rsin B (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 13.3s

Alternatives: 12

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: 76.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. 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. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. fma-define N/A

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

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

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

   6. sub0-neg N/A

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

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

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

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

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

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

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

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

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

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

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

   12. cos-lowering-cos.f64 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. 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. Add Preprocessing

Alternative 3: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. associate-/r/ N/A

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

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

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. sin-lowering-sin.f64 75.8%

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

4. Applied egg-rr 75.8%

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

5. Final simplification 75.8%

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

Alternative 4: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Final simplification 75.8%

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

Alternative 5: 76.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \mathbf{elif}\;b \leq 0.036:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + b \cdot \left(b \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.062)
   (/ 1.0 (/ (/ 1.0 r) (tan b)))
   (if (<= b 0.036)
     (*
      r
      (/
       (*
        b
        (+
         1.0
         (*
          b
          (* b (+ -0.16666666666666666 (* (* b b) 0.008333333333333333))))))
       (cos (+ b a))))
     (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.062) {
		tmp = 1.0 / ((1.0 / r) / tan(b));
	} else if (b <= 0.036) {
		tmp = r * ((b * (1.0 + (b * (b * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / cos((b + a)));
	} else {
		tmp = r * 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 <= (-0.062d0)) then
        tmp = 1.0d0 / ((1.0d0 / r) / tan(b))
    else if (b <= 0.036d0) then
        tmp = r * ((b * (1.0d0 + (b * (b * ((-0.16666666666666666d0) + ((b * b) * 0.008333333333333333d0)))))) / cos((b + a)))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.062) {
		tmp = 1.0 / ((1.0 / r) / Math.tan(b));
	} else if (b <= 0.036) {
		tmp = r * ((b * (1.0 + (b * (b * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / Math.cos((b + a)));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.062:
		tmp = 1.0 / ((1.0 / r) / math.tan(b))
	elif b <= 0.036:
		tmp = r * ((b * (1.0 + (b * (b * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / math.cos((b + a)))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.062)
		tmp = Float64(1.0 / Float64(Float64(1.0 / r) / tan(b)));
	elseif (b <= 0.036)
		tmp = Float64(r * Float64(Float64(b * Float64(1.0 + Float64(b * Float64(b * Float64(-0.16666666666666666 + Float64(Float64(b * b) * 0.008333333333333333)))))) / cos(Float64(b + a))));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.062)
		tmp = 1.0 / ((1.0 / r) / tan(b));
	elseif (b <= 0.036)
		tmp = r * ((b * (1.0 + (b * (b * (-0.16666666666666666 + ((b * b) * 0.008333333333333333)))))) / cos((b + a)));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.062], N[(1.0 / N[(N[(1.0 / r), $MachinePrecision] / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.036], N[(r * N[(N[(b * N[(1.0 + N[(b * N[(b * N[(-0.16666666666666666 + N[(N[(b * b), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.062

   1. Initial program 60.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 60.3%

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

   5. Simplified 60.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot 2}{\frac{\color{blue}{\frac{\cos b}{\sin b}}}{r}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2\right), \color{blue}{\left(\frac{\frac{\cos b}{\sin b}}{r}\right)}\right) \]

      6. metadata-eval N/A

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

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

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

      8. clear-num N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot 2}{\frac{\sin b}{\cos b}}\right), r\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2\right), \left(\frac{\sin b}{\cos b}\right)\right), r\right)\right) \]

      11. metadata-eval N/A

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

      12. quot-tan N/A

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

      13. tan-lowering-tan.f64 60.4%

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

   7. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\tan b}}{r}}} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. tan-lowering-tan.f64 60.5%

