

      4. neg-mul-1 98.7%

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

      5. associate-/r* 98.7%

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

      6. associate-/l* 98.9%

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

      7. *-commutative 98.9%

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

      8. associate-*l/ 98.9%

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

      9. associate-/r/ 98.9%

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

      10. *-commutative 98.9%

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

      11. associate-*r/ 98.9%

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

      12. neg-mul-1 98.9%

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

      13. sin-neg 98.9%

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

      14. remove-double-neg 98.9%

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

      15. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in b around 0 99.0%

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

   if 54 < b

   1. Initial program 48.7%

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

      1. associate-/l* 48.6%

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

      2. remove-double-neg 48.6%

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

      3. sin-neg 48.6%

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

      4. neg-mul-1 48.6%

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

      5. associate-/r* 48.6%

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

      6. associate-/l* 48.7%

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

      7. *-commutative 48.7%

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

      8. associate-*l/ 48.7%

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

      9. associate-/r/ 48.7%

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

      10. *-commutative 48.7%

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

      11. associate-*r/ 48.7%

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

      12. neg-mul-1 48.7%

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

      13. sin-neg 48.7%

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

      14. remove-double-neg 48.7%

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

      15. +-commutative 48.7%

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

   3. Simplified 48.7%

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

      1. associate-*l/ 48.7%

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

      2. associate-/l* 48.7%

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

   6. Applied egg-rr 48.7%

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

   7. Taylor expanded in a around 0 49.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 54:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 54:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -5.4e-6)
   (* r (/ (sin b) (cos b)))
   (if (<= b 54.0) (* r (/ b (cos a))) (/ (* (sin b) r) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -5.4e-6) {
		tmp = r * (sin(b) / cos(b));
	} else if (b <= 54.0) {
		tmp = r * (b / cos(a));
	} else {
		tmp = (sin(b) * r) / cos(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-5.4d-6)) then
        tmp = r * (sin(b) / cos(b))
    else if (b <= 54.0d0) then
        tmp = r * (b / cos(a))
    else
        tmp = (sin(b) * r) / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -5.4e-6) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else if (b <= 54.0) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = (Math.sin(b) * r) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -5.4e-6:
		tmp = r * (math.sin(b) / math.cos(b))
	elif b <= 54.0:
		tmp = r * (b / math.cos(a))
	else:
		tmp = (math.sin(b) * r) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -5.4e-6)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (b <= 54.0)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -5.4e-6)
		tmp = r * (sin(b) / cos(b));
	elseif (b <= 54.0)
		tmp = r * (b / cos(a));
	else
		tmp = (sin(b) * r) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -5.4e-6], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 54.0], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.39999999999999997e-6

   1. Initial program 59.6%

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

      1. associate-/l* 59.6%

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

      2. remove-double-neg 59.6%

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

      3. sin-neg 59.6%

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

      4. neg-mul-1 59.6%

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

      5. associate-/r* 59.6%

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

      6. associate-/l* 59.6%

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

      7. *-commutative 59.6%

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

      8. associate-*l/ 59.8%

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

      9. associate-/r/ 59.8%

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

      10. *-commutative 59.8%

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

      11. associate-*r/ 59.8%

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

      12. neg-mul-1 59.8%

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

      13. sin-neg 59.8%

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

      14. remove-double-neg 59.8%

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

      15. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in a around 0 59.7%

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

   if -5.39999999999999997e-6 < b < 54

   1. Initial program 98.9%

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

      1. associate-/l* 98.7%

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

      2. remove-double-neg 98.7%

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

      3. sin-neg 98.7%

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

      4. neg-mul-1 98.7%

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

      5. associate-/r* 98.7%

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

      6. associate-/l* 98.9%

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

      7. *-commutative 98.9%

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

      8. associate-*l/ 98.9%

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

      9. associate-/r/ 98.9%

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

      10. *-commutative 98.9%

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

      11. associate-*r/ 98.9%

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

      12. neg-mul-1 98.9%

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

      13. sin-neg 98.9%

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

      14. remove-double-neg 98.9%

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

      15. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in b around 0 99.0%

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

   if 54 < b

   1. Initial program 48.7%

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

      1. associate-/l* 48.6%

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

      2. remove-double-neg 48.6%

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

      3. sin-neg 48.6%

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

      4. neg-mul-1 48.6%

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

      5. associate-/r* 48.6%

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

      6. associate-/l* 48.7%

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

      7. *-commutative 48.7%

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

      8. associate-*l/ 48.7%

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

      9. associate-/r/ 48.7%

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

      10. *-commutative 48.7%

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

      11. associate-*r/ 48.7%

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

      12. neg-mul-1 48.7%

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

      13. sin-neg 48.7%

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

      14. remove-double-neg 48.7%

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

      15. +-commutative 48.7%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in a around 0 49.5%

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

      1. *-commutative 49.5%

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

   7. Simplified 49.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 54:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% 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 77.1%

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

   1. associate-/l* 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. associate-/r/ 77.1%

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

6. Applied egg-rr 77.1%

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

7. Final simplification 77.1%

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

Alternative 9: 76.3% 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 77.1%

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

   1. associate-/l* 77.0%

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

   2. remove-double-neg 77.0%

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

   3. sin-neg 77.0%

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

   4. neg-mul-1 77.0%

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

   5. associate-/r* 77.0%

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

   6. associate-/l* 77.1%

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

   7. *-commutative 77.1%

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

   8. associate-*l/ 77.2%

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

   9. associate-/r/ 77.2%

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

   10. *-commutative 77.2%

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

   11. associate-*r/ 77.2%

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

   12. neg-mul-1 77.2%

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

   13. sin-neg 77.2%

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

   14. remove-double-neg 77.2%

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

   15. +-commutative 77.2%

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

3. Simplified 77.2%

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

5. Final simplification 77.2%

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

Alternative 10: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-/l* 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. associate-/r/ 77.1%

