rsin A (should all be same)

Percentage Accurate: 76.5% → 99.5%

Time: 18.2s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. cos-sum 99.6%

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

   2. sub-neg 99.6%

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

6. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

   3. fma-define 99.6%

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

   4. *-commutative 99.6%

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

8. Simplified 99.6%

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

Alternative 2: 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 77.4%

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

   1. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. cos-sum 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   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. 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 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. cos-sum 99.6%

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

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. Final simplification 99.5%

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

Alternative 5: 78.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. cos-sum 99.6%

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

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. Step-by-step derivation

   1. *-commutative 99.5%

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

   2. cos-mult 78.9%

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

   3. clear-num 78.8%

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

   4. +-commutative 78.8%

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

   5. cos-sum 79.1%

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

   6. sub-neg 79.1%

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

   7. distribute-rgt-neg-in 79.1%

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

   8. add-sqr-sqrt 40.1%

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

   9. sqrt-unprod 78.6%

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

   10. sqr-neg 78.6%

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

   11. sqrt-unprod 38.4%

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

   12. add-sqr-sqrt 78.4%

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

   13. cos-diff 78.7%

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

   14. cos-diff 78.4%

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

   15. *-commutative 78.4%

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

   16. *-commutative 78.4%

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

   17. cos-diff 78.7%

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

8. Applied egg-rr 78.7%

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

   1. associate-/r/ 78.8%

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

   2. metadata-eval 78.8%

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

   3. count-2 78.8%

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

10. Simplified 78.8%

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

Alternative 6: 77.5% 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 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. cos-sum 99.6%

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

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. Step-by-step derivation

   1. sin-mult 78.4%

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

   2. div-sub 78.4%

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

   3. cos-sum 79.2%

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

   4. sub-neg 79.2%

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

   5. distribute-rgt-neg-in 79.2%

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

   6. add-sqr-sqrt 40.2%

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

   7. sqrt-unprod 78.5%

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

   8. sqr-neg 78.5%

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

   9. sqrt-unprod 38.3%

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

   10. add-sqr-sqrt 77.6%

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

   11. cos-diff 78.3%

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

8. Applied egg-rr 78.3%

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

   1. +-inverses 78.3%

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

10. Simplified 78.3%

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

11. Taylor expanded in r around 0 78.3%

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

    1. times-frac 78.3%

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

13. Simplified 78.3%

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

Alternative 7: 77.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. cos-sum 99.6%

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

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. Step-by-step derivation

   1. sin-mult 78.4%

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

   2. div-sub 78.4%

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

   3. cos-sum 79.2%

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

   4. sub-neg 79.2%

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

   5. distribute-rgt-neg-in 79.2%

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

   6. add-sqr-sqrt 40.2%

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

   7. sqrt-unprod 78.5%

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

   8. sqr-neg 78.5%

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

   9. sqrt-unprod 38.3%

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

   10. add-sqr-sqrt 77.6%

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

   11. cos-diff 78.3%

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

8. Applied egg-rr 78.3%

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

   1. +-inverses 78.3%

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

10. Simplified 78.3%

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

    1. --rgt-identity 78.3%

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

12. Applied egg-rr 78.3%

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

13. Final simplification 78.3%

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

Alternative 8: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{-7} \lor \neg \left(a \leq 620000000\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -1.85e-7) (not (<= a 620000000.0)))
   (/ (sin b) (/ (cos a) r))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.85e-7) || !(a <= 620000000.0)) {
		tmp = sin(b) / (cos(a) / r);
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.85d-7)) .or. (.not. (a <= 620000000.0d0))) then
        tmp = sin(b) / (cos(a) / r)
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.85e-7) || !(a <= 620000000.0)) {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -1.85e-7) or not (a <= 620000000.0):
		tmp = math.sin(b) / (math.cos(a) / r)
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -1.85e-7) || !(a <= 620000000.0))
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -1.85e-7) || ~((a <= 620000000.0)))
		tmp = sin(b) / (cos(a) / r);
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -1.85e-7], N[Not[LessEqual[a, 620000000.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.85000000000000002e-7 or 6.2e8 < a

