rsin A (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 18.2s

Alternatives: 12

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: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. +-commutative 80.3%

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

3. Simplified 80.3%

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

   1. cos-sum 99.5%

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

   2. fma-neg 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. associate-/l* 80.2%

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

   2. remove-double-neg 80.2%

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

   3. sin-neg 80.2%

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

   4. neg-mul-1 80.2%

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

   5. associate-/r* 80.2%

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

   6. associate-/l* 80.3%

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

   7. *-commutative 80.3%

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

   8. associate-*l/ 80.3%

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

   9. associate-/l* 80.3%

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

   10. sin-neg 80.3%

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

   11. distribute-lft-neg-in 80.3%

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

   12. distribute-rgt-neg-in 80.3%

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

   13. associate-/l* 80.3%

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

   14. metadata-eval 80.3%

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

   15. /-rgt-identity 80.3%

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

   16. +-commutative 80.3%

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

3. Simplified 80.3%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 77.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. +-commutative 80.3%

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

3. Simplified 80.3%

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

   1. cos-sum 99.5%

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

   2. fma-neg 99.6%

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

6. Applied egg-rr 99.6%

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

   1. sin-mult 81.1%

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

   2. div-sub 81.1%

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

   3. cos-sum 81.9%

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

   4. sub-neg 81.9%

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

   5. add-sqr-sqrt 49.0%

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

   6. sqrt-unprod 80.7%

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

   7. sqr-neg 80.7%

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

   8. sqrt-unprod 46.5%

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

   9. add-sqr-sqrt 80.2%

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

   10. cos-diff 80.8%

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

8. Applied egg-rr 80.8%

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

   1. +-inverses 80.8%

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

10. Simplified 80.8%

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

11. Taylor expanded in r around 0 80.8%

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

    1. *-commutative 80.8%

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

    2. associate-*r/ 80.8%

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

    3. *-commutative 80.8%

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

13. Simplified 80.8%

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

14. Final simplification 80.8%

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

Alternative 4: 77.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. +-commutative 80.3%

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

3. Simplified 80.3%

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

   1. cos-sum 99.5%

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

   2. fma-neg 99.6%

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

6. Applied egg-rr 99.6%

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

   1. sin-mult 81.1%

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

   2. div-sub 81.1%

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

   3. cos-sum 81.9%

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

   4. sub-neg 81.9%

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

   5. add-sqr-sqrt 49.0%

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

   6. sqrt-unprod 80.7%

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

   7. sqr-neg 80.7%

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

   8. sqrt-unprod 46.5%

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

   9. add-sqr-sqrt 80.2%

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

   10. cos-diff 80.8%

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

8. Applied egg-rr 80.8%

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

   1. +-inverses 80.8%

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

10. Simplified 80.8%

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, -\color{blue}{0}\right)} \]
11. Step-by-step derivation

    1. associate-/l* 80.7%

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

    2. associate-/r/ 80.8%

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

    3. metadata-eval 80.8%

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

    4. fma-udef 80.8%

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

    5. +-rgt-identity 80.8%

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

12. Applied egg-rr 80.8%

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

13. Taylor expanded in r around 0 80.8%

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

    1. times-frac 80.8%

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

15. Simplified 80.8%

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

16. Final simplification 80.8%

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

Alternative 5: 77.3% accurate, 0.7× speedup

Math

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

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

Derivation

1. Initial program 80.3%

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

   1. +-commutative 80.3%

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

3. Simplified 80.3%

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

   1. cos-sum 99.5%

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

   2. fma-neg 99.6%

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

6. Applied egg-rr 99.6%

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

   1. sin-mult 81.1%

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

   2. div-sub 81.1%

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

   3. cos-sum 81.9%

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

   4. sub-neg 81.9%

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

   5. add-sqr-sqrt 49.0%

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

   6. sqrt-unprod 80.7%

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

   7. sqr-neg 80.7%

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

   8. sqrt-unprod 46.5%

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

   9. add-sqr-sqrt 80.2%

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

   10. cos-diff 80.8%

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

8. Applied egg-rr 80.8%

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

   1. +-inverses 80.8%

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

10. Simplified 80.8%

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, -\color{blue}{0}\right)} \]
11. Step-by-step derivation

