rsin B (should all be same)

Percentage Accurate: 76.5% → 99.5%

Time: 14.1s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. +-commutative N/A

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

   7. +-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. fmm-def N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-in N/A

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

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

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

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

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

   11. neg-sub0 N/A

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

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

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

   13. sin-lowering-sin.f64 99.6%

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

6. Applied egg-rr 99.6%

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

   1. *-commutative N/A

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

   2. sub0-neg N/A

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

   3. distribute-lft-neg-out N/A

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

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

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

   5. *-commutative N/A

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

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

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

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

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

   8. sin-lowering-sin.f64 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(0 - \sin a\right)\right)} \]
10. 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%

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. +-commutative N/A

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

   7. +-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. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. sin-lowering-sin.f64 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. cos-sum N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. sin-lowering-sin.f64 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
6. 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%

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

   3. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 59.1%

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

   5. Simplified 59.1%

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

   if -6.7999999999999999e-5 < a < 2.3e15

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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%

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

   5. Simplified 98.0%

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

   if 2.3e15 < a

   1. Initial program 56.9%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-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. Add Preprocessing

Alternative 5: 75.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000235)
   (* r (/ (sin b) (cos a)))
   (if (<= a 2.3e+15) (* r (tan 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 = r * tan(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 = r * tan(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 = r * Math.tan(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 = r * math.tan(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(r * tan(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 = r * tan(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[(r * N[Tan[b], $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%

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

   3. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 59.1%

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

   5. Simplified 59.1%

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

   if -2.34999999999999993e-4 < a < 2.3e15

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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%

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

   5. Simplified 98.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 98.0%

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

   7. Applied egg-rr 98.0%

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

   if 2.3e15 < a

   1. Initial program 56.9%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-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%

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

Alternative 6: 75.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000155)
   (* r (/ (sin b) (cos a)))
   (if (<= a 2.3e+15) (* r (tan 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 = r * tan(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 = r * tan(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 = r * Math.tan(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 = r * math.tan(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(r * tan(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 = r * tan(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[(r * N[Tan[b], $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%

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

   3. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 59.1%

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

   5. Simplified 59.1%

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

   if -1.55e-4 < a < 2.3e15

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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%

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

   5. Simplified 98.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 98.0%

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

   7. Applied egg-rr 98.0%

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

   if 2.3e15 < a

   1. Initial program 56.9%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-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%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      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%

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

   9. 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%

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

Alternative 7: 75.8% accurate, 1.0× speedup

Math

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

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) (* r (tan 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 = r * tan(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 = r * tan(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 = r * Math.tan(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 = r * math.tan(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(r * tan(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 = r * tan(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[(r * N[Tan[b], $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%

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

   3. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 57.8%

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

   5. Simplified 57.8%

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

   if -7.50000000000000019e-6 < a < 2.3e15

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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%

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

   5. Simplified 98.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 98.0%

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

   7. Applied egg-rr 98.0%

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

4. Final simplification 78.2%

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

Alternative 8: 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%

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. +-commutative N/A

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

   7. +-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. Add Preprocessing

Alternative 9: 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%

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

3. Final simplification 78.5%

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

Alternative 10: 76.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.026:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;\frac{b \cdot \left(r + \left(r \cdot \left(b \cdot b\right)\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.026)
   (* r (tan b))
   (if (<= b 8.0)
     (/
      (*
       b
       (+
        r
        (*
         (* r (* b b))
         (+ -0.16666666666666666 (* (* b b) 0.008333333333333333)))))
      (cos (+ b a)))
     (/ r (/ 1.0 (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.026) {
		tmp = r * tan(b);
	} else if (b <= 8.0) {
		tmp = (b * (r + ((r * (b * b)) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))) / cos((b + a));
	} else {
		tmp = r / (1.0 / tan(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.026d0)) then
        tmp = r * tan(b)
    else if (b <= 8.0d0) then
        tmp = (b * (r + ((r * (b * b)) * ((-0.16666666666666666d0) + ((b * b) * 0.008333333333333333d0))))) / cos((b + a))
    else
        tmp = r / (1.0d0 / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.026) {
		tmp = r * Math.tan(b);
	} else if (b <= 8.0) {
		tmp = (b * (r + ((r * (b * b)) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))) / Math.cos((b + a));
	} else {
		tmp = r / (1.0 / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.026:
		tmp = r * math.tan(b)
	elif b <= 8.0:
		tmp = (b * (r + ((r * (b * b)) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))) / math.cos((b + a))
	else:
		tmp = r / (1.0 / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.026)
		tmp = Float64(r * tan(b));
	elseif (b <= 8.0)
		tmp = Float64(Float64(b * Float64(r + Float64(Float64(r * Float64(b * b)) * Float64(-0.16666666666666666 + Float64(Float64(b * b) * 0.008333333333333333))))) / cos(Float64(b + a)));
	else
		tmp = Float64(r / Float64(1.0 / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.026)
		tmp = r * tan(b);
	elseif (b <= 8.0)
		tmp = (b * (r + ((r * (b * b)) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))) / cos((b + a));
	else
		tmp = r / (1.0 / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.026], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.0], N[(N[(b * N[(r + N[(N[(r * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(b * b), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0259999999999999988