   1. Initial program 59.5%

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

      1. associate-/l* 59.4%

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

      2. remove-double-neg 59.4%

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

      3. remove-double-neg 59.4%

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

      4. +-commutative 59.4%

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

   3. Simplified 59.4%

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

      1. associate-*r/ 59.5%

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

      2. clear-num 59.4%

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

   6. Applied egg-rr 59.4%

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

   7. Taylor expanded in b around 0 58.9%

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{a}}{r \cdot \sin b}} \]
   8. Step-by-step derivation

      1. frac-2neg 58.9%

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

      2. metadata-eval 58.9%

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

      3. div-inv 58.9%

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

      4. distribute-neg-frac2 58.9%

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

      5. distribute-rgt-neg-in 58.9%

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

      6. add-sqr-sqrt 27.3%

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

      7. sqrt-unprod 27.8%

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

      8. sqr-neg 27.8%

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

      9. sqrt-unprod 9.8%

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

      10. add-sqr-sqrt 19.3%

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

      11. clear-num 19.3%

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

   9. Applied egg-rr 19.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{r \cdot \sin b}{\cos a}} \]
   10. Step-by-step derivation

       1. neg-mul-1 19.3%

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

       2. *-commutative 19.3%

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

       3. associate-*r/ 19.3%

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

       4. distribute-rgt-neg-in 19.3%

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

   11. Simplified 19.3%

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

       1. add-sqr-sqrt 8.3%

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

       2. sqrt-unprod 28.9%

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

       3. sqr-neg 28.9%

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

       4. sqrt-unprod 29.2%

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

       5. add-sqr-sqrt 59.0%

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

       6. clear-num 58.9%

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

       7. un-div-inv 59.0%

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

   13. Applied egg-rr 59.0%

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

   if -1.85000000000000002e-7 < a < 6.2e8

   1. Initial program 97.5%

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

      1. associate-/l* 97.6%

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

      2. remove-double-neg 97.6%

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

      3. remove-double-neg 97.6%

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

      4. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in a around 0 97.5%

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

      1. associate-/l* 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{-7} \lor \neg \left(a \leq 620000000\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{-7} \lor \neg \left(a \leq 620000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -1.85e-7) (not (<= a 620000000.0)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.85e-7) || !(a <= 620000000.0)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.85d-7)) .or. (.not. (a <= 620000000.0d0))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.85e-7) || !(a <= 620000000.0)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -1.85e-7) or not (a <= 620000000.0):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -1.85e-7) || !(a <= 620000000.0))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -1.85e-7) || ~((a <= 620000000.0)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -1.85e-7], N[Not[LessEqual[a, 620000000.0]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.85000000000000002e-7 or 6.2e8 < a

   1. Initial program 59.5%

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

      1. associate-/l* 59.4%

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

      2. remove-double-neg 59.4%

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

      3. remove-double-neg 59.4%

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

      4. +-commutative 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in b around 0 58.9%

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

   if -1.85000000000000002e-7 < a < 6.2e8

   1. Initial program 97.5%

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

      1. associate-/l* 97.6%

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

      2. remove-double-neg 97.6%

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

      3. remove-double-neg 97.6%

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

      4. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in a around 0 97.5%

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

      1. associate-/l* 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{-7} \lor \neg \left(a \leq 620000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. cos-sum 99.6%

