rsin A (should all be same)

Percentage Accurate: 76.5% → 99.5%

Time: 15.9s

Alternatives: 23

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

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos b \cdot \cos a\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (- 0.0 (sin b)) (sin a) (* (cos b) (cos a)))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / fma((0.0 - sin(b)), sin(a), (cos(b) * cos(a)));
}

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(Float64(0.0 - sin(b)), sin(a), Float64(cos(b) * cos(a))))
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[(0.0 - N[Sin[b], $MachinePrecision]), $MachinePrecision] * N[Sin[a], $MachinePrecision] + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 78.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 78.5%

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

   1. cos-sum N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos b \cdot \cos a + \color{blue}{\left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right) + \color{blue}{\cos b \cdot \cos a}\right)\right) \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\left(\mathsf{neg}\left(\sin b\right)\right) \cdot \sin a + \color{blue}{\cos b} \cdot \cos a\right)\right) \]

   5. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\mathsf{fma}\left(\mathsf{neg}\left(\sin b\right), \color{blue}{\sin a}, \cos b \cdot \cos a\right)\right)\right) \]

   6. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\sin b\right)\right), \color{blue}{\sin a}, \left(\cos b \cdot \cos a\right)\right)\right) \]

   7. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\left(0 - \sin b\right), \sin \color{blue}{a}, \left(\cos b \cdot \cos a\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \sin b\right), \sin \color{blue}{a}, \left(\cos b \cdot \cos a\right)\right)\right) \]

   9. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(b\right)\right), \sin a, \left(\cos b \cdot \cos a\right)\right)\right) \]

   10. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(b\right)\right), \mathsf{sin.f64}\left(a\right), \left(\cos b \cdot \cos a\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(b\right)\right), \mathsf{sin.f64}\left(a\right), \mathsf{*.f64}\left(\cos b, \cos a\right)\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(b\right)\right), \mathsf{sin.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \cos a\right)\right)\right) \]

   13. cos-lowering-cos.f64 99.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(b\right)\right), \mathsf{sin.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

6. Applied egg-rr 99.6%

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

   1. remove-double-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{1}{\frac{1}{\sin b}}\right)\right), \mathsf{sin.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(1, \left(\frac{1}{\sin b}\right)\right)\right), \mathsf{sin.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \sin b\right)\right)\right), \mathsf{sin.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

   4. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right), \mathsf{sin.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(0 - \color{blue}{\frac{1}{\frac{1}{\sin b}}}, \sin a, \cos b \cdot \cos a\right)} \]
9. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\frac{1}{\sin b}}\right)\right), \mathsf{sin.f64}\left(\color{blue}{a}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

   2. remove-double-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\sin b\right)\right), \mathsf{sin.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{neg.f64}\left(\sin b\right), \mathsf{sin.f64}\left(\color{blue}{a}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

   4. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(b\right)\right), \mathsf{sin.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos b \cdot \cos a\right)} \]
12. 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 78.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 78.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 78.5%

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

   1. cos-sum N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}\right)\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\left(\cos b \cdot \cos a\right), \color{blue}{\left(\sin b \cdot \sin a\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos b, \cos a\right), \left(\color{blue}{\sin b} \cdot \sin a\right)\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \cos a\right), \left(\sin \color{blue}{b} \cdot \sin a\right)\right)\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \left(\sin b \cdot \sin a\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\sin b, \color{blue}{\sin a}\right)\right)\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin \color{blue}{a}\right)\right)\right) \]

   8. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right) \]

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. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(N[Sin[b], $MachinePrecision] * N[(r / 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 78.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 78.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 78.5%

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

   1. associate-*l/ N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

   6. sin-lowering-sin.f64 78.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

6. Applied egg-rr 78.5%

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

   1. cos-sum N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \left(\cos b \cdot \cos a - \sin b \cdot \sin a\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\left(\cos b \cdot \cos a\right), \left(\sin b \cdot \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos b, \cos a\right), \left(\sin b \cdot \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \cos a\right), \left(\sin b \cdot \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \left(\sin b \cdot \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\sin b, \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   8. sin-lowering-sin.f64 99.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= a -6.8e-5)
     (* r (/ (sin b) (cos a)))
     (if (<= a 2.3e+15) (/ t_0 (cos b)) (/ t_0 (cos a))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (a <= -6.8e-5) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 2.3e+15) {
		tmp = t_0 / cos(b);
	} else {
		tmp = t_0 / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (a <= (-6.8d-5)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 2.3d+15) then
        tmp = t_0 / cos(b)
    else
        tmp = t_0 / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (a <= -6.8e-5) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 2.3e+15) {
		tmp = t_0 / Math.cos(b);
	} else {
		tmp = t_0 / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if a <= -6.8e-5:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 2.3e+15:
		tmp = t_0 / math.cos(b)
	else:
		tmp = t_0 / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (a <= -6.8e-5)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 2.3e+15)
		tmp = Float64(t_0 / cos(b));
	else
		tmp = Float64(t_0 / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (a <= -6.8e-5)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 2.3e+15)
		tmp = t_0 / cos(b);
	else
		tmp = t_0 / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.8e-5], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.3e+15], N[(t$95$0 / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.7999999999999999e-5

