rsin A (should all be same)

Percentage Accurate: 76.2% → 99.5%

Time: 20.7s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 75.8%

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

   2. remove-double-neg 75.8%

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

   3. sin-neg 75.8%

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

   4. neg-mul-1 75.8%

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

   5. associate-/r* 75.8%

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

   6. associate-/l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-*l/ 75.9%

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

   9. associate-/l* 75.9%

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

   10. sin-neg 75.9%

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

   11. distribute-lft-neg-in 75.9%

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

   12. distribute-rgt-neg-in 75.9%

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

   13. associate-/l* 75.9%

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

   14. metadata-eval 75.9%

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

   15. /-rgt-identity 75.9%

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

   16. +-commutative 75.9%

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

3. Simplified 75.9%

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

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

Alternative 2: 77.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 75.8%

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

   2. remove-double-neg 75.8%

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

   3. sin-neg 75.8%

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

   4. neg-mul-1 75.8%

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

   5. associate-/r* 75.8%

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

   6. associate-/l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-*l/ 75.9%

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

   9. associate-/l* 75.9%

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

   10. sin-neg 75.9%

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

   11. distribute-lft-neg-in 75.9%

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

   12. distribute-rgt-neg-in 75.9%

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

   13. associate-/l* 75.9%

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

   14. metadata-eval 75.9%

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

   15. /-rgt-identity 75.9%

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

   16. +-commutative 75.9%

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

3. Simplified 75.9%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
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 77.0%

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

   2. cos-sum 77.2%

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

   3. div-sub 77.2%

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

   4. cos-diff 99.4%

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

   5. add-sqr-sqrt 53.3%

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

   6. sqrt-unprod 87.4%

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

   7. sqr-neg 87.4%

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

   8. sqrt-unprod 43.4%

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

   9. add-sqr-sqrt 77.1%

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

   10. sub-neg 77.1%

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

   11. cos-sum 77.4%

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

   12. cos-sum 77.1%

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

8. Applied egg-rr 77.1%

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

   1. +-inverses 77.1%

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

10. Simplified 77.1%

    \[\leadsto \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{0}} \cdot r \]
11. Step-by-step derivation

    1. frac-2neg 77.1%

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

    2. associate-*l/ 77.1%

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

    3. distribute-lft-neg-in 77.1%

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

    4. distribute-rgt-neg-in 77.1%

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

    5. --rgt-identity 77.1%

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

    6. distribute-rgt-neg-in 77.1%

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

12. Applied egg-rr 77.1%

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

    1. times-frac 77.1%

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

14. Simplified 77.1%

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

15. Final simplification 77.1%

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

Alternative 3: 77.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 75.8%

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

   2. remove-double-neg 75.8%

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

   3. sin-neg 75.8%

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

   4. neg-mul-1 75.8%

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

   5. associate-/r* 75.8%

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

   6. associate-/l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-*l/ 75.9%

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

   9. associate-/l* 75.9%

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

   10. sin-neg 75.9%

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

   11. distribute-lft-neg-in 75.9%

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

   12. distribute-rgt-neg-in 75.9%

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

   13. associate-/l* 75.9%

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

   14. metadata-eval 75.9%

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

   15. /-rgt-identity 75.9%

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

   16. +-commutative 75.9%

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

3. Simplified 75.9%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
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 77.0%

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

   2. cos-sum 77.2%

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

   3. div-sub 77.2%

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

   4. cos-diff 99.4%

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

   5. add-sqr-sqrt 53.3%

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

   6. sqrt-unprod 87.4%

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

   7. sqr-neg 87.4%

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

   8. sqrt-unprod 43.4%

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

   9. add-sqr-sqrt 77.1%

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

   10. sub-neg 77.1%

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

   11. cos-sum 77.4%

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

   12. cos-sum 77.1%

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

8. Applied egg-rr 77.1%

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

   1. +-inverses 77.1%

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

10. Simplified 77.1%

    \[\leadsto \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{0}} \cdot r \]
11. Step-by-step derivation

    1. *-commutative 77.1%

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

    2. clear-num 77.1%

       \[\leadsto r \cdot \color{blue}{\frac{1}{\frac{\cos b \cdot \cos a - 0}{\sin b}}} \]

