rsin A (should all be same)

Percentage Accurate: 76.0% → 99.4%

Time: 20.1s

Alternatives: 15

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: 76.0% 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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. remove-double-neg 77.8%

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

   2. associate-/l* 77.8%

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

   3. distribute-frac-neg 77.8%

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

   4. associate-/r/ 77.8%

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

   5. distribute-rgt-neg-in 77.8%

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

   6. sin-neg 77.8%

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

   7. neg-mul-1 77.8%

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

   8. *-commutative 77.8%

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

   9. associate-/l* 77.8%

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

   10. associate-/r/ 77.8%

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

   11. associate-/l/ 77.8%

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

   12. sin-neg 77.8%

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

   13. distribute-lft-neg-in 77.8%

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

   14. distribute-rgt-neg-in 77.8%

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

   15. associate-/l/ 77.8%

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

   16. metadata-eval 77.8%

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

   17. /-rgt-identity 77.8%

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

   18. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. +-commutative 77.8%

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

   2. cos-sum 99.5%

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

   3. div-sub 99.5%

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

   4. *-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in b around 0 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. remove-double-neg 77.8%

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

   2. associate-/l* 77.8%

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

   3. distribute-frac-neg 77.8%

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

   4. associate-/r/ 77.8%

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

   5. distribute-rgt-neg-in 77.8%

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

   6. sin-neg 77.8%

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

   7. neg-mul-1 77.8%

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

   8. *-commutative 77.8%

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

   9. associate-/l* 77.8%

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

   10. associate-/r/ 77.8%

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

   11. associate-/l/ 77.8%

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

   12. sin-neg 77.8%

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

   13. distribute-lft-neg-in 77.8%

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

   14. distribute-rgt-neg-in 77.8%

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

   15. associate-/l/ 77.8%

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

   16. metadata-eval 77.8%

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

   17. /-rgt-identity 77.8%

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

   18. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. +-commutative 77.8%

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

   2. cos-sum 99.5%

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

   3. div-sub 99.5%

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

   4. *-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in r around 0 99.5%

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

   1. associate-*r/ 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\frac{0.5 \cdot \left(2 \cdot \cos \left(a + b\right)\right)}{\sin b} - \sin a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ r (- (/ (* 0.5 (* 2.0 (cos (+ a b)))) (sin b)) (sin a))))

C

double code(double r, double a, double b) {
	return r / (((0.5 * (2.0 * cos((a + b)))) / 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 / (((0.5d0 * (2.0d0 * cos((a + b)))) / sin(b)) - sin(a))
end function

Java

public static double code(double r, double a, double b) {
	return r / (((0.5 * (2.0 * Math.cos((a + b)))) / Math.sin(b)) - Math.sin(a));
}

Python

def code(r, a, b):
	return r / (((0.5 * (2.0 * math.cos((a + b)))) / math.sin(b)) - math.sin(a))

Julia

function code(r, a, b)
	return Float64(r / Float64(Float64(Float64(0.5 * Float64(2.0 * cos(Float64(a + b)))) / sin(b)) - sin(a)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r / (((0.5 * (2.0 * cos((a + b)))) / sin(b)) - sin(a));
end

Wolfram

code[r_, a_, b_] := N[(r / N[(N[(N[(0.5 * N[(2.0 * N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision] - N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

   1. remove-double-neg 77.8%

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

   2. associate-/l* 77.8%

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

   3. distribute-frac-neg 77.8%

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

   4. associate-/r/ 77.8%

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

   5. distribute-rgt-neg-in 77.8%

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

   6. sin-neg 77.8%

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

   7. neg-mul-1 77.8%

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

   8. *-commutative 77.8%

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

   9. associate-/l* 77.8%

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

   10. associate-/r/ 77.8%

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

   11. associate-/l/ 77.8%

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

   12. sin-neg 77.8%

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

   13. distribute-lft-neg-in 77.8%

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

   14. distribute-rgt-neg-in 77.8%

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

   15. associate-/l/ 77.8%

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

   16. metadata-eval 77.8%

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

   17. /-rgt-identity 77.8%

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

   18. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. +-commutative 77.8%

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

   2. cos-sum 99.5%

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

   3. div-sub 99.5%

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

   4. *-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in b around 0 99.5%

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

   1. cos-mult 79.0%

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

   2. clear-num 79.0%

      \[\leadsto \frac{r}{\frac{\color{blue}{\frac{1}{\frac{2}{\cos \left(a + b\right) + \cos \left(a - b\right)}}}}{\sin b} - \sin a} \]

   3. +-commutative 79.0%

      \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \color{blue}{\left(b + a\right)} + \cos \left(a - b\right)}}}{\sin b} - \sin a} \]