   1. Initial program 59.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 59.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 59.8%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 59.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 59.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 59.0%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

       3. associate-/r/ N/A

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

       4. div-inv N/A

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

       5. associate-*l/ N/A

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

       6. remove-double-div N/A

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

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin b \cdot \frac{1}{\cos a}\right), \color{blue}{r}\right) \]

       8. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin b}{\cos a}\right), r\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin b, \cos a\right), r\right) \]

       10. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \cos a\right), r\right) \]

       11. cos-lowering-cos.f64 59.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), r\right) \]

   11. Applied egg-rr 59.1%

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

   if -6.7999999999999999e-5 < a < 2.3e15

   1. Initial program 98.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 98.0%

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

   5. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos b}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{b}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos b\right) \]

      4. cos-lowering-cos.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(b\right)\right) \]

   7. Simplified 98.0%

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

   if 2.3e15 < a

   1. Initial program 57.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 57.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \color{blue}{\cos a}\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 56.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 56.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000235:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000235)
   (* r (/ (sin b) (cos a)))
   (if (<= a 2.3e+15) (* (sin b) (/ r (cos b))) (/ (* r (sin b)) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000235) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 2.3e+15) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = (r * sin(b)) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.000235d0)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 2.3d+15) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = (r * sin(b)) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000235) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 2.3e+15) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000235:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 2.3e+15:
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = (r * math.sin(b)) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000235)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 2.3e+15)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000235)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 2.3e+15)
		tmp = sin(b) * (r / cos(b));
	else
		tmp = (r * sin(b)) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000235], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.3e+15], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.34999999999999993e-4

   1. Initial program 59.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 59.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 59.8%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 59.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 59.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 59.0%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

       3. associate-/r/ N/A

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

       4. div-inv N/A

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

       5. associate-*l/ N/A

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

       6. remove-double-div N/A

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

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin b \cdot \frac{1}{\cos a}\right), \color{blue}{r}\right) \]

       8. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin b}{\cos a}\right), r\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin b, \cos a\right), r\right) \]

       10. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \cos a\right), r\right) \]

       11. cos-lowering-cos.f64 59.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), r\right) \]

   11. Applied egg-rr 59.1%

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

   if -2.34999999999999993e-4 < a < 2.3e15

   1. Initial program 98.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 98.0%

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

      1. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

      6. sin-lowering-sin.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{r}{\cos b}\right)}, \mathsf{sin.f64}\left(b\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos b\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right) \]

      2. cos-lowering-cos.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(b\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   9. Simplified 98.0%

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

   if 2.3e15 < a

   1. Initial program 57.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 57.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \color{blue}{\cos a}\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 56.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 56.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000235:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000155:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000155)
   (* r (/ (sin b) (cos a)))
   (if (<= a 2.3e+15) (* (sin b) (/ r (cos b))) (/ r (/ (cos a) (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000155) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 2.3e+15) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = r / (cos(a) / 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 (a <= (-0.000155d0)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 2.3d+15) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = r / (cos(a) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000155) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 2.3e+15) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = r / (Math.cos(a) / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000155:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 2.3e+15:
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = r / (math.cos(a) / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000155)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 2.3e+15)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000155)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 2.3e+15)
		tmp = sin(b) * (r / cos(b));
	else
		tmp = r / (cos(a) / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000155], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.3e+15], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.55e-4

   1. Initial program 59.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 59.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 59.8%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 59.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 59.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 59.0%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

       3. associate-/r/ N/A

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

       4. div-inv N/A

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

       5. associate-*l/ N/A

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

       6. remove-double-div N/A

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

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin b \cdot \frac{1}{\cos a}\right), \color{blue}{r}\right) \]

       8. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin b}{\cos a}\right), r\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin b, \cos a\right), r\right) \]

       10. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \cos a\right), r\right) \]

       11. cos-lowering-cos.f64 59.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), r\right) \]

   11. Applied egg-rr 59.1%

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

   if -1.55e-4 < a < 2.3e15

   1. Initial program 98.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 98.0%

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

      1. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

      6. sin-lowering-sin.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{r}{\cos b}\right)}, \mathsf{sin.f64}\left(b\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos b\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right) \]

      2. cos-lowering-cos.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(b\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   9. Simplified 98.0%

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

   if 2.3e15 < a

   1. Initial program 57.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 57.0%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 56.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 56.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 56.5%

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

       1. frac-times N/A

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

       2. *-lft-identity N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(r, \color{blue}{\left(\cos a \cdot \frac{1}{\sin b}\right)}\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(r, \left(\frac{\cos a}{\color{blue}{\sin b}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\cos a, \color{blue}{\sin b}\right)\right) \]

       6. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \sin \color{blue}{b}\right)\right) \]

       7. sin-lowering-sin.f64 56.6%

          \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{sin.f64}\left(b\right)\right)\right) \]