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

   9. Applied egg-rr 60.5%

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

   if -0.062 < b < 0.0359999999999999973

   1. Initial program 98.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 98.2%

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

   5. Simplified 98.2%

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

   if 0.0359999999999999973 < b

   1. Initial program 53.4%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 52.4%

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

   5. Simplified 52.4%

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

      1. *-commutative N/A

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

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

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

      3. quot-tan N/A

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

      4. tan-lowering-tan.f64 52.6%

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

   7. Applied egg-rr 52.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \mathbf{elif}\;b \leq 0.036:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + b \cdot \left(b \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \mathbf{elif}\;b \leq 0.0052:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + -0.16666666666666666 \cdot \left(b \cdot b\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.062)
   (/ 1.0 (/ (/ 1.0 r) (tan b)))
   (if (<= b 0.0052)
     (* r (/ (* b (+ 1.0 (* -0.16666666666666666 (* b b)))) (cos (+ b a))))
     (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.062) {
		tmp = 1.0 / ((1.0 / r) / tan(b));
	} else if (b <= 0.0052) {
		tmp = r * ((b * (1.0 + (-0.16666666666666666 * (b * b)))) / cos((b + a)));
	} else {
		tmp = r * 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 <= (-0.062d0)) then
        tmp = 1.0d0 / ((1.0d0 / r) / tan(b))
    else if (b <= 0.0052d0) then
        tmp = r * ((b * (1.0d0 + ((-0.16666666666666666d0) * (b * b)))) / cos((b + a)))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.062) {
		tmp = 1.0 / ((1.0 / r) / Math.tan(b));
	} else if (b <= 0.0052) {
		tmp = r * ((b * (1.0 + (-0.16666666666666666 * (b * b)))) / Math.cos((b + a)));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.062:
		tmp = 1.0 / ((1.0 / r) / math.tan(b))
	elif b <= 0.0052:
		tmp = r * ((b * (1.0 + (-0.16666666666666666 * (b * b)))) / math.cos((b + a)))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.062)
		tmp = Float64(1.0 / Float64(Float64(1.0 / r) / tan(b)));
	elseif (b <= 0.0052)
		tmp = Float64(r * Float64(Float64(b * Float64(1.0 + Float64(-0.16666666666666666 * Float64(b * b)))) / cos(Float64(b + a))));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.062)
		tmp = 1.0 / ((1.0 / r) / tan(b));
	elseif (b <= 0.0052)
		tmp = r * ((b * (1.0 + (-0.16666666666666666 * (b * b)))) / cos((b + a)));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.062], N[(1.0 / N[(N[(1.0 / r), $MachinePrecision] / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0052], N[(r * N[(N[(b * N[(1.0 + N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.062

   1. Initial program 60.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 60.3%

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

   5. Simplified 60.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot 2}{\frac{\color{blue}{\frac{\cos b}{\sin b}}}{r}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2\right), \color{blue}{\left(\frac{\frac{\cos b}{\sin b}}{r}\right)}\right) \]

      6. metadata-eval N/A

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

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

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

      8. clear-num N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot 2}{\frac{\sin b}{\cos b}}\right), r\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2\right), \left(\frac{\sin b}{\cos b}\right)\right), r\right)\right) \]

      11. metadata-eval N/A

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

      12. quot-tan N/A

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

      13. tan-lowering-tan.f64 60.4%

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

   7. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\tan b}}{r}}} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. tan-lowering-tan.f64 60.5%

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

   9. Applied egg-rr 60.5%

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

   if -0.062 < b < 0.0051999999999999998

   1. Initial program 98.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 98.2%

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

   5. Simplified 98.2%

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

   if 0.0051999999999999998 < b

   1. Initial program 53.4%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 52.4%

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

   5. Simplified 52.4%

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

      1. *-commutative N/A

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

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

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

      3. quot-tan N/A

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

      4. tan-lowering-tan.f64 52.6%

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

   7. Applied egg-rr 52.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \mathbf{elif}\;b \leq 0.0052:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + -0.16666666666666666 \cdot \left(b \cdot b\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \mathbf{elif}\;b \leq 0.00037:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.062)
   (/ 1.0 (/ (/ 1.0 r) (tan b)))
   (if (<= b 0.00037) (* r (/ b (cos (+ b a)))) (* r (tan b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.062) {
		tmp = 1.0 / ((1.0 / r) / tan(b));
	} else if (b <= 0.00037) {
		tmp = r * (b / cos((b + a)));
	} else {
		tmp = r * 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 <= (-0.062d0)) then
        tmp = 1.0d0 / ((1.0d0 / r) / tan(b))
    else if (b <= 0.00037d0) then
        tmp = r * (b / cos((b + a)))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.062) {
		tmp = 1.0 / ((1.0 / r) / Math.tan(b));
	} else if (b <= 0.00037) {
		tmp = r * (b / Math.cos((b + a)));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.062:
		tmp = 1.0 / ((1.0 / r) / math.tan(b))
	elif b <= 0.00037:
		tmp = r * (b / math.cos((b + a)))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.062)
		tmp = Float64(1.0 / Float64(Float64(1.0 / r) / tan(b)));
	elseif (b <= 0.00037)
		tmp = Float64(r * Float64(b / cos(Float64(b + a))));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -0.062)
		tmp = 1.0 / ((1.0 / r) / tan(b));
	elseif (b <= 0.00037)
		tmp = r * (b / cos((b + a)));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.062], N[(1.0 / N[(N[(1.0 / r), $MachinePrecision] / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.00037], N[(r * N[(b / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.062

   1. Initial program 60.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 60.3%

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

   5. Simplified 60.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot 2}{\frac{\color{blue}{\frac{\cos b}{\sin b}}}{r}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2\right), \color{blue}{\left(\frac{\frac{\cos b}{\sin b}}{r}\right)}\right) \]

      6. metadata-eval N/A

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

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

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

      8. clear-num N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot 2}{\frac{\sin b}{\cos b}}\right), r\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2\right), \left(\frac{\sin b}{\cos b}\right)\right), r\right)\right) \]

      11. metadata-eval N/A

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

      12. quot-tan N/A

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

      13. tan-lowering-tan.f64 60.4%

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

   7. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\tan b}}{r}}} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. tan-lowering-tan.f64 60.5%