    3. un-div-inv 77.1%

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

    4. --rgt-identity 77.1%

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

    5. associate-/l* 77.1%

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

12. Applied egg-rr 77.1%

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

13. Final simplification 77.1%

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

Alternative 4: 77.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 75.8%

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

   2. remove-double-neg 75.8%

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

   3. sin-neg 75.8%

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

   4. neg-mul-1 75.8%

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

   5. associate-/r* 75.8%

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

   6. associate-/l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-*l/ 75.9%

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

   9. associate-/l* 75.9%

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

   10. sin-neg 75.9%

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

   11. distribute-lft-neg-in 75.9%

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

   12. distribute-rgt-neg-in 75.9%

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

   13. associate-/l* 75.9%

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

   14. metadata-eval 75.9%

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

   15. /-rgt-identity 75.9%

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

   16. +-commutative 75.9%

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

3. Simplified 75.9%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
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 77.0%

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

   2. cos-sum 77.2%

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

   3. div-sub 77.2%

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

   4. cos-diff 99.4%

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

   5. add-sqr-sqrt 53.3%

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

   6. sqrt-unprod 87.4%

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

   7. sqr-neg 87.4%

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

   8. sqrt-unprod 43.4%

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

   9. add-sqr-sqrt 77.1%

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

   10. sub-neg 77.1%

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

   11. cos-sum 77.4%

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

   12. cos-sum 77.1%

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

8. Applied egg-rr 77.1%

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

   1. +-inverses 77.1%

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

10. Simplified 77.1%

    \[\leadsto \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{0}} \cdot r \]
11. Step-by-step derivation

    1. --rgt-identity 77.1%

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

    2. associate-*l/ 77.1%

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

    3. *-commutative 77.1%

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

12. Applied egg-rr 77.1%

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

13. Final simplification 77.1%

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

Alternative 5: 77.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 75.8%

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

   2. remove-double-neg 75.8%

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

   3. sin-neg 75.8%

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

   4. neg-mul-1 75.8%

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

   5. associate-/r* 75.8%

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

   6. associate-/l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-*l/ 75.9%

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

   9. associate-/l* 75.9%

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

   10. sin-neg 75.9%

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

   11. distribute-lft-neg-in 75.9%

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

   12. distribute-rgt-neg-in 75.9%

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

   13. associate-/l* 75.9%

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

   14. metadata-eval 75.9%

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

   15. /-rgt-identity 75.9%

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

   16. +-commutative 75.9%

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

3. Simplified 75.9%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
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 77.0%

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

   2. cos-sum 77.2%

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

   3. div-sub 77.2%

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

   4. cos-diff 99.4%

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

   5. add-sqr-sqrt 53.3%

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

   6. sqrt-unprod 87.4%

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

   7. sqr-neg 87.4%

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

   8. sqrt-unprod 43.4%

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

   9. add-sqr-sqrt 77.1%

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

   10. sub-neg 77.1%

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

   11. cos-sum 77.4%

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

   12. cos-sum 77.1%

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

8. Applied egg-rr 77.1%

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

   1. +-inverses 77.1%

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

10. Simplified 77.1%

    \[\leadsto \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{0}} \cdot r \]
11. Step-by-step derivation

    1. --rgt-identity 77.1%

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

    2. associate-*l/ 77.1%

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

    3. *-commutative 77.1%

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

    4. associate-/r* 77.1%

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

12. Applied egg-rr 77.1%

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

13. Final simplification 77.1%

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

Alternative 6: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0065 \lor \neg \left(a \leq 0.00185\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.0065) (not (<= a 0.00185)))
   (/ r (/ (cos a) (sin b)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.0065) || !(a <= 0.00185)) {
		tmp = r / (cos(a) / sin(b));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-0.0065d0)) .or. (.not. (a <= 0.00185d0))) then
        tmp = r / (cos(a) / sin(b))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.0065) || !(a <= 0.00185)) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.0065) or not (a <= 0.00185):
		tmp = r / (math.cos(a) / math.sin(b))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.0065) || !(a <= 0.00185))
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.0065) || ~((a <= 0.00185)))
		tmp = r / (cos(a) / sin(b));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.0065], N[Not[LessEqual[a, 0.00185]], $MachinePrecision]], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0064999999999999997 or 0.0018500000000000001 < a

   1. Initial program 56.6%

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

      1. associate-/l* 56.7%

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

      2. +-commutative 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in b around 0 56.4%