   11. Applied egg-rr 56.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000155:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos a)))))
   (if (<= a -7.5e-6) t_0 (if (<= a 2.3e+15) (* (sin b) (/ r (cos b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(a));
	double tmp;
	if (a <= -7.5e-6) {
		tmp = t_0;
	} else if (a <= 2.3e+15) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * (sin(b) / cos(a))
    if (a <= (-7.5d-6)) then
        tmp = t_0
    else if (a <= 2.3d+15) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * (Math.sin(b) / Math.cos(a));
	double tmp;
	if (a <= -7.5e-6) {
		tmp = t_0;
	} else if (a <= 2.3e+15) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / math.cos(a))
	tmp = 0
	if a <= -7.5e-6:
		tmp = t_0
	elif a <= 2.3e+15:
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(a)))
	tmp = 0.0
	if (a <= -7.5e-6)
		tmp = t_0;
	elseif (a <= 2.3e+15)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / cos(a));
	tmp = 0.0;
	if (a <= -7.5e-6)
		tmp = t_0;
	elseif (a <= 2.3e+15)
		tmp = sin(b) * (r / cos(b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.5e-6], t$95$0, If[LessEqual[a, 2.3e+15], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.50000000000000019e-6 or 2.3e15 < a

   1. Initial program 58.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 58.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 58.4%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 58.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 58.3%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 57.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 57.8%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

       3. associate-/r/ N/A

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

       4. div-inv N/A

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

       5. associate-*l/ N/A

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

       6. remove-double-div N/A

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

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin b \cdot \frac{1}{\cos a}\right), \color{blue}{r}\right) \]

       8. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin b}{\cos a}\right), r\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin b, \cos a\right), r\right) \]

       10. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \cos a\right), r\right) \]

       11. cos-lowering-cos.f64 57.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), r\right) \]

   11. Applied egg-rr 57.8%

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

   if -7.50000000000000019e-6 < a < 2.3e15

   1. Initial program 98.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 98.0%

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

      1. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

      6. sin-lowering-sin.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{r}{\cos b}\right)}, \mathsf{sin.f64}\left(b\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos b\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right) \]

      2. cos-lowering-cos.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(b\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   9. Simplified 98.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos a)))))
   (if (<= a -3.8e-5) t_0 (if (<= a 2.3e+15) (* (sin b) (/ r (cos b))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(a));
	double tmp;
	if (a <= -3.8e-5) {
		tmp = t_0;
	} else if (a <= 2.3e+15) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * (r / cos(a))
    if (a <= (-3.8d-5)) then
        tmp = t_0
    else if (a <= 2.3d+15) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * (r / Math.cos(a));
	double tmp;
	if (a <= -3.8e-5) {
		tmp = t_0;
	} else if (a <= 2.3e+15) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / math.cos(a))
	tmp = 0
	if a <= -3.8e-5:
		tmp = t_0
	elif a <= 2.3e+15:
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(a)))
	tmp = 0.0
	if (a <= -3.8e-5)
		tmp = t_0;
	elseif (a <= 2.3e+15)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * (r / cos(a));
	tmp = 0.0;
	if (a <= -3.8e-5)
		tmp = t_0;
	elseif (a <= 2.3e+15)
		tmp = sin(b) * (r / cos(b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.8e-5], t$95$0, If[LessEqual[a, 2.3e+15], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8000000000000002e-5 or 2.3e15 < a

   1. Initial program 58.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 58.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 58.4%

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

      1. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

      6. sin-lowering-sin.f64 58.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. Applied egg-rr 58.3%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{r}{\cos a}\right)}, \mathsf{sin.f64}\left(b\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right) \]

      2. cos-lowering-cos.f64 57.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   9. Simplified 57.8%

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

   if -3.8000000000000002e-5 < a < 2.3e15

   1. Initial program 98.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 98.0%

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

      1. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

      6. sin-lowering-sin.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{r}{\cos b}\right)}, \mathsf{sin.f64}\left(b\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos b\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right) \]

      2. cos-lowering-cos.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(b\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   9. Simplified 98.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% 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 78.5%

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

3. Final simplification 78.5%

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

Alternative 10: 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 78.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 78.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 78.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin b}{\cos \left(b + a\right)}\right), \color{blue}{r}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin b, \cos \left(b + a\right)\right), r\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \cos \left(b + a\right)\right), r\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), r\right) \]

   7. +-lowering-+.f64 78.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), r\right) \]

6. Applied egg-rr 78.5%

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

7. Final simplification 78.5%

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

Alternative 11: 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 78.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 78.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 78.5%

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

   1. associate-*l/ N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

   6. sin-lowering-sin.f64 78.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

6. Applied egg-rr 78.5%

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

7. Final simplification 78.5%

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

Alternative 12: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 78.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 78.5%

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

   1. associate-*l/ N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

   6. sin-lowering-sin.f64 78.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

6. Applied egg-rr 78.5%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{r}{\cos a}\right)}, \mathsf{sin.f64}\left(b\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos a\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right) \]