   4. cos-diff 78.8%

      \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \color{blue}{\left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}}}}{\sin b} - \sin a} \]

   5. *-commutative 78.8%

      \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \left(\cos a \cdot \cos b + \color{blue}{\sin b \cdot \sin a}\right)}}}{\sin b} - \sin a} \]

   6. add-sqr-sqrt 44.5%

      \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \left(\cos a \cdot \cos b + \color{blue}{\sqrt{\sin b \cdot \sin a} \cdot \sqrt{\sin b \cdot \sin a}}\right)}}}{\sin b} - \sin a} \]

   7. sqrt-unprod 79.0%

      \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \left(\cos a \cdot \cos b + \color{blue}{\sqrt{\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)}}\right)}}}{\sin b} - \sin a} \]

   8. sqr-neg 79.0%

      \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \left(\cos a \cdot \cos b + \sqrt{\color{blue}{\left(-\sin b \cdot \sin a\right) \cdot \left(-\sin b \cdot \sin a\right)}}\right)}}}{\sin b} - \sin a} \]

   9. sqrt-unprod 45.4%

      \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \left(\cos a \cdot \cos b + \color{blue}{\sqrt{-\sin b \cdot \sin a} \cdot \sqrt{-\sin b \cdot \sin a}}\right)}}}{\sin b} - \sin a} \]

   10. add-sqr-sqrt 78.9%

       \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \left(\cos a \cdot \cos b + \color{blue}{\left(-\sin b \cdot \sin a\right)}\right)}}}{\sin b} - \sin a} \]

   11. sub-neg 78.9%

       \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \color{blue}{\left(\cos a \cdot \cos b - \sin b \cdot \sin a\right)}}}}{\sin b} - \sin a} \]

   12. *-commutative 78.9%

       \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \left(\color{blue}{\cos b \cdot \cos a} - \sin b \cdot \sin a\right)}}}{\sin b} - \sin a} \]

   13. cos-sum 78.8%

       \[\leadsto \frac{r}{\frac{\frac{1}{\frac{2}{\cos \left(b + a\right) + \color{blue}{\cos \left(b + a\right)}}}}{\sin b} - \sin a} \]

8. Applied egg-rr 78.8%

   \[\leadsto \frac{r}{\frac{\color{blue}{\frac{1}{\frac{2}{\cos \left(b + a\right) + \cos \left(b + a\right)}}}}{\sin b} - \sin a} \]
9. Step-by-step derivation

   1. associate-/r/ 78.8%

      \[\leadsto \frac{r}{\frac{\color{blue}{\frac{1}{2} \cdot \left(\cos \left(b + a\right) + \cos \left(b + a\right)\right)}}{\sin b} - \sin a} \]

   2. metadata-eval 78.8%

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

   3. count-2 78.8%

      \[\leadsto \frac{r}{\frac{0.5 \cdot \color{blue}{\left(2 \cdot \cos \left(b + a\right)\right)}}{\sin b} - \sin a} \]

10. Simplified 78.8%

    \[\leadsto \frac{r}{\frac{\color{blue}{0.5 \cdot \left(2 \cdot \cos \left(b + a\right)\right)}}{\sin b} - \sin a} \]

11. Final simplification 78.8%

    \[\leadsto \frac{r}{\frac{0.5 \cdot \left(2 \cdot \cos \left(a + b\right)\right)}{\sin b} - \sin a} \]

Alternative 4: 77.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. /-rgt-identity 77.8%

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

   2. metadata-eval 77.8%

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

   3. associate-*l/ 77.8%

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

   4. metadata-eval 77.8%

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

   5. /-rgt-identity 77.8%

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

   6. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-neg 99.5%

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

   4. *-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

   1. fma-udef 99.5%

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

   2. *-commutative 99.5%

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

   3. +-commutative 99.5%

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

7. Applied egg-rr 78.8%

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

   1. associate--r- 78.8%

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

   2. +-inverses 78.8%

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

   3. +-lft-identity 78.8%

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

   4. *-commutative 78.8%

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

9. Simplified 78.8%

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

10. Final simplification 78.8%

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

Alternative 5: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.88) (not (<= a 26000000.0)))
   (* (sin b) (/ r (cos a)))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.88) || !(a <= 26000000.0)) {
		tmp = sin(b) * (r / 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 ((a <= (-0.88d0)) .or. (.not. (a <= 26000000.0d0))) then
        tmp = sin(b) * (r / 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 ((a <= -0.88) || !(a <= 26000000.0)) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.88) or not (a <= 26000000.0):
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.88) || !(a <= 26000000.0))
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.88) || ~((a <= 26000000.0)))
		tmp = sin(b) * (r / cos(a));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.88], N[Not[LessEqual[a, 26000000.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.880000000000000004 or 2.6e7 < a

