rsin B (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 15.0s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 80.2%

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

3. Simplified 80.2%

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

   1. +-commutative N/A

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

   2. cos-sum N/A

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. sin-lowering-sin.f64 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin b \cdot \sin a}} \]
7. 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 80.1%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{\cos b}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) (cos b))))
   (if (<= b -2e-6) t_0 (if (<= b 6.8e-9) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / cos(b);
	double tmp;
	if (b <= -2e-6) {
		tmp = t_0;
	} else if (b <= 6.8e-9) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r * sin(b)) / cos(b)
    if (b <= (-2d-6)) then
        tmp = t_0
    else if (b <= 6.8d-9) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (r * Math.sin(b)) / Math.cos(b);
	double tmp;
	if (b <= -2e-6) {
		tmp = t_0;
	} else if (b <= 6.8e-9) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r * math.sin(b)) / math.cos(b)
	tmp = 0
	if b <= -2e-6:
		tmp = t_0
	elif b <= 6.8e-9:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / cos(b))
	tmp = 0.0
	if (b <= -2e-6)
		tmp = t_0;
	elseif (b <= 6.8e-9)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r * sin(b)) / cos(b);
	tmp = 0.0;
	if (b <= -2e-6)
		tmp = t_0;
	elseif (b <= 6.8e-9)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e-6], t$95$0, If[LessEqual[b, 6.8e-9], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.99999999999999991e-6 or 6.7999999999999997e-9 < b

   1. Initial program 59.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 60.2%

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

   5. Simplified 60.2%

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

   if -1.99999999999999991e-6 < b < 6.7999999999999997e-9

   1. Initial program 99.2%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 99.3%

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

   3. Simplified 99.3%

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

      1. +-commutative N/A

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

      2. cos-sum N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   8. 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 99.3%

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

   9. Simplified 99.3%

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

Alternative 4: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos b)))))
   (if (<= b -1.1e-6) t_0 (if (<= b 6.8e-9) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(b));
	double tmp;
	if (b <= -1.1e-6) {
		tmp = t_0;
	} else if (b <= 6.8e-9) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * (sin(b) / cos(b))
    if (b <= (-1.1d-6)) then
        tmp = t_0
    else if (b <= 6.8d-9) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * (Math.sin(b) / Math.cos(b));
	double tmp;
	if (b <= -1.1e-6) {
		tmp = t_0;
	} else if (b <= 6.8e-9) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / math.cos(b))
	tmp = 0
	if b <= -1.1e-6:
		tmp = t_0
	elif b <= 6.8e-9:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(b)))
	tmp = 0.0
	if (b <= -1.1e-6)
		tmp = t_0;
	elseif (b <= 6.8e-9)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / cos(b));
	tmp = 0.0;
	if (b <= -1.1e-6)
		tmp = t_0;
	elseif (b <= 6.8e-9)
		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[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1e-6], t$95$0, If[LessEqual[b, 6.8e-9], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001e-6 or 6.7999999999999997e-9 < b

   1. Initial program 59.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   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 60.1%

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

   5. Simplified 60.1%

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

   if -1.1000000000000001e-6 < b < 6.7999999999999997e-9

   1. Initial program 99.2%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 99.3%

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

   3. Simplified 99.3%

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

      1. +-commutative N/A

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

      2. cos-sum N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   8. 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 99.3%

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

   9. Simplified 99.3%

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

Alternative 5: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 80.2%

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

3. Simplified 80.2%

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

Alternative 6: 76.3% 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 80.1%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
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. 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 80.2%