   2. cos-lowering-cos.f64 54.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

9. Simplified 54.0%

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

10. Final simplification 54.0%

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

Alternative 13: 55.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+27}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + \left(b \cdot b\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot \left(0.008333333333333333 + \left(b \cdot b\right) \cdot -0.0001984126984126984\right)\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -9e+27)
   (* r (sin b))
   (if (<= b 3.8e+25)
     (/
      (*
       r
       (*
        b
        (+
         1.0
         (*
          (* b b)
          (+
           -0.16666666666666666
           (*
            (* b b)
            (+ 0.008333333333333333 (* (* b b) -0.0001984126984126984))))))))
      (cos (+ b a)))
     (/ r (/ 1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -9e+27) {
		tmp = r * sin(b);
	} else if (b <= 3.8e+25) {
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * (0.008333333333333333 + ((b * b) * -0.0001984126984126984)))))))) / cos((b + a));
	} else {
		tmp = r / (1.0 / 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 <= (-9d+27)) then
        tmp = r * sin(b)
    else if (b <= 3.8d+25) then
        tmp = (r * (b * (1.0d0 + ((b * b) * ((-0.16666666666666666d0) + ((b * b) * (0.008333333333333333d0 + ((b * b) * (-0.0001984126984126984d0))))))))) / cos((b + a))
    else
        tmp = r / (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -9e+27) {
		tmp = r * Math.sin(b);
	} else if (b <= 3.8e+25) {
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * (0.008333333333333333 + ((b * b) * -0.0001984126984126984)))))))) / Math.cos((b + a));
	} else {
		tmp = r / (1.0 / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -9e+27:
		tmp = r * math.sin(b)
	elif b <= 3.8e+25:
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * (0.008333333333333333 + ((b * b) * -0.0001984126984126984)))))))) / math.cos((b + a))
	else:
		tmp = r / (1.0 / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -9e+27)
		tmp = Float64(r * sin(b));
	elseif (b <= 3.8e+25)
		tmp = Float64(Float64(r * Float64(b * Float64(1.0 + Float64(Float64(b * b) * Float64(-0.16666666666666666 + Float64(Float64(b * b) * Float64(0.008333333333333333 + Float64(Float64(b * b) * -0.0001984126984126984)))))))) / cos(Float64(b + a)));
	else
		tmp = Float64(r / Float64(1.0 / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -9e+27)
		tmp = r * sin(b);
	elseif (b <= 3.8e+25)
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * (0.008333333333333333 + ((b * b) * -0.0001984126984126984)))))))) / cos((b + a));
	else
		tmp = r / (1.0 / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -9e+27], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e+25], N[(N[(r * N[(b * N[(1.0 + N[(N[(b * b), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(b * b), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.9999999999999998e27

   1. Initial program 60.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 60.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 60.1%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 60.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 10.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 10.2%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 10.5%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 10.5%

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

   if -8.9999999999999998e27 < b < 3.8e25

   1. Initial program 97.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 97.9%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{-1}{6} + {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({b}^{2}\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {b}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({b}^{2} \cdot \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({b}^{2}\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(b \cdot b\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      17. *-lowering-*.f64 93.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 93.8%

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + \left(b \cdot b\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot \left(0.008333333333333333 + \left(b \cdot b\right) \cdot -0.0001984126984126984\right)\right)\right)\right)}}{\cos \left(b + a\right)} \]

   if 3.8e25 < b

   1. Initial program 54.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 54.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 54.0%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 53.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 11.4%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 10.9%

       \[\leadsto \color{blue}{1} \cdot \frac{r}{\frac{1}{\sin b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+27}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + \left(b \cdot b\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot \left(0.008333333333333333 + \left(b \cdot b\right) \cdot -0.0001984126984126984\right)\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+17}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot \left(r + \left(b \cdot b\right) \cdot \left(r \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2e+17)
   (* r (sin b))
   (if (<= b 15.5)
     (/
      (*
       b
       (+
        r
        (*
         (* b b)
         (* r (+ -0.16666666666666666 (* (* b b) 0.008333333333333333))))))
      (cos (+ b a)))
     (/ r (/ 1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2e+17) {
		tmp = r * sin(b);
	} else if (b <= 15.5) {
		tmp = (b * (r + ((b * b) * (r * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / cos((b + a));
	} else {
		tmp = r / (1.0 / 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 <= (-2d+17)) then
        tmp = r * sin(b)
    else if (b <= 15.5d0) then
        tmp = (b * (r + ((b * b) * (r * ((-0.16666666666666666d0) + ((b * b) * 0.008333333333333333d0)))))) / cos((b + a))
    else
        tmp = r / (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -2e+17) {
		tmp = r * Math.sin(b);
	} else if (b <= 15.5) {
		tmp = (b * (r + ((b * b) * (r * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / Math.cos((b + a));
	} else {
		tmp = r / (1.0 / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2e+17:
		tmp = r * math.sin(b)
	elif b <= 15.5:
		tmp = (b * (r + ((b * b) * (r * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / math.cos((b + a))
	else:
		tmp = r / (1.0 / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2e+17)
		tmp = Float64(r * sin(b));
	elseif (b <= 15.5)
		tmp = Float64(Float64(b * Float64(r + Float64(Float64(b * b) * Float64(r * Float64(-0.16666666666666666 + Float64(Float64(b * b) * 0.008333333333333333)))))) / cos(Float64(b + a)));
	else
		tmp = Float64(r / Float64(1.0 / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2e+17)
		tmp = r * sin(b);
	elseif (b <= 15.5)
		tmp = (b * (r + ((b * b) * (r * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / cos((b + a));
	else
		tmp = r / (1.0 / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2e+17], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 15.5], N[(N[(b * N[(r + N[(N[(b * b), $MachinePrecision] * N[(r * N[(-0.16666666666666666 + N[(N[(b * b), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2e17