   1. Initial program 63.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 62.3%

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

   5. Simplified 62.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 62.3%

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

   7. Applied egg-rr 62.3%

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

   if -0.0259999999999999988 < b < 8

   1. Initial program 98.3%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-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. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

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

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

   7. Simplified 98.3%

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

   if 8 < b

   1. Initial program 54.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 52.9%

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

   5. Simplified 52.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. quot-tan N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \tan b\right)\right) \]

      8. tan-lowering-tan.f64 52.9%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(b\right)\right)\right) \]

   7. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.026:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;\frac{b \cdot \left(r + \left(r \cdot \left(b \cdot b\right)\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.032:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + \left(b \cdot b\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.032)
   (* r (tan b))
   (if (<= b 8.0)
     (*
      r
      (/
       (*
        b
        (+
         1.0
         (*
          (* b b)
          (+ -0.16666666666666666 (* (* b b) 0.008333333333333333)))))
       (cos (+ b a))))
     (/ r (/ 1.0 (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.032) {
		tmp = r * tan(b);
	} else if (b <= 8.0) {
		tmp = r * ((b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))) / cos((b + a)));
	} else {
		tmp = r / (1.0 / tan(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.032d0)) then
        tmp = r * tan(b)
    else if (b <= 8.0d0) then
        tmp = r * ((b * (1.0d0 + ((b * b) * ((-0.16666666666666666d0) + ((b * b) * 0.008333333333333333d0))))) / cos((b + a)))
    else
        tmp = r / (1.0d0 / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.032) {
		tmp = r * Math.tan(b);
	} else if (b <= 8.0) {
		tmp = r * ((b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))) / Math.cos((b + a)));
	} else {
		tmp = r / (1.0 / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.032:
		tmp = r * math.tan(b)
	elif b <= 8.0:
		tmp = r * ((b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))) / math.cos((b + a)))
	else:
		tmp = r / (1.0 / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.032)
		tmp = Float64(r * tan(b));
	elseif (b <= 8.0)
		tmp = Float64(r * Float64(Float64(b * Float64(1.0 + Float64(Float64(b * b) * Float64(-0.16666666666666666 + Float64(Float64(b * b) * 0.008333333333333333))))) / cos(Float64(b + a))));
	else
		tmp = Float64(r / Float64(1.0 / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.032)
		tmp = r * tan(b);
	elseif (b <= 8.0)
		tmp = r * ((b * (1.0 + ((b * b) * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))) / cos((b + a)));
	else
		tmp = r / (1.0 / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.032], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.0], N[(r * N[(N[(b * N[(1.0 + N[(N[(b * b), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(b * b), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.032000000000000001

   1. Initial program 63.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 62.3%

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

   5. Simplified 62.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 62.3%

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

   7. Applied egg-rr 62.3%

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

   if -0.032000000000000001 < b < 8

   1. Initial program 98.3%

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

   3. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 98.3%

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

   5. Simplified 98.3%

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

   if 8 < b

   1. Initial program 54.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 52.9%

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

   5. Simplified 52.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. quot-tan N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \tan b\right)\right) \]

      8. tan-lowering-tan.f64 52.9%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(b\right)\right)\right) \]

   7. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.032:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + \left(b \cdot b\right) \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.006:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + \left(b \cdot b\right) \cdot -0.16666666666666666\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.006)
   (* r (tan b))
   (if (<= b 6.6e+18)
     (/ (* r (* b (+ 1.0 (* (* b b) -0.16666666666666666)))) (cos (+ b a)))
     (/ r (/ 1.0 (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.006) {
		tmp = r * tan(b);
	} else if (b <= 6.6e+18) {
		tmp = (r * (b * (1.0 + ((b * b) * -0.16666666666666666)))) / cos((b + a));
	} else {
		tmp = r / (1.0 / tan(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.006d0)) then
        tmp = r * tan(b)
    else if (b <= 6.6d+18) then
        tmp = (r * (b * (1.0d0 + ((b * b) * (-0.16666666666666666d0))))) / cos((b + a))
    else
        tmp = r / (1.0d0 / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.006) {
		tmp = r * Math.tan(b);
	} else if (b <= 6.6e+18) {
		tmp = (r * (b * (1.0 + ((b * b) * -0.16666666666666666)))) / Math.cos((b + a));
	} else {
		tmp = r / (1.0 / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.006:
		tmp = r * math.tan(b)
	elif b <= 6.6e+18:
		tmp = (r * (b * (1.0 + ((b * b) * -0.16666666666666666)))) / math.cos((b + a))
	else:
		tmp = r / (1.0 / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.006)
		tmp = Float64(r * tan(b));
	elseif (b <= 6.6e+18)
		tmp = Float64(Float64(r * Float64(b * Float64(1.0 + Float64(Float64(b * b) * -0.16666666666666666)))) / cos(Float64(b + a)));
	else
		tmp = Float64(r / Float64(1.0 / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.006)
		tmp = r * tan(b);
	elseif (b <= 6.6e+18)
		tmp = (r * (b * (1.0 + ((b * b) * -0.16666666666666666)))) / cos((b + a));
	else
		tmp = r / (1.0 / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.006], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6e+18], N[(N[(r * N[(b * N[(1.0 + N[(N[(b * b), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0060000000000000001