   1. Initial program 58.5%

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

      1. /-rgt-identity 58.5%

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

      2. metadata-eval 58.5%

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

      3. associate-*l/ 58.5%

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

      4. metadata-eval 58.5%

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

      5. /-rgt-identity 58.5%

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

      6. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in b around 0 58.5%

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

   if -0.880000000000000004 < a < 2.6e7

   1. Initial program 97.0%

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

      1. /-rgt-identity 97.0%

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

      2. metadata-eval 97.0%

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

      3. associate-*l/ 97.0%

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

      4. metadata-eval 97.0%

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

      5. /-rgt-identity 97.0%

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

      6. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in a around 0 96.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 6: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.88) (not (<= a 26000000.0)))
   (* (sin b) (/ r (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.88) || !(a <= 26000000.0)) {
		tmp = sin(b) * (r / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-0.88d0)) .or. (.not. (a <= 26000000.0d0))) then
        tmp = sin(b) * (r / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.88) || !(a <= 26000000.0)) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.88) or not (a <= 26000000.0):
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.88) || !(a <= 26000000.0))
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.88) || ~((a <= 26000000.0)))
		tmp = sin(b) * (r / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.88], N[Not[LessEqual[a, 26000000.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $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 < -0.880000000000000004 or 2.6e7 < a

   1. Initial program 58.5%

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

      1. /-rgt-identity 58.5%

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

      2. metadata-eval 58.5%

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

      3. associate-*l/ 58.5%

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

      4. metadata-eval 58.5%

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

      5. /-rgt-identity 58.5%

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

      6. +-commutative 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in b around 0 58.5%

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

   if -0.880000000000000004 < a < 2.6e7

   1. Initial program 97.0%

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

      1. remove-double-neg 97.0%

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

      2. associate-/l* 97.0%

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

      3. distribute-frac-neg 97.0%

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

      4. associate-/r/ 97.0%

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

      5. distribute-rgt-neg-in 97.0%

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

      6. sin-neg 97.0%

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

      7. neg-mul-1 97.0%

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

      8. *-commutative 97.0%

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

      9. associate-/l* 97.0%

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

      10. associate-/r/ 97.0%

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

      11. associate-/l/ 97.0%

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

      12. sin-neg 97.0%

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

      13. distribute-lft-neg-in 97.0%

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

      14. distribute-rgt-neg-in 97.0%

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

      15. associate-/l/ 97.0%

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

      16. metadata-eval 97.0%

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

      17. /-rgt-identity 97.0%

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

      18. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in a around 0 96.8%

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

      1. div-inv 96.8%

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

      2. *-commutative 96.8%

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

      3. clear-num 96.9%

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

   6. Applied egg-rr 96.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 7: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.88) (not (<= a 26000000.0)))
   (/ r (/ (cos a) (sin b)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.88) || !(a <= 26000000.0)) {
		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 <= (-0.88d0)) .or. (.not. (a <= 26000000.0d0))) 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 <= -0.88) || !(a <= 26000000.0)) {
		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 <= -0.88) or not (a <= 26000000.0):
		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 <= -0.88) || !(a <= 26000000.0))
		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 <= -0.88) || ~((a <= 26000000.0)))
		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, -0.88], N[Not[LessEqual[a, 26000000.0]], $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 < -0.880000000000000004 or 2.6e7 < a

   1. Initial program 58.5%

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

      1. remove-double-neg 58.5%

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

      2. associate-/l* 58.6%

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

      3. distribute-frac-neg 58.6%

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

      4. associate-/r/ 58.5%

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

      5. distribute-rgt-neg-in 58.5%

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

      6. sin-neg 58.5%

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

      7. neg-mul-1 58.5%

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

      8. *-commutative 58.5%

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

      9. associate-/l* 58.5%

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

      10. associate-/r/ 58.6%

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

      11. associate-/l/ 58.6%

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

      12. sin-neg 58.6%

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

      13. distribute-lft-neg-in 58.6%

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

      14. distribute-rgt-neg-in 58.6%

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

      15. associate-/l/ 58.6%

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

      16. metadata-eval 58.6%

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

      17. /-rgt-identity 58.6%

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

      18. +-commutative 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in b around 0 58.5%