   1. Initial program 60.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 60.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 60.8%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 60.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 10.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 10.0%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 10.4%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 10.4%

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

   if -2e17 < b < 15.5

   1. Initial program 98.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 98.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 98.4%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(b \cdot \left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)\right)}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, a\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \left({b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\left({b}^{2}\right), \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{-1}{6} \cdot r + \left(\frac{1}{120} \cdot {b}^{2}\right) \cdot r\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(r \cdot \left(\frac{-1}{6} + \frac{1}{120} \cdot {b}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(r \cdot \left(\frac{1}{120} \cdot {b}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(r \cdot \left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(r \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(r, \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(r, \left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(r, \left(\frac{1}{120} \cdot {b}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(r, \left(\frac{-1}{6} + \frac{1}{120} \cdot {b}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {b}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{-1}{6}, \left({b}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({b}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(b \cdot b\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      19. *-lowering-*.f64 95.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 95.7%

      \[\leadsto \frac{\color{blue}{b \cdot \left(r + \left(b \cdot b\right) \cdot \left(r \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)\right)}}{\cos \left(b + a\right)} \]

   if 15.5 < b

   1. Initial program 54.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 54.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 54.2%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 54.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 54.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 11.5%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 10.6%

       \[\leadsto \color{blue}{1} \cdot \frac{r}{\frac{1}{\sin b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+17}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot \left(r + \left(b \cdot b\right) \cdot \left(r \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 11:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot \left(-0.16666666666666666 + b \cdot \left(b \cdot 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -8.5e+16)
   (* r (sin b))
   (if (<= b 11.0)
     (*
      (/ r (cos (+ b a)))
      (*
       b
       (+
        1.0
        (*
         b
         (* b (+ -0.16666666666666666 (* b (* b 0.008333333333333333))))))))
     (/ r (/ 1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -8.5e+16) {
		tmp = r * sin(b);
	} else if (b <= 11.0) {
		tmp = (r / cos((b + a))) * (b * (1.0 + (b * (b * (-0.16666666666666666 + (b * (b * 0.008333333333333333)))))));
	} else {
		tmp = r / (1.0 / 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 <= (-8.5d+16)) then
        tmp = r * sin(b)
    else if (b <= 11.0d0) then
        tmp = (r / cos((b + a))) * (b * (1.0d0 + (b * (b * ((-0.16666666666666666d0) + (b * (b * 0.008333333333333333d0)))))))
    else
        tmp = r / (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -8.5e+16) {
		tmp = r * Math.sin(b);
	} else if (b <= 11.0) {
		tmp = (r / Math.cos((b + a))) * (b * (1.0 + (b * (b * (-0.16666666666666666 + (b * (b * 0.008333333333333333)))))));
	} else {
		tmp = r / (1.0 / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -8.5e+16:
		tmp = r * math.sin(b)
	elif b <= 11.0:
		tmp = (r / math.cos((b + a))) * (b * (1.0 + (b * (b * (-0.16666666666666666 + (b * (b * 0.008333333333333333)))))))
	else:
		tmp = r / (1.0 / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -8.5e+16)
		tmp = Float64(r * sin(b));
	elseif (b <= 11.0)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * Float64(b * Float64(1.0 + Float64(b * Float64(b * Float64(-0.16666666666666666 + Float64(b * Float64(b * 0.008333333333333333))))))));
	else
		tmp = Float64(r / Float64(1.0 / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -8.5e+16)
		tmp = r * sin(b);
	elseif (b <= 11.0)
		tmp = (r / cos((b + a))) * (b * (1.0 + (b * (b * (-0.16666666666666666 + (b * (b * 0.008333333333333333)))))));
	else
		tmp = r / (1.0 / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -8.5e+16], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 11.0], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(1.0 + N[(b * N[(b * N[(-0.16666666666666666 + N[(b * N[(b * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5e16

   1. Initial program 60.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 60.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 60.8%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 60.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 10.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 10.0%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 10.4%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 10.4%

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

   if -8.5e16 < b < 11

   1. Initial program 98.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 98.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 98.4%