   1. Initial program 63.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 62.3%

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

   5. Simplified 62.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 62.3%

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

   7. Applied egg-rr 62.3%

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

   if -0.0060000000000000001 < b < 6.6e18

   1. Initial program 97.7%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 97.8%

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

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   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. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 97.5%

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

   7. Simplified 97.5%

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

   if 6.6e18 < b

   1. Initial program 54.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 53.7%

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

   5. Simplified 53.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. quot-tan N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \tan b\right)\right) \]

      8. tan-lowering-tan.f64 53.7%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(b\right)\right)\right) \]

   7. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.006:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + \left(b \cdot b\right) \cdot -0.16666666666666666\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0068:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+18}:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + \left(b \cdot b\right) \cdot -0.16666666666666666\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.0068)
   (* r (tan b))
   (if (<= b 6.6e+18)
     (* r (/ (* b (+ 1.0 (* (* b b) -0.16666666666666666))) (cos (+ b a))))
     (/ r (/ 1.0 (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0068) {
		tmp = r * tan(b);
	} else if (b <= 6.6e+18) {
		tmp = r * ((b * (1.0 + ((b * b) * -0.16666666666666666))) / cos((b + a)));
	} else {
		tmp = r / (1.0 / tan(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.0068d0)) then
        tmp = r * tan(b)
    else if (b <= 6.6d+18) then
        tmp = r * ((b * (1.0d0 + ((b * b) * (-0.16666666666666666d0)))) / cos((b + a)))
    else
        tmp = r / (1.0d0 / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0068) {
		tmp = r * Math.tan(b);
	} else if (b <= 6.6e+18) {
		tmp = r * ((b * (1.0 + ((b * b) * -0.16666666666666666))) / Math.cos((b + a)));
	} else {
		tmp = r / (1.0 / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.0068:
		tmp = r * math.tan(b)
	elif b <= 6.6e+18:
		tmp = r * ((b * (1.0 + ((b * b) * -0.16666666666666666))) / math.cos((b + a)))
	else:
		tmp = r / (1.0 / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.0068)
		tmp = Float64(r * tan(b));
	elseif (b <= 6.6e+18)
		tmp = Float64(r * Float64(Float64(b * Float64(1.0 + Float64(Float64(b * b) * -0.16666666666666666))) / cos(Float64(b + a))));
	else
		tmp = Float64(r / Float64(1.0 / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.0068)
		tmp = r * tan(b);
	elseif (b <= 6.6e+18)
		tmp = r * ((b * (1.0 + ((b * b) * -0.16666666666666666))) / cos((b + a)));
	else
		tmp = r / (1.0 / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.0068], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6e+18], N[(r * N[(N[(b * N[(1.0 + N[(N[(b * b), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.00679999999999999962

   1. Initial program 63.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 62.3%

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

   5. Simplified 62.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 62.3%

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

   7. Applied egg-rr 62.3%

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

   if -0.00679999999999999962 < b < 6.6e18

   1. Initial program 97.7%

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

   3. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 97.5%

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

   5. Simplified 97.5%

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

   if 6.6e18 < b

   1. Initial program 54.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 53.7%

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

   5. Simplified 53.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. quot-tan N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \tan b\right)\right) \]

      8. tan-lowering-tan.f64 53.7%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(b\right)\right)\right) \]

