rsin A (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 14.9s

Alternatives: 17

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: 76.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. +-commutative 71.7%

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

3. Simplified 71.7%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. +-commutative 71.7%

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

3. Simplified 71.7%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. associate-/l* 71.6%

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

   2. remove-double-neg 71.6%

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

   3. remove-double-neg 71.6%

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

   4. +-commutative 71.6%

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

3. Simplified 71.6%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 4: 77.6% accurate, 0.7× speedup

Math

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

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

Derivation

1. Initial program 71.7%

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

   1. +-commutative 71.7%

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

3. Simplified 71.7%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.6%

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

6. Applied egg-rr 99.6%

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

   1. add-sqr-sqrt 44.7%

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

   2. sqrt-unprod 84.6%

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

   3. sqr-neg 84.6%

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

   4. sqrt-unprod 39.9%

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

   5. add-sqr-sqrt 71.4%

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

   6. sin-mult 73.0%

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

   7. div-sub 73.0%

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

   8. cos-diff 72.0%

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

   9. add-sqr-sqrt 40.2%

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

   10. sqrt-unprod 72.7%

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

   11. sqr-neg 72.7%

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

   12. sqrt-unprod 32.5%

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

   13. add-sqr-sqrt 73.6%

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

   14. cancel-sign-sub-inv 73.6%

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

   15. cos-sum 72.9%

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

8. Applied egg-rr 72.9%

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

   1. +-inverses 72.9%

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

10. Simplified 72.9%

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

    1. fma-undefine 72.9%

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

    2. +-rgt-identity 72.9%

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

12. Applied egg-rr 72.9%

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

Alternative 5: 77.6% 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 71.7%

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

   1. +-commutative 71.7%

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

3. Simplified 71.7%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.6%

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

6. Applied egg-rr 99.6%

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

   1. add-sqr-sqrt 44.7%

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

   2. sqrt-unprod 84.6%

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

   3. sqr-neg 84.6%

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

   4. sqrt-unprod 39.9%

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

   5. add-sqr-sqrt 71.4%

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

   6. sin-mult 73.0%

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

   7. div-sub 73.0%

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

   8. cos-diff 72.0%

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

   9. add-sqr-sqrt 40.2%

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

   10. sqrt-unprod 72.7%

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

   11. sqr-neg 72.7%

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

   12. sqrt-unprod 32.5%

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

   13. add-sqr-sqrt 73.6%

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

   14. cancel-sign-sub-inv 73.6%

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

   15. cos-sum 72.9%

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

8. Applied egg-rr 72.9%

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

   1. +-inverses 72.9%

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

10. Simplified 72.9%

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

11. Taylor expanded in r around 0 72.9%

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

    1. *-commutative 72.9%

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

    2. *-commutative 72.9%

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

    3. associate-*r/ 72.8%

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

    4. *-commutative 72.8%

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

13. Simplified 72.8%

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

14. Final simplification 72.8%

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

Alternative 6: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00055 \lor \neg \left(b \leq 0.52\right):\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00055) (not (<= b 0.52)))
   (/ r (/ (cos b) (sin b)))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00055) || !(b <= 0.52)) {
		tmp = r / (cos(b) / sin(b));
	} else {
		tmp = (r * b) / cos((b + 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 <= (-0.00055d0)) .or. (.not. (b <= 0.52d0))) then
        tmp = r / (cos(b) / sin(b))
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00055) || !(b <= 0.52)) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00055) or not (b <= 0.52):
		tmp = r / (math.cos(b) / math.sin(b))
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.00055) || !(b <= 0.52))
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.00055) || ~((b <= 0.52)))
		tmp = r / (cos(b) / sin(b));
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00055], N[Not[LessEqual[b, 0.52]], $MachinePrecision]], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.50000000000000033e-4 or 0.52000000000000002 < b

   1. Initial program 46.8%

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

      1. +-commutative 46.8%

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

   3. Simplified 46.8%

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

      1. cos-sum 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. fma-define 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. frac-2neg 99.3%