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

      1. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

      6. sin-lowering-sin.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(1 + {b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(b \cdot b\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {b}^{2}} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \color{blue}{\left(b \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(\frac{1}{120} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(\frac{1}{120} \cdot {b}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {b}^{2}}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{6}, \left({b}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(b \cdot b\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{6}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{6}, \left(b \cdot \left(\frac{1}{120} \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{120} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 95.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 95.6%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(b \cdot \left(-0.16666666666666666 + b \cdot \left(b \cdot 0.008333333333333333\right)\right)\right)\right)\right)} \]

   if 11 < b

   1. Initial program 54.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 54.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 54.2%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 54.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 54.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 11.5%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 10.6%

       \[\leadsto \color{blue}{1} \cdot \frac{r}{\frac{1}{\sin b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 11:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot \left(-0.16666666666666666 + b \cdot \left(b \cdot 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+27}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{\left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right) \cdot \left(r \cdot b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -9e+27)
   (* r (sin b))
   (if (<= b 3.8e+25)
     (/ (* (+ 1.0 (* b (* b -0.16666666666666666))) (* r b)) (cos (+ b a)))
     (/ r (/ 1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -9e+27) {
		tmp = r * sin(b);
	} else if (b <= 3.8e+25) {
		tmp = ((1.0 + (b * (b * -0.16666666666666666))) * (r * b)) / cos((b + a));
	} else {
		tmp = r / (1.0 / 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 <= (-9d+27)) then
        tmp = r * sin(b)
    else if (b <= 3.8d+25) then
        tmp = ((1.0d0 + (b * (b * (-0.16666666666666666d0)))) * (r * b)) / cos((b + a))
    else
        tmp = r / (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -9e+27) {
		tmp = r * Math.sin(b);
	} else if (b <= 3.8e+25) {
		tmp = ((1.0 + (b * (b * -0.16666666666666666))) * (r * b)) / Math.cos((b + a));
	} else {
		tmp = r / (1.0 / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -9e+27:
		tmp = r * math.sin(b)
	elif b <= 3.8e+25:
		tmp = ((1.0 + (b * (b * -0.16666666666666666))) * (r * b)) / math.cos((b + a))
	else:
		tmp = r / (1.0 / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -9e+27)
		tmp = Float64(r * sin(b));
	elseif (b <= 3.8e+25)
		tmp = Float64(Float64(Float64(1.0 + Float64(b * Float64(b * -0.16666666666666666))) * Float64(r * b)) / cos(Float64(b + a)));
	else
		tmp = Float64(r / Float64(1.0 / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -9e+27)
		tmp = r * sin(b);
	elseif (b <= 3.8e+25)
		tmp = ((1.0 + (b * (b * -0.16666666666666666))) * (r * b)) / cos((b + a));
	else
		tmp = r / (1.0 / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -9e+27], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e+25], N[(N[(N[(1.0 + N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * b), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.9999999999999998e27

   1. Initial program 60.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 60.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 60.1%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 60.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 10.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 10.2%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 10.5%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 10.5%

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

   if -8.9999999999999998e27 < b < 3.8e25

   1. Initial program 97.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 97.9%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(b \cdot \left(r + \frac{-1}{6} \cdot \left({b}^{2} \cdot r\right)\right)\right)}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \left(\frac{-1}{6} \cdot \left({b}^{2} \cdot r\right) + r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \left(\left({b}^{2} \cdot r\right) \cdot \frac{-1}{6} + r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \left({b}^{2} \cdot \left(r \cdot \frac{-1}{6}\right) + r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \left({b}^{2} \cdot \left(\frac{-1}{6} \cdot r\right) + r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({b}^{2} \cdot \left(\frac{-1}{6} \cdot r\right)\right) \cdot b + r \cdot b\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, a\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({b}^{2} \cdot \left(\frac{-1}{6} \cdot r\right)\right) \cdot b + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({b}^{2} \cdot \frac{-1}{6}\right) \cdot r\right) \cdot b + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{-1}{6} \cdot {b}^{2}\right) \cdot r\right) \cdot b + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {b}^{2}\right) \cdot \left(r \cdot b\right) + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {b}^{2}\right) \cdot \left(b \cdot r\right) + b \cdot r\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      11. distribute-lft1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {b}^{2} + 1\right) \cdot \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, a\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{-1}{6} \cdot {b}^{2}\right) \cdot \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {b}^{2}\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, a\right)}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {b}^{2}\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(b \cdot b\right)\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot b\right) \cdot b\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(b \cdot \left(\frac{-1}{6} \cdot b\right)\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\frac{-1}{6} \cdot b\right)\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \frac{-1}{6}\right)\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \frac{-1}{6}\right)\right)\right), \left(b \cdot r\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \frac{-1}{6}\right)\right)\right), \left(r \cdot b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

      22. *-lowering-*.f64 93.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \frac{-1}{6}\right)\right)\right), \mathsf{*.f64}\left(r, b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 93.3%

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

   if 3.8e25 < b

   1. Initial program 54.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 54.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 54.0%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 53.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 11.4%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 10.9%