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

   if -0.0064999999999999997 < a < 0.0018500000000000001

   1. Initial program 97.7%

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

      1. associate-/l* 97.5%

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

      2. remove-double-neg 97.5%

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

      3. sin-neg 97.5%

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

      4. neg-mul-1 97.5%

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

      5. associate-/r* 97.5%

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

      6. associate-/l* 97.7%

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

      7. *-commutative 97.7%

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

      8. associate-*l/ 97.7%

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

      9. associate-/l* 97.7%

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

      10. sin-neg 97.7%

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

      11. distribute-lft-neg-in 97.7%

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

      12. distribute-rgt-neg-in 97.7%

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

      13. associate-/l* 97.7%

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

      14. metadata-eval 97.7%

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

      15. /-rgt-identity 97.7%

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

      16. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in a around 0 97.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0065 \lor \neg \left(a \leq 0.00185\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0065:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.0018:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.0065)
   (* r (/ (sin b) (cos a)))
   (if (<= a 0.0018) (* r (/ (sin b) (cos b))) (* (sin b) (/ r (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0065) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 0.0018) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = sin(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 (a <= (-0.0065d0)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 0.0018d0) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = sin(b) * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0065) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 0.0018) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.0065:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 0.0018:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = math.sin(b) * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.0065)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 0.0018)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.0065)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 0.0018)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = sin(b) * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.0065], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0018], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0064999999999999997

   1. Initial program 51.3%

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

      1. associate-/l* 51.4%

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

      2. remove-double-neg 51.4%

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

      3. sin-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. associate-/r* 51.4%

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

      6. associate-/l* 51.3%

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

      7. *-commutative 51.3%

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

      8. associate-*l/ 51.4%

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

      9. associate-/l* 51.4%

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

      10. sin-neg 51.4%

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

      11. distribute-lft-neg-in 51.4%

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

      12. distribute-rgt-neg-in 51.4%

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

      13. associate-/l* 51.4%

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

      14. metadata-eval 51.4%

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

      15. /-rgt-identity 51.4%

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

      16. +-commutative 51.4%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in b around 0 52.1%

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

   if -0.0064999999999999997 < a < 0.0018

   1. Initial program 97.7%

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

      1. associate-/l* 97.5%

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

      2. remove-double-neg 97.5%

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

      3. sin-neg 97.5%

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

      4. neg-mul-1 97.5%

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

      5. associate-/r* 97.5%

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

      6. associate-/l* 97.7%

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

      7. *-commutative 97.7%

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

      8. associate-*l/ 97.7%

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

      9. associate-/l* 97.7%

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

      10. sin-neg 97.7%

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

      11. distribute-lft-neg-in 97.7%

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

      12. distribute-rgt-neg-in 97.7%

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

      13. associate-/l* 97.7%

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

      14. metadata-eval 97.7%

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

      15. /-rgt-identity 97.7%

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

      16. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in a around 0 97.7%

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

   if 0.0018 < a

   1. Initial program 61.9%

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

      1. +-commutative 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in b around 0 60.7%

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

      1. associate-/l* 60.8%

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

      2. associate-/r/ 60.8%

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

   7. Applied egg-rr 60.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0065:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.0018:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.36 \cdot 10^{+14} \lor \neg \left(b \leq 3.6 \cdot 10^{+15}\right):\\ \;\;\;\;\left|\sin b \cdot r\right|\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.36e+14) (not (<= b 3.6e+15)))
   (fabs (* (sin b) r))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.36e+14) || !(b <= 3.6e+15)) {
		tmp = fabs((sin(b) * r));
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.36d+14)) .or. (.not. (b <= 3.6d+15))) then
        tmp = abs((sin(b) * r))
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.36e+14) || !(b <= 3.6e+15)) {
		tmp = Math.abs((Math.sin(b) * r));
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.36e+14) or not (b <= 3.6e+15):
		tmp = math.fabs((math.sin(b) * r))
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.36e+14) || !(b <= 3.6e+15))
		tmp = abs(Float64(sin(b) * r));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.36e+14) || ~((b <= 3.6e+15)))
		tmp = abs((sin(b) * r));
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.36e+14], N[Not[LessEqual[b, 3.6e+15]], $MachinePrecision]], N[Abs[N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]], $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.36e14 or 3.6e15 < b

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in b around 0 11.0%

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

   6. Taylor expanded in a around 0 11.2%

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

      1. add-sqr-sqrt 5.3%

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

      2. sqrt-unprod 7.1%

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

      3. pow2 7.1%

         \[\leadsto \sqrt{\color{blue}{{\left(r \cdot \sin b\right)}^{2}}} \]

      4. *-commutative 7.1%

         \[\leadsto \sqrt{{\color{blue}{\left(\sin b \cdot r\right)}}^{2}} \]