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

   if -0.880000000000000004 < a < 2.6e7

   1. Initial program 97.0%

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

      1. remove-double-neg 97.0%

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

      2. associate-/l* 97.0%

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

      3. distribute-frac-neg 97.0%

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

      4. associate-/r/ 97.0%

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

      5. distribute-rgt-neg-in 97.0%

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

      6. sin-neg 97.0%

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

      7. neg-mul-1 97.0%

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

      8. *-commutative 97.0%

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

      9. associate-/l* 97.0%

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

      10. associate-/r/ 97.0%

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

      11. associate-/l/ 97.0%

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

      12. sin-neg 97.0%

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

      13. distribute-lft-neg-in 97.0%

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

      14. distribute-rgt-neg-in 97.0%

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

      15. associate-/l/ 97.0%

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

      16. metadata-eval 97.0%

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

      17. /-rgt-identity 97.0%

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

      18. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in a around 0 96.8%

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

      1. div-inv 96.8%

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

      2. *-commutative 96.8%

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

      3. clear-num 96.9%

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

   6. Applied egg-rr 96.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.88 \lor \neg \left(a \leq 26000000\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 8: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. /-rgt-identity 77.8%

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

   2. metadata-eval 77.8%

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

   3. associate-*l/ 77.8%

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

   4. metadata-eval 77.8%

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

   5. /-rgt-identity 77.8%

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

   6. +-commutative 77.8%

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

3. Simplified 77.8%

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

4. Final simplification 77.8%

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

Alternative 9: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. remove-double-neg 77.8%

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

   2. associate-/l* 77.8%

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

   3. distribute-frac-neg 77.8%

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

   4. associate-/r/ 77.8%

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

   5. distribute-rgt-neg-in 77.8%

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

   6. sin-neg 77.8%

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

   7. neg-mul-1 77.8%

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

   8. *-commutative 77.8%

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

   9. associate-/l* 77.8%

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

   10. associate-/r/ 77.8%

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

   11. associate-/l/ 77.8%

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

   12. sin-neg 77.8%

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

   13. distribute-lft-neg-in 77.8%

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

   14. distribute-rgt-neg-in 77.8%

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

   15. associate-/l/ 77.8%

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

   16. metadata-eval 77.8%

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

   17. /-rgt-identity 77.8%

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

   18. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. div-inv 77.7%

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

   2. *-commutative 77.7%

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

   3. clear-num 77.8%

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

5. Applied egg-rr 77.8%

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

6. Final simplification 77.8%

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

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 77.8%

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

   1. /-rgt-identity 77.8%

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

   2. metadata-eval 77.8%

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

   3. associate-*l/ 77.8%

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

   4. metadata-eval 77.8%

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

   5. /-rgt-identity 77.8%

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

   6. +-commutative 77.8%

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

3. Simplified 77.8%

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

4. Taylor expanded in b around 0 55.1%

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

5. Final simplification 55.1%

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

Alternative 11: 55.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -32000000 \lor \neg \left(b \leq 4000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -32000000.0) (not (<= b 4000000000.0)))
   (* r (sin b))
   (* b (/ r (cos (+ a b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -32000000.0) || !(b <= 4000000000.0)) {
		tmp = r * sin(b);
	} else {
		tmp = b * (r / cos((a + 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 <= (-32000000.0d0)) .or. (.not. (b <= 4000000000.0d0))) then
        tmp = r * sin(b)
    else
        tmp = b * (r / cos((a + b)))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -32000000.0) || !(b <= 4000000000.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = b * (r / Math.cos((a + b)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -32000000.0) or not (b <= 4000000000.0):
		tmp = r * math.sin(b)
	else:
		tmp = b * (r / math.cos((a + b)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -32000000.0) || !(b <= 4000000000.0))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(b * Float64(r / cos(Float64(a + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -32000000.0) || ~((b <= 4000000000.0)))
		tmp = r * sin(b);
	else
		tmp = b * (r / cos((a + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -32000000.0], N[Not[LessEqual[b, 4000000000.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2e7 or 4e9 < b

   1. Initial program 55.7%

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

      1. /-rgt-identity 55.7%

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

      2. metadata-eval 55.7%

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

      3. associate-*l/ 55.7%

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

      4. metadata-eval 55.7%

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

      5. /-rgt-identity 55.7%

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

      6. +-commutative 55.7%

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

   3. Simplified 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*r/ 55.7%

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

      3. associate-/l* 55.7%

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

   5. Applied egg-rr 55.7%

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

   6. Taylor expanded in b around 0 5.5%

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

      1. associate-*r* 5.5%

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

      2. mul-1-neg 5.5%

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

   8. Simplified 5.5%

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

   9. Taylor expanded in a around 0 11.5%

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

   if -3.2e7 < b < 4e9

   1. Initial program 97.5%

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

      1. /-rgt-identity 97.5%

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

      2. metadata-eval 97.5%

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

      3. associate-*l/ 97.6%

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

      4. metadata-eval 97.6%

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

      5. /-rgt-identity 97.6%

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

      6. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in b around 0 95.0%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -32000000 \lor \neg \left(b \leq 4000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(a + b\right)}\\ \end{array} \]