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

      2. fma-undefine 99.2%

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

      3. cancel-sign-sub-inv 99.2%

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

      4. cos-sum 46.8%

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

      5. div-inv 46.8%

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

      6. distribute-rgt-neg-in 46.8%

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

      7. add-sqr-sqrt 18.3%

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

      8. sqrt-unprod 22.8%

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

      9. sqr-neg 22.8%

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

      10. sqrt-unprod 4.4%

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

      11. add-sqr-sqrt 8.0%

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

   8. Applied egg-rr 8.0%

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

      1. associate-*r/ 8.0%

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

      2. *-rgt-identity 8.0%

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

      3. associate-/l* 8.0%

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

      4. distribute-neg-frac2 8.0%

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

      5. distribute-neg-frac 8.0%

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

   10. Simplified 8.0%

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

       1. clear-num 8.0%

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

       2. un-div-inv 8.0%

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

       3. cos-sum 2.3%

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

       4. fma-neg 2.3%

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

       5. fma-define 2.3%

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

       6. distribute-lft-neg-in 2.3%

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

       7. add-sqr-sqrt 0.8%

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

       8. sqrt-unprod 5.7%

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

       9. sqr-neg 5.7%

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

       10. sqrt-unprod 4.9%

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

       11. add-sqr-sqrt 8.4%

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

       12. cos-diff 7.7%

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

       13. add-sqr-sqrt 3.4%

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

       14. sqrt-unprod 31.9%

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

       15. sqr-neg 31.9%

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

       16. sqrt-unprod 28.4%

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

       17. add-sqr-sqrt 47.1%

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

   12. Applied egg-rr 47.1%

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

   13. Taylor expanded in a around 0 48.1%

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

   if -5.50000000000000033e-4 < b < 0.52000000000000002

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in b around 0 97.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00055 \lor \neg \left(b \leq 0.52\right):\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0006 \lor \neg \left(b \leq 0.52\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0006) (not (<= b 0.52)))
   (* (sin b) (/ r (cos b)))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0006) || !(b <= 0.52)) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = (r * b) / cos((b + 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 <= (-0.0006d0)) .or. (.not. (b <= 0.52d0))) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0006) || !(b <= 0.52)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0006) or not (b <= 0.52):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0006) || !(b <= 0.52))
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.0006) || ~((b <= 0.52)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0006], N[Not[LessEqual[b, 0.52]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.99999999999999947e-4 or 0.52000000000000002 < b

   1. Initial program 46.8%

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

      1. +-commutative 46.8%

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

   3. Simplified 46.8%

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

      1. cos-sum 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. fma-define 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-undefine 99.2%

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

   8. Applied egg-rr 99.2%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \color{blue}{e^{\mathsf{log1p}\left(\cos a\right)} - 1}, \left(-\sin b\right) \cdot \sin a\right)} \]
   9. Step-by-step derivation

      1. expm1-define 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in a around 0 48.1%

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

       1. *-commutative 48.1%

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

       2. *-lft-identity 48.1%

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

       3. times-frac 48.1%

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

       4. /-rgt-identity 48.1%

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

   13. Simplified 48.1%

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

   if -5.99999999999999947e-4 < b < 0.52000000000000002

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in b around 0 97.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0006 \lor \neg \left(b \leq 0.52\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00055 \lor \neg \left(b \leq 0.52\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00055) (not (<= b 0.52)))
   (* r (/ (sin b) (cos b)))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00055) || !(b <= 0.52)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = (r * b) / cos((b + 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 <= (-0.00055d0)) .or. (.not. (b <= 0.52d0))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00055) || !(b <= 0.52)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00055) or not (b <= 0.52):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.00055) || !(b <= 0.52))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.00055) || ~((b <= 0.52)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00055], N[Not[LessEqual[b, 0.52]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.50000000000000033e-4 or 0.52000000000000002 < b

   1. Initial program 46.8%

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

      1. associate-/l* 46.8%

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

      2. remove-double-neg 46.8%

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

      3. remove-double-neg 46.8%

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

      4. +-commutative 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in a around 0 48.1%

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

   if -5.50000000000000033e-4 < b < 0.52000000000000002

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in b around 0 97.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00055 \lor \neg \left(b \leq 0.52\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00065:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 0.52:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.00065)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 0.52) (/ (* r b) (cos (+ b a))) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.00065) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 0.52) {
		tmp = (r * b) / cos((b + 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 (b <= (-0.00065d0)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 0.52d0) then
        tmp = (r * b) / cos((b + 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 (b <= -0.00065) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 0.52) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.00065:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 0.52:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = (r * math.sin(b)) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.00065)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 0.52)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.00065)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 0.52)
		tmp = (r * b) / cos((b + a));
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.00065], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.52], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.4999999999999997e-4

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

   3. Simplified 49.7%

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

      1. cos-sum 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. fma-define 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. frac-2neg 99.4%

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

      2. fma-undefine 99.2%

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

      3. cancel-sign-sub-inv 99.2%

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

      4. cos-sum 49.7%

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

      5. div-inv 49.6%

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

      6. distribute-rgt-neg-in 49.6%

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

      7. add-sqr-sqrt 21.9%

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

      8. sqrt-unprod 24.9%

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

      9. sqr-neg 24.9%

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

      10. sqrt-unprod 2.9%

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

      11. add-sqr-sqrt 6.6%

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

   8. Applied egg-rr 6.6%

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

      1. associate-*r/ 6.6%

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

      2. *-rgt-identity 6.6%

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

      3. associate-/l* 6.6%

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

      4. distribute-neg-frac2 6.6%

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

      5. distribute-neg-frac 6.6%

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

   10. Simplified 6.6%

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

       1. clear-num 6.6%

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

       2. un-div-inv 6.6%

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

       3. cos-sum 2.4%

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

       4. fma-neg 2.4%

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

       5. fma-define 2.4%

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

       6. distribute-lft-neg-in 2.4%

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

       7. add-sqr-sqrt 1.0%

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

       8. sqrt-unprod 5.7%

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

       9. sqr-neg 5.7%

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

       10. sqrt-unprod 4.7%

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

       11. add-sqr-sqrt 8.7%

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

       12. cos-diff 6.6%

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

       13. add-sqr-sqrt 3.6%

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

       14. sqrt-unprod 31.4%

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

       15. sqr-neg 31.4%

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

       16. sqrt-unprod 27.7%

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

       17. add-sqr-sqrt 49.7%

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

   12. Applied egg-rr 49.7%

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

   13. Taylor expanded in a around 0 50.7%

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

   if -6.4999999999999997e-4 < b < 0.52000000000000002

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in b around 0 97.4%

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

   if 0.52000000000000002 < b

   1. Initial program 44.6%

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

      1. associate-/l* 44.5%

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

      2. remove-double-neg 44.5%

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

      3. remove-double-neg 44.5%

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

      4. +-commutative 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in a around 0 46.1%