   8. Applied egg-rr 7.1%

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

      1. unpow2 7.1%

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

      2. rem-sqrt-square 11.0%

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

   10. Simplified 11.0%

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

   if -1.36e14 < b < 3.6e15

   1. Initial program 95.6%

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

      1. associate-/l* 95.5%

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

      2. remove-double-neg 95.5%

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

      3. sin-neg 95.5%

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

      4. neg-mul-1 95.5%

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

      5. associate-/r* 95.5%

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

      6. associate-/l* 95.6%

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

      7. *-commutative 95.6%

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

      8. associate-*l/ 95.6%

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

      9. associate-/l* 95.6%

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

      10. sin-neg 95.6%

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

      11. distribute-lft-neg-in 95.6%

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

      12. distribute-rgt-neg-in 95.6%

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

      13. associate-/l* 95.6%

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

      14. metadata-eval 95.6%

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

      15. /-rgt-identity 95.6%

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

      16. +-commutative 95.6%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in b around 0 94.3%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.36 \cdot 10^{+14} \lor \neg \left(b \leq 3.6 \cdot 10^{+15}\right):\\ \;\;\;\;\left|\sin b \cdot r\right|\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% 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 75.9%

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

   1. associate-/l* 75.8%

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

   2. +-commutative 75.8%

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

3. Simplified 75.8%

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

   1. associate-/r/ 75.9%

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

6. Applied egg-rr 75.9%

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

7. Final simplification 75.9%

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

Alternative 10: 76.2% 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 75.9%

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

   1. associate-/l* 75.8%

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

   2. remove-double-neg 75.8%

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

   3. sin-neg 75.8%

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

   4. neg-mul-1 75.8%

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

   5. associate-/r* 75.8%

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

   6. associate-/l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-*l/ 75.9%

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

   9. associate-/l* 75.9%

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

   10. sin-neg 75.9%

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

   11. distribute-lft-neg-in 75.9%

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

   12. distribute-rgt-neg-in 75.9%

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

   13. associate-/l* 75.9%

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

   14. metadata-eval 75.9%

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

   15. /-rgt-identity 75.9%

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

   16. +-commutative 75.9%

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

3. Simplified 75.9%

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

5. Final simplification 75.9%

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

Alternative 11: 76.0% 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 75.9%

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

   1. associate-/l* 75.8%

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

   2. remove-double-neg 75.8%

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

   3. sin-neg 75.8%

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

   4. neg-mul-1 75.8%

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

   5. associate-/r* 75.8%

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

   6. associate-/l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-*l/ 75.9%

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

   9. associate-/l* 75.9%

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

   10. sin-neg 75.9%

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

   11. distribute-lft-neg-in 75.9%

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

   12. distribute-rgt-neg-in 75.9%

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

   13. associate-/l* 75.9%

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

   14. metadata-eval 75.9%

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

   15. /-rgt-identity 75.9%

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

   16. +-commutative 75.9%

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

3. Simplified 75.9%

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

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

   2. add-sqr-sqrt 55.5%

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

   3. sqrt-unprod 88.4%

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

   4. sqr-neg 88.4%

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

   5. sqrt-unprod 42.2%

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

   6. add-sqr-sqrt 75.6%

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

   7. cos-diff 76.1%

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

8. Applied egg-rr 76.1%

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

9. Final simplification 76.1%

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

Alternative 12: 54.6% 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 75.9%

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

   1. +-commutative 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in b around 0 57.2%

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

   1. associate-/l* 57.2%

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

   2. associate-/r/ 57.2%

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

7. Applied egg-rr 57.2%

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

8. Final simplification 57.2%

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

Alternative 13: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -72000 \lor \neg \left(b \leq 2800000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -72000.0) (not (<= b 2800000000.0)))
   (* (sin b) r)
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -72000.0) || !(b <= 2800000000.0)) {
		tmp = sin(b) * r;
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-72000.0d0)) .or. (.not. (b <= 2800000000.0d0))) then
        tmp = sin(b) * r
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -72000.0) || !(b <= 2800000000.0)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -72000.0) or not (b <= 2800000000.0):
		tmp = math.sin(b) * r
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -72000.0) || !(b <= 2800000000.0))
		tmp = Float64(sin(b) * r);
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -72000.0) || ~((b <= 2800000000.0)))
		tmp = sin(b) * r;
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -72000.0], N[Not[LessEqual[b, 2800000000.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -72000 or 2.8e9 < b