       \[\leadsto \color{blue}{1} \cdot \frac{r}{\frac{1}{\sin b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+27}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{\left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right) \cdot \left(r \cdot b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+27}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -9e+27)
   (* r (sin b))
   (if (<= b 3.8e+25)
     (/ (* r (* b (+ 1.0 (* b (* b -0.16666666666666666))))) (cos (+ b a)))
     (/ r (/ 1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -9e+27) {
		tmp = r * sin(b);
	} else if (b <= 3.8e+25) {
		tmp = (r * (b * (1.0 + (b * (b * -0.16666666666666666))))) / cos((b + a));
	} else {
		tmp = r / (1.0 / 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 <= (-9d+27)) then
        tmp = r * sin(b)
    else if (b <= 3.8d+25) then
        tmp = (r * (b * (1.0d0 + (b * (b * (-0.16666666666666666d0)))))) / cos((b + a))
    else
        tmp = r / (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -9e+27) {
		tmp = r * Math.sin(b);
	} else if (b <= 3.8e+25) {
		tmp = (r * (b * (1.0 + (b * (b * -0.16666666666666666))))) / Math.cos((b + a));
	} else {
		tmp = r / (1.0 / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -9e+27:
		tmp = r * math.sin(b)
	elif b <= 3.8e+25:
		tmp = (r * (b * (1.0 + (b * (b * -0.16666666666666666))))) / math.cos((b + a))
	else:
		tmp = r / (1.0 / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -9e+27)
		tmp = Float64(r * sin(b));
	elseif (b <= 3.8e+25)
		tmp = Float64(Float64(r * Float64(b * Float64(1.0 + Float64(b * Float64(b * -0.16666666666666666))))) / cos(Float64(b + a)));
	else
		tmp = Float64(r / Float64(1.0 / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -9e+27)
		tmp = r * sin(b);
	elseif (b <= 3.8e+25)
		tmp = (r * (b * (1.0 + (b * (b * -0.16666666666666666))))) / cos((b + a));
	else
		tmp = r / (1.0 / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -9e+27], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e+25], N[(N[(r * N[(b * N[(1.0 + N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.9999999999999998e27

   1. Initial program 60.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 60.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 60.1%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 60.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 10.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 10.2%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 10.5%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 10.5%

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

   if -8.9999999999999998e27 < b < 3.8e25

   1. Initial program 97.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 97.9%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {b}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(b \cdot b\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot b\right) \cdot b\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \left(\frac{-1}{6} \cdot b\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\frac{-1}{6} \cdot b\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

      8. *-lowering-*.f64 93.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 93.3%

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

   if 3.8e25 < b

   1. Initial program 54.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 54.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 54.0%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 53.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 11.4%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 10.9%

       \[\leadsto \color{blue}{1} \cdot \frac{r}{\frac{1}{\sin b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+27}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+28}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+25}:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1e+28)
   (* r (sin b))
   (if (<= b 4e+25)
     (* (/ r (cos (+ b a))) (* b (+ 1.0 (* b (* b -0.16666666666666666)))))
     (/ r (/ 1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1e+28) {
		tmp = r * sin(b);
	} else if (b <= 4e+25) {
		tmp = (r / cos((b + a))) * (b * (1.0 + (b * (b * -0.16666666666666666))));
	} else {
		tmp = r / (1.0 / 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 <= (-1d+28)) then
        tmp = r * sin(b)
    else if (b <= 4d+25) then
        tmp = (r / cos((b + a))) * (b * (1.0d0 + (b * (b * (-0.16666666666666666d0)))))
    else
        tmp = r / (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1e+28) {
		tmp = r * Math.sin(b);
	} else if (b <= 4e+25) {
		tmp = (r / Math.cos((b + a))) * (b * (1.0 + (b * (b * -0.16666666666666666))));
	} else {
		tmp = r / (1.0 / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1e+28:
		tmp = r * math.sin(b)
	elif b <= 4e+25:
		tmp = (r / math.cos((b + a))) * (b * (1.0 + (b * (b * -0.16666666666666666))))
	else:
		tmp = r / (1.0 / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1e+28)
		tmp = Float64(r * sin(b));
	elseif (b <= 4e+25)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * Float64(b * Float64(1.0 + Float64(b * Float64(b * -0.16666666666666666)))));
	else
		tmp = Float64(r / Float64(1.0 / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1e+28)
		tmp = r * sin(b);
	elseif (b <= 4e+25)
		tmp = (r / cos((b + a))) * (b * (1.0 + (b * (b * -0.16666666666666666))));
	else
		tmp = r / (1.0 / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1e+28], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+25], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(1.0 + N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.99999999999999958e27

   1. Initial program 60.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 60.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 60.1%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 60.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 10.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 10.2%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 10.5%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 10.5%

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

   if -9.99999999999999958e27 < b < 4.00000000000000036e25

   1. Initial program 97.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 97.9%

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

      1. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

      6. sin-lowering-sin.f64 97.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(1 + \frac{-1}{6} \cdot {b}^{2}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {b}^{2}\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({b}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(b \cdot b\right) \cdot \frac{-1}{6}\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \color{blue}{\left(b \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \left(\frac{-1}{6} \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{-1}{6} \cdot b\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 93.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   9. Simplified 93.2%