    1. associate-/l* 80.7%

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

    2. associate-/r/ 80.8%

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

    3. metadata-eval 80.8%

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

    4. fma-udef 80.8%

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

    5. +-rgt-identity 80.8%

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

12. Applied egg-rr 80.8%

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

13. Final simplification 80.8%

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

Alternative 6: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. associate-/l* 80.2%

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

   2. +-commutative 80.2%

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

3. Simplified 80.2%

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

   1. associate-/r/ 80.3%

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

6. Applied egg-rr 80.3%

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

7. Final simplification 80.3%

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

Alternative 7: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-6} \lor \neg \left(b \leq 5.6 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.8e-6) (not (<= b 5.6e-5)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8e-6) || !(b <= 5.6e-5)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-4.8d-6)) .or. (.not. (b <= 5.6d-5))) then
        tmp = r * tan(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8e-6) || !(b <= 5.6e-5)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.8e-6) or not (b <= 5.6e-5):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.8e-6) || !(b <= 5.6e-5))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.8e-6) || ~((b <= 5.6e-5)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.8e-6], N[Not[LessEqual[b, 5.6e-5]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.7999999999999998e-6 or 5.59999999999999992e-5 < b

   1. Initial program 59.2%

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

      1. associate-/l* 59.1%

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

      2. remove-double-neg 59.1%

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

      3. sin-neg 59.1%

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

      4. neg-mul-1 59.1%

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

      5. associate-/r* 59.1%

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

      6. associate-/l* 59.2%

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

      7. *-commutative 59.2%

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

      8. associate-*l/ 59.2%

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

      9. associate-/l* 59.2%

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

      10. sin-neg 59.2%

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

      11. distribute-lft-neg-in 59.2%

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

      12. distribute-rgt-neg-in 59.2%

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

      13. associate-/l* 59.2%

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

      14. metadata-eval 59.2%

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

      15. /-rgt-identity 59.2%

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

      16. +-commutative 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in a around 0 58.2%

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

      1. expm1-log1p-u 41.4%

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

      2. expm1-udef 41.3%

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

      3. quot-tan 41.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\tan b}\right)} - 1\right) \cdot r \]

   7. Applied egg-rr 41.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\tan b\right)} - 1\right)} \cdot r \]
   8. Step-by-step derivation

      1. expm1-def 41.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

      2. expm1-log1p 58.3%

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

   9. Simplified 58.3%

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

   if -4.7999999999999998e-6 < b < 5.59999999999999992e-5

   1. Initial program 99.4%

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

      1. associate-/l* 99.4%

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

      2. remove-double-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. neg-mul-1 99.4%

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

      5. associate-/r* 99.4%

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

      6. associate-/l* 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*l/ 99.5%

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

      9. associate-/l* 99.5%

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

      10. sin-neg 99.5%

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

      11. distribute-lft-neg-in 99.5%

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

      12. distribute-rgt-neg-in 99.5%

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

      13. associate-/l* 99.5%

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

      14. metadata-eval 99.5%

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

      15. /-rgt-identity 99.5%

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

      16. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 99.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-6} \lor \neg \left(b \leq 5.6 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00011 \lor \neg \left(b \leq 5.7 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00011) (not (<= b 5.7e-5)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00011) || !(b <= 5.7e-5)) {
		tmp = r * tan(b);
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.00011d0)) .or. (.not. (b <= 5.7d-5))) then
        tmp = r * tan(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00011) || !(b <= 5.7e-5)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00011) or not (b <= 5.7e-5):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.00011) || !(b <= 5.7e-5))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.00011) || ~((b <= 5.7e-5)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00011], N[Not[LessEqual[b, 5.7e-5]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.10000000000000004e-4 or 5.7000000000000003e-5 < b

   1. Initial program 59.2%

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

      1. associate-/l* 59.1%

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

      2. remove-double-neg 59.1%

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

      3. sin-neg 59.1%

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

      4. neg-mul-1 59.1%

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

      5. associate-/r* 59.1%

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

      6. associate-/l* 59.2%

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

      7. *-commutative 59.2%

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

      8. associate-*l/ 59.2%

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

      9. associate-/l* 59.2%

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

      10. sin-neg 59.2%

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

      11. distribute-lft-neg-in 59.2%

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

      12. distribute-rgt-neg-in 59.2%

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

      13. associate-/l* 59.2%

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

      14. metadata-eval 59.2%

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

      15. /-rgt-identity 59.2%

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

      16. +-commutative 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in a around 0 58.2%