   7. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0068:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+18}:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + \left(b \cdot b\right) \cdot -0.16666666666666666\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -4.6e-5)
   (* r (tan b))
   (if (<= b 8.0) (/ (* r b) (cos a)) (/ r (/ 1.0 (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -4.6e-5) {
		tmp = r * tan(b);
	} else if (b <= 8.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = r / (1.0 / tan(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 <= (-4.6d-5)) then
        tmp = r * tan(b)
    else if (b <= 8.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = r / (1.0d0 / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -4.6e-5) {
		tmp = r * Math.tan(b);
	} else if (b <= 8.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = r / (1.0 / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -4.6e-5:
		tmp = r * math.tan(b)
	elif b <= 8.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = r / (1.0 / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -4.6e-5)
		tmp = Float64(r * tan(b));
	elseif (b <= 8.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(r / Float64(1.0 / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -4.6e-5)
		tmp = r * tan(b);
	elseif (b <= 8.0)
		tmp = (r * b) / cos(a);
	else
		tmp = r / (1.0 / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -4.6e-5], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.6e-5

   1. Initial program 63.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 62.0%

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

   5. Simplified 62.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 62.0%

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

   7. Applied egg-rr 62.0%

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

   if -4.6e-5 < b < 8

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   4. 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 98.9%

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

   5. Simplified 98.9%

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

   if 8 < b

   1. Initial program 54.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 52.9%

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

   5. Simplified 52.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. quot-tan N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \tan b\right)\right) \]

      8. tan-lowering-tan.f64 52.9%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(b\right)\right)\right) \]

   7. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -3e-5)
   (* r (tan b))
   (if (<= b 8.0) (* r (/ b (cos a))) (/ r (/ 1.0 (tan b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -3e-5) {
		tmp = r * tan(b);
	} else if (b <= 8.0) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r / (1.0 / tan(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 <= (-3d-5)) then
        tmp = r * tan(b)
    else if (b <= 8.0d0) then
        tmp = r * (b / cos(a))
    else
        tmp = r / (1.0d0 / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -3e-5) {
		tmp = r * Math.tan(b);
	} else if (b <= 8.0) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r / (1.0 / Math.tan(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -3e-5:
		tmp = r * math.tan(b)
	elif b <= 8.0:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r / (1.0 / math.tan(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -3e-5)
		tmp = Float64(r * tan(b));
	elseif (b <= 8.0)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r / Float64(1.0 / tan(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -3e-5)
		tmp = r * tan(b);
	elseif (b <= 8.0)
		tmp = r * (b / cos(a));
	else
		tmp = r / (1.0 / tan(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -3e-5], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.0], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000008e-5

   1. Initial program 63.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 62.0%

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

   5. Simplified 62.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 62.0%

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

   7. Applied egg-rr 62.0%

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

   if -3.00000000000000008e-5 < b < 8

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. cos-lowering-cos.f64 98.8%

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

   5. Simplified 98.8%

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

   if 8 < b

   1. Initial program 54.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 52.9%

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

   5. Simplified 52.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. quot-tan N/A

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \tan b\right)\right) \]

      8. tan-lowering-tan.f64 52.9%

         \[\leadsto \mathsf{/.f64}\left(r, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(b\right)\right)\right) \]

   7. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{r}{\frac{1}{\tan b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -1.52 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -1.52e-5) t_0 (if (<= b 8.0) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -1.52e-5) {
		tmp = t_0;
	} else if (b <= 8.0) {
		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 * tan(b)
    if (b <= (-1.52d-5)) then
        tmp = t_0
    else if (b <= 8.0d0) 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.tan(b);
	double tmp;
	if (b <= -1.52e-5) {
		tmp = t_0;
	} else if (b <= 8.0) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -1.52e-5:
		tmp = t_0
	elif b <= 8.0:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -1.52e-5)
		tmp = t_0;
	elseif (b <= 8.0)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -1.52e-5)
		tmp = t_0;
	elseif (b <= 8.0)
		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[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.52e-5], t$95$0, If[LessEqual[b, 8.0], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.52e-5 or 8 < b

   1. Initial program 59.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   4. 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 57.5%

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

   5. Simplified 57.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 57.5%

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

   7. Applied egg-rr 57.5%

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

   if -1.52e-5 < b < 8

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0

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

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

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

      2. cos-lowering-cos.f64 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.52 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 8:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
4. 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 61.1%

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

5. Simplified 61.1%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. quot-tan N/A

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

   5. tan-lowering-tan.f64 61.1%

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

7. Applied egg-rr 61.1%

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

8. Final simplification 61.1%

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

Alternative 18: 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%

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. +-commutative N/A

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

   7. +-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

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

   1. cos-lowering-cos.f64 54.1%

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

7. Simplified 54.1%

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

8. Taylor expanded in a around 0

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

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

10. Simplified 37.2%

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

Alternative 19: 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%

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

3. Taylor expanded in b around 0

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

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

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

   2. cos-lowering-cos.f64 50.6%

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

5. Simplified 50.6%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 33.9%

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

8. Simplified 33.9%

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

Reproduce

herbie shell --seed 2024159 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))