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

   2. sub-neg 99.6%

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

6. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

   3. fma-define 99.6%

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

   4. *-commutative 99.6%

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

8. Simplified 99.6%

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

   1. associate-/l* 99.5%

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

   2. fma-undefine 99.5%

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

   3. add-sqr-sqrt 48.6%

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

   4. sqrt-unprod 86.8%

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

   5. sqr-neg 86.8%

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

   6. sqrt-unprod 38.2%

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

   7. add-sqr-sqrt 77.4%

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

   8. *-commutative 77.4%

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

   9. +-commutative 77.4%

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

   10. cos-diff 77.6%

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

10. Applied egg-rr 77.6%

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

    1. *-commutative 77.6%

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

    2. associate-*l/ 77.6%

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

    3. associate-*r/ 77.6%

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

12. Simplified 77.6%

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

Alternative 11: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

Alternative 12: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

5. Taylor expanded in b around 0 56.1%

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

Alternative 13: 55.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -21000:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \mathbf{elif}\;b \leq 115:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -21000.0)
   (/ 1.0 (/ 1.0 (* r (sin b))))
   (if (<= b 115.0) (* b (/ r (cos a))) (* r (- (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -21000.0) {
		tmp = 1.0 / (1.0 / (r * sin(b)));
	} else if (b <= 115.0) {
		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 <= (-21000.0d0)) then
        tmp = 1.0d0 / (1.0d0 / (r * sin(b)))
    else if (b <= 115.0d0) 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 <= -21000.0) {
		tmp = 1.0 / (1.0 / (r * Math.sin(b)));
	} else if (b <= 115.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = r * -Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -21000.0:
		tmp = 1.0 / (1.0 / (r * math.sin(b)))
	elif b <= 115.0:
		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 <= -21000.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(r * sin(b))));
	elseif (b <= 115.0)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(r * Float64(-sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -21000.0)
		tmp = 1.0 / (1.0 / (r * sin(b)));
	elseif (b <= 115.0)
		tmp = b * (r / cos(a));
	else
		tmp = r * -sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -21000.0], N[(1.0 / N[(1.0 / N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 115.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -21000

   1. Initial program 53.4%

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

      1. associate-/l* 53.4%

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

      2. remove-double-neg 53.4%

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

      3. remove-double-neg 53.4%

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

      4. +-commutative 53.4%

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

   3. Simplified 53.4%

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

      1. associate-*r/ 53.4%

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

      2. clear-num 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in b around 0 12.2%

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

   8. Taylor expanded in a around 0 13.0%

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

   if -21000 < b < 115

   1. Initial program 99.2%

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

      1. associate-/l* 99.2%

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

      2. remove-double-neg 99.2%

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

      3. remove-double-neg 99.2%

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

      4. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in b around 0 98.2%

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

      1. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

   if 115 < b

   1. Initial program 54.3%

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

      1. associate-/l* 54.4%

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

      2. remove-double-neg 54.4%

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

      3. remove-double-neg 54.4%

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

      4. +-commutative 54.4%

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

   3. Simplified 54.4%

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

      1. associate-*r/ 54.3%

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

      2. clear-num 54.1%

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

   6. Applied egg-rr 54.1%

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

   7. Taylor expanded in b around 0 9.4%

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{a}}{r \cdot \sin b}} \]
   8. Step-by-step derivation

      1. frac-2neg 9.4%

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

      2. metadata-eval 9.4%

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

      3. div-inv 9.4%

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

      4. distribute-neg-frac2 9.4%

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

      5. distribute-rgt-neg-in 9.4%

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

      6. add-sqr-sqrt 4.6%

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

      7. sqrt-unprod 12.4%

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

      8. sqr-neg 12.4%

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

      9. sqrt-unprod 7.8%

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

      10. add-sqr-sqrt 13.9%

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

      11. clear-num 13.9%

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

   9. Applied egg-rr 13.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{r \cdot \sin b}{\cos a}} \]
   10. Step-by-step derivation