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

   if 4.00000000000000036e25 < b

   1. Initial program 54.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 54.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 54.0%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 53.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 11.4%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 10.9%

       \[\leadsto \color{blue}{1} \cdot \frac{r}{\frac{1}{\sin b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+28}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+25}:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+27}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+25}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -9e+27)
   (* r (sin b))
   (if (<= b 4e+25) (/ (* r b) (cos a)) (/ r (/ 1.0 (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -9e+27) {
		tmp = r * sin(b);
	} else if (b <= 4e+25) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = r / (1.0 / 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 <= (-9d+27)) then
        tmp = r * sin(b)
    else if (b <= 4d+25) then
        tmp = (r * b) / cos(a)
    else
        tmp = r / (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -9e+27) {
		tmp = r * Math.sin(b);
	} else if (b <= 4e+25) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = r / (1.0 / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -9e+27:
		tmp = r * math.sin(b)
	elif b <= 4e+25:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = r / (1.0 / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -9e+27)
		tmp = Float64(r * sin(b));
	elseif (b <= 4e+25)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(r / Float64(1.0 / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -9e+27)
		tmp = r * sin(b);
	elseif (b <= 4e+25)
		tmp = (r * b) / cos(a);
	else
		tmp = r / (1.0 / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -9e+27], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+25], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.9999999999999998e27

   1. Initial program 60.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 60.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 60.1%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 60.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 10.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 10.2%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 10.5%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 10.5%

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

   if -8.9999999999999998e27 < b < 4.00000000000000036e25

   1. Initial program 97.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 97.9%

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

   5. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

      4. cos-lowering-cos.f64 92.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 92.9%

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

   if 4.00000000000000036e25 < b

   1. Initial program 54.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 54.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 54.0%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 53.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 11.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 11.4%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 10.9%

       \[\leadsto \color{blue}{1} \cdot \frac{r}{\frac{1}{\sin b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+27}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+25}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 55.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -4 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+25}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -4e+28) t_0 (if (<= b 4e+25) (/ (* r b) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -4e+28) {
		tmp = t_0;
	} else if (b <= 4e+25) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-4d+28)) then
        tmp = t_0
    else if (b <= 4d+25) then
        tmp = (r * b) / cos(a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -4e+28) {
		tmp = t_0;
	} else if (b <= 4e+25) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -4e+28:
		tmp = t_0
	elif b <= 4e+25:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -4e+28)
		tmp = t_0;
	elseif (b <= 4e+25)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -4e+28)
		tmp = t_0;
	elseif (b <= 4e+25)
		tmp = (r * b) / cos(a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4e+28], t$95$0, If[LessEqual[b, 4e+25], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999999983e28 or 4.00000000000000036e25 < b

   1. Initial program 57.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 57.0%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 56.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 10.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 10.8%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 10.7%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 10.7%

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

   if -3.99999999999999983e28 < b < 4.00000000000000036e25

   1. Initial program 97.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 97.9%

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

   5. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

      4. cos-lowering-cos.f64 92.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 92.9%

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

Alternative 21: 55.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -4 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+25}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -4e+28) t_0 (if (<= b 4e+25) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -4e+28) {
		tmp = t_0;
	} else if (b <= 4e+25) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-4d+28)) then
        tmp = t_0
    else if (b <= 4d+25) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -4e+28) {
		tmp = t_0;
	} else if (b <= 4e+25) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -4e+28:
		tmp = t_0
	elif b <= 4e+25:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -4e+28)
		tmp = t_0;
	elseif (b <= 4e+25)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -4e+28)
		tmp = t_0;
	elseif (b <= 4e+25)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4e+28], t$95$0, If[LessEqual[b, 4e+25], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999999983e28 or 4.00000000000000036e25 < b

   1. Initial program 57.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 57.0%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

      12. sin-lowering-sin.f64 56.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. cos-lowering-cos.f64 10.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   9. Simplified 10.8%

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

   10. Taylor expanded in a around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

       2. sin-lowering-sin.f64 10.7%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   12. Simplified 10.7%

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

   if -3.99999999999999983e28 < b < 4.00000000000000036e25

   1. Initial program 97.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 97.9%

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

   5. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

      4. cos-lowering-cos.f64 92.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 92.9%

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

      1. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos a}\right), \color{blue}{b}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos a\right), b\right) \]

      4. cos-lowering-cos.f64 92.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right), b\right) \]

   9. Applied egg-rr 92.8%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+28}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+25}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 39.2% 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 78.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 78.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 78.5%

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

   1. associate-*l/ N/A

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

   2. associate-/r/ N/A

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

   3. *-lft-identity N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\cos \left(b + a\right)}\right), \color{blue}{\left(\frac{r}{\frac{1}{\sin b}}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \cos \left(b + a\right)\right), \left(\frac{\color{blue}{r}}{\frac{1}{\sin b}}\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \left(\frac{r}{\frac{1}{\sin b}}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \color{blue}{\left(\frac{1}{\sin b}\right)}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \color{blue}{\sin b}\right)\right)\right) \]

   12. sin-lowering-sin.f64 78.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

6. Applied egg-rr 78.3%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\cos a}\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. cos-lowering-cos.f64 53.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(a\right)\right), \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

9. Simplified 53.9%

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

10. Taylor expanded in a around 0

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

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

    2. sin-lowering-sin.f64 37.2%

       \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

12. Simplified 37.2%

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

Alternative 23: 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 78.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 78.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 78.5%

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

5. Taylor expanded in b around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

   4. cos-lowering-cos.f64 50.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

7. Simplified 50.7%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 33.9%

      \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{b}\right) \]

10. Simplified 33.9%

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

Reproduce

herbie shell --seed 2024159 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))