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

      1. *-commutative 46.1%

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

   7. Simplified 46.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00065:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 0.52:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

3. Final simplification 71.7%

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

Alternative 11: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. +-commutative 71.7%

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

3. Simplified 71.7%

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

   1. *-commutative 71.7%

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

   2. associate-/l* 71.7%

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

6. Applied egg-rr 71.7%

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

Alternative 12: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. associate-/l* 71.6%

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

   2. remove-double-neg 71.6%

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

   3. remove-double-neg 71.6%

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

   4. +-commutative 71.6%

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

3. Simplified 71.6%

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

Alternative 13: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. associate-/l* 71.6%

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

   2. remove-double-neg 71.6%

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

   3. remove-double-neg 71.6%

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

   4. +-commutative 71.6%

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

3. Simplified 71.6%

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

5. Taylor expanded in b around 0 52.6%

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

Alternative 14: 53.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3450:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 3450.0) (/ (* r b) (cos a)) (* r (sin b))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 3450.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = r * sin(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 <= 3450.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = r * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 3450.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = r * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 3450.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = r * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 3450.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(r * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 3450.0)
		tmp = (r * b) / cos(a);
	else
		tmp = r * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 3450.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3450

   1. Initial program 82.8%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. remove-double-neg 82.7%

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

      4. +-commutative 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in b around 0 67.8%

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

   if 3450 < b

   1. Initial program 43.9%

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

      1. +-commutative 43.9%

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

   3. Simplified 43.9%

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

      1. associate-*r/ 43.8%

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

      2. add-cube-cbrt 43.1%

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

      3. associate-*l* 43.1%

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

      4. pow2 43.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2}} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \]

   6. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)} \]

   7. Taylor expanded in b around 0 10.3%

      \[\leadsto {\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\color{blue}{\cos a}}\right) \]

   8. Taylor expanded in a around 0 11.0%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3450:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3450:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b 3450.0) (* b (/ r (cos a))) (* r (sin b))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= 3450.0) {
		tmp = b * (r / cos(a));
	} else {
		tmp = r * sin(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 <= 3450.0d0) then
        tmp = b * (r / cos(a))
    else
        tmp = r * sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= 3450.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = r * Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= 3450.0:
		tmp = b * (r / math.cos(a))
	else:
		tmp = r * math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= 3450.0)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(r * sin(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= 3450.0)
		tmp = b * (r / cos(a));
	else
		tmp = r * sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, 3450.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3450

   1. Initial program 82.8%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. remove-double-neg 82.7%

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

      4. +-commutative 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in b around 0 67.8%

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

      1. associate-/l* 67.8%

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

   7. Simplified 67.8%

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

   if 3450 < b

   1. Initial program 43.9%

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

      1. +-commutative 43.9%

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

   3. Simplified 43.9%

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

      1. associate-*r/ 43.8%

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

      2. add-cube-cbrt 43.1%

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

      3. associate-*l* 43.1%

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

      4. pow2 43.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2}} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \]

   6. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)} \]

   7. Taylor expanded in b around 0 10.3%

      \[\leadsto {\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\color{blue}{\cos a}}\right) \]

   8. Taylor expanded in a around 0 11.0%

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

Alternative 16: 39.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. +-commutative 71.7%

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

3. Simplified 71.7%

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

   1. associate-*r/ 71.6%

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

   2. add-cube-cbrt 70.5%

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

   3. associate-*l* 70.5%

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

   4. pow2 70.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2}} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \]

6. Applied egg-rr 70.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)} \]

7. Taylor expanded in b around 0 51.9%

   \[\leadsto {\left(\sqrt[3]{r}\right)}^{2} \cdot \left(\sqrt[3]{r} \cdot \frac{\sin b}{\color{blue}{\cos a}}\right) \]

8. Taylor expanded in a around 0 37.0%

   \[\leadsto \color{blue}{r \cdot \sin b} \]
9. Add Preprocessing

Alternative 17: 35.1% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. associate-/l* 71.6%

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

   2. remove-double-neg 71.6%

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

   3. remove-double-neg 71.6%

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

   4. +-commutative 71.6%

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

3. Simplified 71.6%

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

5. Taylor expanded in b around 0 49.5%

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

   1. associate-/l* 49.5%

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

7. Simplified 49.5%

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

8. Taylor expanded in a around 0 33.6%

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

   1. *-commutative 33.6%

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

10. Simplified 33.6%

    \[\leadsto \color{blue}{r \cdot b} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))