       1. neg-mul-1 13.9%

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

       2. *-commutative 13.9%

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

       3. associate-*r/ 13.9%

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

       4. distribute-rgt-neg-in 13.9%

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

   11. Simplified 13.9%

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

   12. Taylor expanded in a around 0 15.0%

       \[\leadsto \sin b \cdot \left(-\color{blue}{r}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -21000:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \mathbf{elif}\;b \leq 115:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -22000:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 390:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -22000.0)
   (* r (sin b))
   (if (<= b 390.0) (* b (/ r (cos a))) (* r (- (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -22000.0) {
		tmp = r * sin(b);
	} else if (b <= 390.0) {
		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 <= (-22000.0d0)) then
        tmp = r * sin(b)
    else if (b <= 390.0d0) 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 <= -22000.0) {
		tmp = r * Math.sin(b);
	} else if (b <= 390.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = r * -Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -22000.0:
		tmp = r * math.sin(b)
	elif b <= 390.0:
		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 <= -22000.0)
		tmp = Float64(r * sin(b));
	elseif (b <= 390.0)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(r * Float64(-sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -22000.0)
		tmp = r * sin(b);
	elseif (b <= 390.0)
		tmp = b * (r / cos(a));
	else
		tmp = r * -sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -22000.0], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 390.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -22000

   1. Initial program 53.4%

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

      1. associate-/l* 53.4%

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

      2. remove-double-neg 53.4%

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

      3. remove-double-neg 53.4%

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

      4. +-commutative 53.4%

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

   3. Simplified 53.4%

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

      1. associate-*r/ 53.4%

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

      2. clear-num 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in b around 0 12.2%

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

   8. Taylor expanded in a around 0 13.0%

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

   if -22000 < b < 390

   1. Initial program 99.2%

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

      1. associate-/l* 99.2%

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

      2. remove-double-neg 99.2%

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

      3. remove-double-neg 99.2%

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

      4. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in b around 0 98.2%

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

      1. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

   if 390 < b

   1. Initial program 54.3%

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

      1. associate-/l* 54.4%

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

      2. remove-double-neg 54.4%

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

      3. remove-double-neg 54.4%

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

      4. +-commutative 54.4%

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

   3. Simplified 54.4%

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

      1. associate-*r/ 54.3%

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

      2. clear-num 54.1%

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

   6. Applied egg-rr 54.1%

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

   7. Taylor expanded in b around 0 9.4%

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{a}}{r \cdot \sin b}} \]
   8. Step-by-step derivation

      1. frac-2neg 9.4%

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

      2. metadata-eval 9.4%

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

      3. div-inv 9.4%

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

      4. distribute-neg-frac2 9.4%

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

      5. distribute-rgt-neg-in 9.4%

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

      6. add-sqr-sqrt 4.6%

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

      7. sqrt-unprod 12.4%

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

      8. sqr-neg 12.4%

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

      9. sqrt-unprod 7.8%

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

      10. add-sqr-sqrt 13.9%

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

      11. clear-num 13.9%

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

   9. Applied egg-rr 13.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{r \cdot \sin b}{\cos a}} \]
   10. Step-by-step derivation

       1. neg-mul-1 13.9%

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

       2. *-commutative 13.9%

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

       3. associate-*r/ 13.9%

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

       4. distribute-rgt-neg-in 13.9%

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

   11. Simplified 13.9%

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

   12. Taylor expanded in a around 0 15.0%

       \[\leadsto \sin b \cdot \left(-\color{blue}{r}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -22000:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 390:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

   1. associate-*r/ 77.4%

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

   2. clear-num 77.2%

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

6. Applied egg-rr 77.2%

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

7. Taylor expanded in b around 0 55.9%

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

8. Taylor expanded in a around 0 37.6%

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

Alternative 16: 34.9% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 77.4%

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

   2. remove-double-neg 77.4%

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

   3. remove-double-neg 77.4%

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

   4. +-commutative 77.4%

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

3. Simplified 77.4%

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

5. Taylor expanded in b around 0 52.8%

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

   1. associate-/l* 52.8%

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

7. Simplified 52.8%

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

8. Taylor expanded in a around 0 34.2%

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

   1. *-commutative 34.2%

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

10. Simplified 34.2%

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

Reproduce

herbie shell --seed 2024156 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))