Alternative 12: 55.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \lor \neg \left(b \leq 1550000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.5) (not (<= b 1550000000.0)))
   (* r (sin b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.5) || !(b <= 1550000000.0)) {
		tmp = r * sin(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.5d0)) .or. (.not. (b <= 1550000000.0d0))) then
        tmp = r * sin(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.5) || !(b <= 1550000000.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.5) or not (b <= 1550000000.0):
		tmp = r * math.sin(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.5) || !(b <= 1550000000.0))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.5) || ~((b <= 1550000000.0)))
		tmp = r * sin(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.5], N[Not[LessEqual[b, 1550000000.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.5 or 1.55e9 < b

   1. Initial program 56.1%

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

      1. /-rgt-identity 56.1%

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

      2. metadata-eval 56.1%

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

      3. associate-*l/ 56.0%

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

      4. metadata-eval 56.0%

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

      5. /-rgt-identity 56.0%

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

      6. +-commutative 56.0%

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

   3. Simplified 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*r/ 56.1%

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

      3. associate-/l* 56.0%

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

   5. Applied egg-rr 56.0%

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

   6. Taylor expanded in b around 0 5.4%

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

      1. associate-*r* 5.4%

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

      2. mul-1-neg 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in a around 0 11.5%

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

   if -1.5 < b < 1.55e9

   1. Initial program 97.5%

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

      1. /-rgt-identity 97.5%

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

      2. metadata-eval 97.5%

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

      3. associate-*l/ 97.6%

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

      4. metadata-eval 97.6%

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

      5. /-rgt-identity 97.6%

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

      6. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in b around 0 95.5%

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

      1. associate-/l* 95.4%

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

   6. Simplified 95.4%

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

      1. associate-/r/ 95.5%

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

   8. Applied egg-rr 95.5%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \lor \neg \left(b \leq 1550000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 13: 39.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. /-rgt-identity 77.8%

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

   2. metadata-eval 77.8%

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

   3. associate-*l/ 77.8%

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

   4. metadata-eval 77.8%

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

   5. /-rgt-identity 77.8%

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

   6. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. *-commutative 77.8%

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

   2. associate-*r/ 77.8%

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

   3. associate-/l* 77.7%

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

5. Applied egg-rr 77.7%

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

6. Taylor expanded in b around 0 52.8%

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

   1. associate-*r* 52.8%

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

   2. mul-1-neg 52.8%

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

8. Simplified 52.8%

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

9. Taylor expanded in a around 0 36.4%

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

10. Final simplification 36.4%

    \[\leadsto r \cdot \sin b \]

Alternative 14: 35.3% 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 77.8%

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

   1. remove-double-neg 77.8%

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

   2. associate-/l* 77.8%

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

   3. distribute-frac-neg 77.8%

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

   4. associate-/r/ 77.8%

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

   5. distribute-rgt-neg-in 77.8%

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

   6. sin-neg 77.8%

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

   7. neg-mul-1 77.8%

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

   8. *-commutative 77.8%

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

   9. associate-/l* 77.8%

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

   10. associate-/r/ 77.8%

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

   11. associate-/l/ 77.8%

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

   12. sin-neg 77.8%

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

   13. distribute-lft-neg-in 77.8%

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

   14. distribute-rgt-neg-in 77.8%

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

   15. associate-/l/ 77.8%

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

   16. metadata-eval 77.8%

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

   17. /-rgt-identity 77.8%

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

   18. +-commutative 77.8%

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

3. Simplified 77.8%

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

4. Taylor expanded in a around 0 58.7%

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

5. Taylor expanded in b around 0 33.5%

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

6. Final simplification 33.5%

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

Alternative 15: 34.7% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. /-rgt-identity 77.8%

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

   2. metadata-eval 77.8%

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

   3. associate-*l/ 77.8%

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

   4. metadata-eval 77.8%

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

   5. /-rgt-identity 77.8%

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

   6. +-commutative 77.8%

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

3. Simplified 77.8%

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

4. Taylor expanded in b around 0 51.6%

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

   1. associate-/l* 51.5%

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

6. Simplified 51.5%

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

7. Taylor expanded in a around 0 32.5%

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

   1. *-commutative 32.5%

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

9. Simplified 32.5%

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

10. Final simplification 32.5%

    \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023301 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))