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

   9. Applied egg-rr 60.5%

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

   if -0.062 < b < 3.6999999999999999e-4

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 98.0%

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

   if 3.6999999999999999e-4 < b

   1. Initial program 53.4%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 52.4%

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

   5. Simplified 52.4%

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

      1. *-commutative N/A

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

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

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

      3. quot-tan N/A

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

      4. tan-lowering-tan.f64 52.6%

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

   7. Applied egg-rr 52.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\tan b}}\\ \mathbf{elif}\;b \leq 0.00037:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -0.062) t_0 (if (<= b 9.5e-6) (* r (/ b (cos (+ b a)))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -0.062) {
		tmp = t_0;
	} else if (b <= 9.5e-6) {
		tmp = r * (b / cos((b + 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 <= (-0.062d0)) then
        tmp = t_0
    else if (b <= 9.5d-6) then
        tmp = r * (b / cos((b + 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 <= -0.062) {
		tmp = t_0;
	} else if (b <= 9.5e-6) {
		tmp = r * (b / Math.cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -0.062:
		tmp = t_0
	elif b <= 9.5e-6:
		tmp = r * (b / math.cos((b + a)))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -0.062)
		tmp = t_0;
	elseif (b <= 9.5e-6)
		tmp = Float64(r * Float64(b / cos(Float64(b + 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 <= -0.062)
		tmp = t_0;
	elseif (b <= 9.5e-6)
		tmp = r * (b / cos((b + 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, -0.062], t$95$0, If[LessEqual[b, 9.5e-6], N[(r * N[(b / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.062 or 9.5000000000000005e-6 < b

   1. Initial program 57.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 56.7%

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

   5. Simplified 56.7%

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

      1. *-commutative N/A

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

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

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

      3. quot-tan N/A

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

      4. tan-lowering-tan.f64 56.8%

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

   7. Applied egg-rr 56.8%

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

   if -0.062 < b < 9.5000000000000005e-6

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 98.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -0.062) t_0 (if (<= b 4e-5) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -0.062) {
		tmp = t_0;
	} else if (b <= 4e-5) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-0.062d0)) then
        tmp = t_0
    else if (b <= 4d-5) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -0.062) {
		tmp = t_0;
	} else if (b <= 4e-5) {
		tmp = b * (r / 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 <= -0.062:
		tmp = t_0
	elif b <= 4e-5:
		tmp = b * (r / 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 <= -0.062)
		tmp = t_0;
	elseif (b <= 4e-5)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -0.062)
		tmp = t_0;
	elseif (b <= 4e-5)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.062], t$95$0, If[LessEqual[b, 4e-5], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.062 or 4.00000000000000033e-5 < b

   1. Initial program 57.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. cos-lowering-cos.f64 56.7%

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

   5. Simplified 56.7%

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

      1. *-commutative N/A

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

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

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

      3. quot-tan N/A

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

      4. tan-lowering-tan.f64 56.8%

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

   7. Applied egg-rr 56.8%

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

   if -0.062 < b < 4.00000000000000033e-5

   1. Initial program 98.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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. associate-/l* N/A

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

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

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

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

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

      4. cos-lowering-cos.f64 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.062:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.4% 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 75.8%

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

3. Taylor expanded in a around 0

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

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

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

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

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

   3. cos-lowering-cos.f64 60.6%

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

5. Simplified 60.6%

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

   1. *-commutative N/A

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

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

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

   3. quot-tan N/A

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

   4. tan-lowering-tan.f64 60.7%

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

7. Applied egg-rr 60.7%

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

8. Final simplification 60.7%

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

Alternative 11: 34.7% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{b}{1 + b \cdot \left(b \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (/ b (+ 1.0 (* b (* b -0.5))))))

C

double code(double r, double a, double b) {
	return r * (b / (1.0 + (b * (b * -0.5))));
}

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 / (1.0d0 + (b * (b * (-0.5d0)))))
end function

Java

public static double code(double r, double a, double b) {
	return r * (b / (1.0 + (b * (b * -0.5))));
}

Python

def code(r, a, b):
	return r * (b / (1.0 + (b * (b * -0.5))))

Julia

function code(r, a, b)
	return Float64(r * Float64(b / Float64(1.0 + Float64(b * Float64(b * -0.5)))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (b / (1.0 + (b * (b * -0.5))));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(b / N[(1.0 + N[(b * N[(b * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in a around 0

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

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

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

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

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

   3. cos-lowering-cos.f64 60.6%

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

5. Simplified 60.6%

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

6. Taylor expanded in b around 0

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

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

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 32.2%

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

8. Simplified 32.2%

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

9. Taylor expanded in b around 0

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

11. Simplified 32.1%

    \[\leadsto r \cdot \frac{\color{blue}{b}}{1 + b \cdot \left(b \cdot -0.5\right)} \]
12. Add Preprocessing

Alternative 12: 34.3% 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 75.8%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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. associate-/l* N/A

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

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

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

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

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

   4. cos-lowering-cos.f64 46.9%

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

5. Simplified 46.9%

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

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 32.1%

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

8. Simplified 32.1%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))