rsin B (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 14.5s

Alternatives: 10

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: 76.8% 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, 0.4× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

   11. neg-sub0 N/A

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

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

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

   13. sin-lowering-sin.f64 99.5%

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

4. Applied egg-rr 99.5%

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

   1. sub0-neg N/A

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

   2. distribute-rgt-neg-out N/A

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

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

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

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

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

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

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

   6. sin-lowering-sin.f64 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 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 79.6%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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. Final simplification 99.5%

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

Alternative 3: 76.8% 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 79.6%

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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

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

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

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

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

   10. +-commutative N/A

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

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

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

   12. sin-lowering-sin.f64 79.7%

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

4. Applied egg-rr 79.7%

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

5. Final simplification 79.7%

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

Alternative 4: 76.8% 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 79.6%

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

3. Final simplification 79.6%

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

Alternative 5: 76.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.0023:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0185:\\ \;\;\;\;\frac{r \cdot \frac{b}{\mathsf{fma}\left(b, b \cdot 0.16666666666666666, 1\right)}}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -0.0023)
     t_0
     (if (<= b 0.0185)
       (/ (* r (/ b (fma b (* b 0.16666666666666666) 1.0))) (cos (+ b a)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -0.0023) {
		tmp = t_0;
	} else if (b <= 0.0185) {
		tmp = (r * (b / fma(b, (b * 0.16666666666666666), 1.0))) / 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 <= -0.0023)
		tmp = t_0;
	elseif (b <= 0.0185)
		tmp = Float64(Float64(r * Float64(b / fma(b, Float64(b * 0.16666666666666666), 1.0))) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0023], t$95$0, If[LessEqual[b, 0.0185], N[(N[(r * N[(b / N[(b * N[(b * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0023 or 0.0184999999999999991 < b

   1. Initial program 57.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 58.2%

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

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   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 58.3%

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

   7. Applied egg-rr 58.3%

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

   if -0.0023 < b < 0.0184999999999999991

   1. Initial program 99.1%

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

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

      4. associate-/r* N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

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

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. /-lowering-/.f64 N/A

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

      15. sin-lowering-sin.f64 99.0%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in b around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 99.0%

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

   7. Simplified 99.0%

      \[\leadsto \frac{\frac{r}{\cos \left(b + a\right)}}{\color{blue}{\frac{\mathsf{fma}\left(b, b \cdot 0.16666666666666666, 1\right)}{b}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\frac{1}{\frac{b \cdot \left(b \cdot \frac{1}{6}\right) + 1}{b}}} \]

      2. clear-num N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \frac{b}{\color{blue}{b \cdot \left(b \cdot \frac{1}{6}\right) + 1}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{r \cdot \frac{b}{b \cdot \left(b \cdot \frac{1}{6}\right) + 1}}{\color{blue}{\cos \left(b + a\right)}} \]

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 99.2%

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

   9. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{r \cdot \frac{b}{\mathsf{fma}\left(b, b \cdot 0.16666666666666666, 1\right)}}{\cos \left(b + a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0023:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 0.0185:\\ \;\;\;\;\frac{r \cdot \frac{b}{\mathsf{fma}\left(b, b \cdot 0.16666666666666666, 1\right)}}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -0.0064:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0043:\\ \;\;\;\;r \cdot \frac{b \cdot \mathsf{fma}\left(b, b \cdot -0.16666666666666666, 1\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -0.0064)
     t_0
     (if (<= b 0.0043)
       (* r (/ (* b (fma b (* b -0.16666666666666666) 1.0)) (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -0.0064) {
		tmp = t_0;
	} else if (b <= 0.0043) {
		tmp = r * ((b * fma(b, (b * -0.16666666666666666), 1.0)) / 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 <= -0.0064)
		tmp = t_0;
	elseif (b <= 0.0043)
		tmp = Float64(r * Float64(Float64(b * fma(b, Float64(b * -0.16666666666666666), 1.0)) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0064], t$95$0, If[LessEqual[b, 0.0043], N[(r * N[(N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00640000000000000031 or 0.0043 < b

   1. Initial program 57.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 58.2%

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

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   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 58.3%

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

   7. Applied egg-rr 58.3%

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

   if -0.00640000000000000031 < b < 0.0043

   1. Initial program 99.1%

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.1%

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

   5. Simplified 99.1%

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \mathsf{fma}\left(b, b \cdot -0.16666666666666666, 1\right)}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0064:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 0.0043:\\ \;\;\;\;r \cdot \frac{b \cdot \mathsf{fma}\left(b, b \cdot -0.16666666666666666, 1\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -2.25 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-24}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -2.25e-7) t_0 (if (<= b 9.8e-24) (* 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 <= -2.25e-7) {
		tmp = t_0;
	} else if (b <= 9.8e-24) {
		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 <= (-2.25d-7)) then
        tmp = t_0
    else if (b <= 9.8d-24) 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 <= -2.25e-7) {
		tmp = t_0;
	} else if (b <= 9.8e-24) {
		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 <= -2.25e-7:
		tmp = t_0
	elif b <= 9.8e-24:
		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 <= -2.25e-7)
		tmp = t_0;
	elseif (b <= 9.8e-24)
		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 <= -2.25e-7)
		tmp = t_0;
	elseif (b <= 9.8e-24)
		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, -2.25e-7], t$95$0, If[LessEqual[b, 9.8e-24], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.2499999999999999e-7 or 9.8000000000000002e-24 < b

   1. Initial program 58.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 59.2%

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

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
   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 59.3%

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

   7. Applied egg-rr 59.3%

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

   if -2.2499999999999999e-7 < b < 9.8000000000000002e-24

   1. Initial program 99.6%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-24}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.9% accurate, 2.1× 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 79.6%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 61.7%

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

5. Simplified 61.7%

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

   1. associate-/l* N/A

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

   2. *-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 61.7%

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

7. Applied egg-rr 61.7%

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

8. Final simplification 61.7%

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

Alternative 9: 39.1% accurate, 2.1× 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 79.6%

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

   11. neg-sub0 N/A

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

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

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

   13. sin-lowering-sin.f64 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in b around 0

   \[\leadsto \mathsf{*.f64}\left(r, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{fma.f64}\left(\color{blue}{1}, \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(a\right)\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 59.4%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

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

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

   3. sin-lowering-sin.f64 39.7%

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

10. Simplified 39.7%

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

11. Final simplification 39.7%

    \[\leadsto r \cdot \sin b \]
12. Add Preprocessing

Alternative 10: 35.1% accurate, 36.7× 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 79.6%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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 54.9%

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

5. Simplified 54.9%

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

6. Taylor expanded in a around 0

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

8. Simplified 36.5%

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

9. Final simplification 36.5%

   \[\leadsto r \cdot b \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024193 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))