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

      1. expm1-log1p-u 41.4%

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

      2. expm1-udef 41.3%

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

      3. quot-tan 41.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\tan b}\right)} - 1\right) \cdot r \]

   7. Applied egg-rr 41.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\tan b\right)} - 1\right)} \cdot r \]
   8. Step-by-step derivation

      1. expm1-def 41.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

      2. expm1-log1p 58.3%

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

   9. Simplified 58.3%

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

   if -1.10000000000000004e-4 < b < 5.7000000000000003e-5

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

   3. Simplified 99.4%

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

      1. cos-sum 99.8%

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

      2. fma-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in b around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-/l* 99.4%

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

      3. associate-/r/ 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00011 \lor \neg \left(b \leq 5.7 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.5% 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 80.3%

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

   1. associate-/l* 80.2%

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

   2. +-commutative 80.2%

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

3. Simplified 80.2%

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

5. Taylor expanded in b around 0 57.2%

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

6. Taylor expanded in a around 0 41.1%

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

   1. *-commutative 41.1%

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

8. Simplified 41.1%

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

9. Final simplification 41.1%

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

Alternative 10: 59.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. associate-/l* 80.2%

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

   2. remove-double-neg 80.2%

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

   3. sin-neg 80.2%

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

   4. neg-mul-1 80.2%

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

   5. associate-/r* 80.2%

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

   6. associate-/l* 80.3%

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

   7. *-commutative 80.3%

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

   8. associate-*l/ 80.3%

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

   9. associate-/l* 80.3%

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

   10. sin-neg 80.3%

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

   11. distribute-lft-neg-in 80.3%

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

   12. distribute-rgt-neg-in 80.3%

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

   13. associate-/l* 80.3%

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

   14. metadata-eval 80.3%

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

   15. /-rgt-identity 80.3%

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

   16. +-commutative 80.3%

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

3. Simplified 80.3%

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

5. Taylor expanded in a around 0 63.0%

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

   1. expm1-log1p-u 55.0%

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

   2. expm1-udef 34.4%

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

   3. quot-tan 34.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\tan b}\right)} - 1\right) \cdot r \]

7. Applied egg-rr 34.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\tan b\right)} - 1\right)} \cdot r \]
8. Step-by-step derivation

   1. expm1-def 55.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

   2. expm1-log1p 63.0%

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

9. Simplified 63.0%

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

10. Final simplification 63.0%

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

Alternative 11: 35.0% 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 80.3%

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

   1. associate-/l* 80.2%

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

   2. +-commutative 80.2%

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

3. Simplified 80.2%

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

5. Taylor expanded in b around 0 55.0%

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

   1. +-commutative 55.0%

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

   2. neg-mul-1 55.0%

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

   3. unsub-neg 55.0%

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

   4. fma-def 55.0%

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

   5. distribute-rgt-out-- 55.0%

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

   6. metadata-eval 55.0%

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

7. Simplified 55.0%

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

8. Taylor expanded in a around 0 37.8%

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

9. Final simplification 37.8%

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

Alternative 12: 34.5% 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 80.3%

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

   1. associate-/l* 80.2%

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

   2. remove-double-neg 80.2%

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

   3. sin-neg 80.2%

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

   4. neg-mul-1 80.2%

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

   5. associate-/r* 80.2%

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

   6. associate-/l* 80.3%

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

   7. *-commutative 80.3%

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

   8. associate-*l/ 80.3%

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

   9. associate-/l* 80.3%

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

   10. sin-neg 80.3%

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

   11. distribute-lft-neg-in 80.3%

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

   12. distribute-rgt-neg-in 80.3%

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

   13. associate-/l* 80.3%

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

   14. metadata-eval 80.3%

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

   15. /-rgt-identity 80.3%

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

   16. +-commutative 80.3%

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

3. Simplified 80.3%

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

5. Taylor expanded in b around 0 54.1%

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

6. Taylor expanded in a around 0 37.3%

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

   1. *-commutative 37.3%

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

8. Simplified 37.3%

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

9. Final simplification 37.3%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))