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

6. Applied egg-rr 77.1%

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

7. Taylor expanded in b around 0 57.3%

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

8. Final simplification 57.3%

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

Alternative 11: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-/l* 77.0%

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

   2. remove-double-neg 77.0%

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

   3. sin-neg 77.0%

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

   4. neg-mul-1 77.0%

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

   5. associate-/r* 77.0%

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

   6. associate-/l* 77.1%

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

   7. *-commutative 77.1%

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

   8. associate-*l/ 77.2%

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

   9. associate-/r/ 77.2%

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

   10. *-commutative 77.2%

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

   11. associate-*r/ 77.2%

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

   12. neg-mul-1 77.2%

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

   13. sin-neg 77.2%

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

   14. remove-double-neg 77.2%

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

   15. +-commutative 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in b around 0 57.3%

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

6. Final simplification 57.3%

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

Alternative 12: 50.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-/l* 77.0%

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

   2. remove-double-neg 77.0%

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

   3. sin-neg 77.0%

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

   4. neg-mul-1 77.0%

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

   5. associate-/r* 77.0%

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

   6. associate-/l* 77.1%

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

   7. *-commutative 77.1%

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

   8. associate-*l/ 77.2%

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

   9. associate-/r/ 77.2%

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

   10. *-commutative 77.2%

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

   11. associate-*r/ 77.2%

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

   12. neg-mul-1 77.2%

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

   13. sin-neg 77.2%

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

   14. remove-double-neg 77.2%

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

   15. +-commutative 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in b around 0 53.6%

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

6. Final simplification 53.6%

   \[\leadsto r \cdot \frac{b}{\cos a} \]
7. Add Preprocessing

Alternative 13: 34.7% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ r (+ (* b -0.3333333333333333) (/ 1.0 b))))

C

double code(double r, double a, double b) {
	return r / ((b * -0.3333333333333333) + (1.0 / 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 * (-0.3333333333333333d0)) + (1.0d0 / b))
end function

Java

public static double code(double r, double a, double b) {
	return r / ((b * -0.3333333333333333) + (1.0 / b));
}

Python

def code(r, a, b):
	return r / ((b * -0.3333333333333333) + (1.0 / b))

Julia

function code(r, a, b)
	return Float64(r / Float64(Float64(b * -0.3333333333333333) + Float64(1.0 / b)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r / ((b * -0.3333333333333333) + (1.0 / b));
end

Wolfram

code[r_, a_, b_] := N[(r / N[(N[(b * -0.3333333333333333), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.1%

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

   1. associate-/l* 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

5. Taylor expanded in b around 0 54.1%

   \[\leadsto \frac{r}{\color{blue}{-1 \cdot \sin a + \left(b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \frac{\cos a}{b}\right)}} \]
6. Step-by-step derivation

   1. +-commutative 54.1%

      \[\leadsto \frac{r}{\color{blue}{\left(b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \frac{\cos a}{b}\right) + -1 \cdot \sin a}} \]

   2. neg-mul-1 54.1%

      \[\leadsto \frac{r}{\left(b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \frac{\cos a}{b}\right) + \color{blue}{\left(-\sin a\right)}} \]

   3. unsub-neg 54.1%

      \[\leadsto \frac{r}{\color{blue}{\left(b \cdot \left(-0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a\right) + \frac{\cos a}{b}\right) - \sin a}} \]

   4. fma-define 54.1%

      \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(b, -0.5 \cdot \cos a - -0.16666666666666666 \cdot \cos a, \frac{\cos a}{b}\right)} - \sin a} \]

   5. distribute-rgt-out-- 54.1%

      \[\leadsto \frac{r}{\mathsf{fma}\left(b, \color{blue}{\cos a \cdot \left(-0.5 - -0.16666666666666666\right)}, \frac{\cos a}{b}\right) - \sin a} \]

   6. metadata-eval 54.1%

      \[\leadsto \frac{r}{\mathsf{fma}\left(b, \cos a \cdot \color{blue}{-0.3333333333333333}, \frac{\cos a}{b}\right) - \sin a} \]

7. Simplified 54.1%

   \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(b, \cos a \cdot -0.3333333333333333, \frac{\cos a}{b}\right) - \sin a}} \]

8. Taylor expanded in a around 0 35.1%

   \[\leadsto \color{blue}{\frac{r}{-0.3333333333333333 \cdot b + \frac{1}{b}}} \]

9. Final simplification 35.1%

   \[\leadsto \frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]
10. Add Preprocessing

Alternative 14: 34.2% 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 77.1%

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

   1. associate-/l* 77.0%

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

   2. remove-double-neg 77.0%

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

   3. sin-neg 77.0%

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

   4. neg-mul-1 77.0%

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

   5. associate-/r* 77.0%

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

   6. associate-/l* 77.1%

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

   7. *-commutative 77.1%

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

   8. associate-*l/ 77.2%

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

   9. associate-/r/ 77.2%

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

   10. *-commutative 77.2%

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

   11. associate-*r/ 77.2%

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

   12. neg-mul-1 77.2%

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

   13. sin-neg 77.2%

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

   14. remove-double-neg 77.2%

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

   15. +-commutative 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in b around 0 53.6%

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

6. Taylor expanded in a around 0 34.7%

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

7. Final simplification 34.7%

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

Reproduce

herbie shell --seed 2024034 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))