   1. Initial program 51.0%

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

      1. +-commutative 51.0%

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

   3. Simplified 51.0%

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

   5. Taylor expanded in b around 0 11.1%

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

   6. Taylor expanded in a around 0 11.4%

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

   if -72000 < b < 2.8e9

   1. Initial program 96.8%

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

      1. associate-/l* 96.7%

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

      2. remove-double-neg 96.7%

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

      3. sin-neg 96.7%

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

      4. neg-mul-1 96.7%

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

      5. associate-/r* 96.7%

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

      6. associate-/l* 96.8%

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

      7. *-commutative 96.8%

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

      8. associate-*l/ 96.8%

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

      9. associate-/l* 96.8%

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

      10. sin-neg 96.8%

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

      11. distribute-lft-neg-in 96.8%

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

      12. distribute-rgt-neg-in 96.8%

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

      13. associate-/l* 96.8%

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

      14. metadata-eval 96.8%

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

      15. /-rgt-identity 96.8%

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

      16. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in b around 0 96.1%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -72000 \lor \neg \left(b \leq 2800000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -32000:\\ \;\;\;\;\frac{1}{\frac{1}{t_0}}\\ \mathbf{elif}\;b \leq 58000000:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= b -32000.0)
     (/ 1.0 (/ 1.0 t_0))
     (if (<= b 58000000.0) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (b <= -32000.0) {
		tmp = 1.0 / (1.0 / t_0);
	} else if (b <= 58000000.0) {
		tmp = r * (b / cos(a));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * r
    if (b <= (-32000.0d0)) then
        tmp = 1.0d0 / (1.0d0 / t_0)
    else if (b <= 58000000.0d0) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * r;
	double tmp;
	if (b <= -32000.0) {
		tmp = 1.0 / (1.0 / t_0);
	} else if (b <= 58000000.0) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * r
	tmp = 0
	if b <= -32000.0:
		tmp = 1.0 / (1.0 / t_0)
	elif b <= 58000000.0:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (b <= -32000.0)
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	elseif (b <= 58000000.0)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * r;
	tmp = 0.0;
	if (b <= -32000.0)
		tmp = 1.0 / (1.0 / t_0);
	elseif (b <= 58000000.0)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -32000.0], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 58000000.0], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if b < -32000

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in b around 0 12.3%

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

      1. clear-num 12.3%

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

      2. inv-pow 12.3%

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

      3. associate-/r* 12.3%

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

   7. Applied egg-rr 12.3%

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

      1. unpow-1 12.3%

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

      2. associate-/l/ 12.3%

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

      3. *-commutative 12.3%

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

   9. Simplified 12.3%

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

   10. Taylor expanded in a around 0 12.0%

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

   if -32000 < b < 5.8e7

   1. Initial program 96.8%

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

      1. associate-/l* 96.7%

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

      2. remove-double-neg 96.7%

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

      3. sin-neg 96.7%

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

      4. neg-mul-1 96.7%

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

      5. associate-/r* 96.7%

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

      6. associate-/l* 96.8%

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

      7. *-commutative 96.8%

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

      8. associate-*l/ 96.8%

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

      9. associate-/l* 96.8%

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

      10. sin-neg 96.8%

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

      11. distribute-lft-neg-in 96.8%

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

      12. distribute-rgt-neg-in 96.8%

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

      13. associate-/l* 96.8%

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

      14. metadata-eval 96.8%

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

      15. /-rgt-identity 96.8%

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

      16. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in b around 0 96.1%

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

   if 5.8e7 < b

   1. Initial program 57.0%

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

      1. +-commutative 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in b around 0 9.9%

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

   6. Taylor expanded in a around 0 10.9%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -32000:\\ \;\;\;\;\frac{1}{\frac{1}{\sin b \cdot r}}\\ \mathbf{elif}\;b \leq 58000000:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. +-commutative 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in b around 0 57.2%

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

6. Taylor expanded in a around 0 38.3%

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

7. Final simplification 38.3%

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

Alternative 16: 34.6% 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 75.9%

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

   1. associate-/l* 75.8%

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

   2. remove-double-neg 75.8%

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

   3. sin-neg 75.8%

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

   4. neg-mul-1 75.8%

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

   5. associate-/r* 75.8%

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

   6. associate-/l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-*l/ 75.9%

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

   9. associate-/l* 75.9%

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

   10. sin-neg 75.9%

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

   11. distribute-lft-neg-in 75.9%

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

   12. distribute-rgt-neg-in 75.9%

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

   13. associate-/l* 75.9%

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

   14. metadata-eval 75.9%

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

   15. /-rgt-identity 75.9%

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

   16. +-commutative 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in b around 0 53.8%

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

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 2024022 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))