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

4. Applied egg-rr 80.2%

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

5. Final simplification 80.2%

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

Alternative 7: 76.3% 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 80.1%

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

3. Final simplification 80.1%

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

Alternative 8: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

3. Taylor expanded in b around 0

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

   1. cos-lowering-cos.f64 57.8%

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

5. Simplified 57.8%

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

Alternative 9: 55.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -1160000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.8:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + \left(b \cdot b\right) \cdot \left(-0.16666666666666666 + b \cdot \left(b \cdot \left(0.008333333333333333 + \left(b \cdot b\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -1160000000.0)
     t_0
     (if (<= b 4.8)
       (/
        (*
         r
         (*
          b
          (+
           1.0
           (*
            (* b b)
            (+
             -0.16666666666666666
             (*
              b
              (*
               b
               (+
                0.008333333333333333
                (* (* b b) -0.0001984126984126984)))))))))
        (cos (+ b a)))
       (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -1160000000.0) {
		tmp = t_0;
	} else if (b <= 4.8) {
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + (b * (b * (0.008333333333333333 + ((b * b) * -0.0001984126984126984))))))))) / cos((b + a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-1160000000.0d0)) then
        tmp = t_0
    else if (b <= 4.8d0) then
        tmp = (r * (b * (1.0d0 + ((b * b) * ((-0.16666666666666666d0) + (b * (b * (0.008333333333333333d0 + ((b * b) * (-0.0001984126984126984d0)))))))))) / cos((b + a))
    else
        tmp = 1.0d0 / (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -1160000000.0) {
		tmp = t_0;
	} else if (b <= 4.8) {
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + (b * (b * (0.008333333333333333 + ((b * b) * -0.0001984126984126984))))))))) / Math.cos((b + a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -1160000000.0:
		tmp = t_0
	elif b <= 4.8:
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + (b * (b * (0.008333333333333333 + ((b * b) * -0.0001984126984126984))))))))) / math.cos((b + a))
	else:
		tmp = 1.0 / (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -1160000000.0)
		tmp = t_0;
	elseif (b <= 4.8)
		tmp = Float64(Float64(r * Float64(b * Float64(1.0 + Float64(Float64(b * b) * Float64(-0.16666666666666666 + Float64(b * Float64(b * Float64(0.008333333333333333 + Float64(Float64(b * b) * -0.0001984126984126984))))))))) / cos(Float64(b + a)));
	else
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -1160000000.0)
		tmp = t_0;
	elseif (b <= 4.8)
		tmp = (r * (b * (1.0 + ((b * b) * (-0.16666666666666666 + (b * (b * (0.008333333333333333 + ((b * b) * -0.0001984126984126984))))))))) / cos((b + a));
	else
		tmp = 1.0 / (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1160000000.0], t$95$0, If[LessEqual[b, 4.8], N[(N[(r * N[(b * N[(1.0 + N[(N[(b * b), $MachinePrecision] * N[(-0.16666666666666666 + N[(b * N[(b * N[(0.008333333333333333 + N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.16e9

   1. Initial program 55.1%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 11.7%

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

   7. Simplified 11.7%

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

   8. Taylor expanded in a around 0

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

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

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

      2. sin-lowering-sin.f64 13.4%

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

   10. Simplified 13.4%

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

   if -1.16e9 < b < 4.79999999999999982

   1. Initial program 98.0%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 97.3%

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

   7. Simplified 97.3%

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

   if 4.79999999999999982 < b

   1. Initial program 63.4%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 63.6%

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

   3. Simplified 63.6%

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

      1. +-commutative N/A

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

      2. cos-sum N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 68.3%

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

       1. clear-num N/A

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

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

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

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

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

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

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

       5. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

       11. sin-lowering-sin.f64 68.2%

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

   11. Applied egg-rr 68.2%

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

   12. Taylor expanded in b around 0

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

   14. Simplified 13.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{1}}{r \cdot \sin b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -0.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5000000000:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + b \cdot \left(b \cdot \left(-0.16666666666666666 + \left(b \cdot b\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -0.58)
     t_0
     (if (<= b 5000000000.0)
       (/
        (*
         r
         (*
          b
          (+
           1.0
           (*
            b
            (* b (+ -0.16666666666666666 (* (* b b) 0.008333333333333333)))))))
        (cos (+ b a)))
       (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 5000000000.0) {
		tmp = (r * (b * (1.0 + (b * (b * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))))) / cos((b + a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-0.58d0)) then
        tmp = t_0
    else if (b <= 5000000000.0d0) then
        tmp = (r * (b * (1.0d0 + (b * (b * ((-0.16666666666666666d0) + ((b * b) * 0.008333333333333333d0))))))) / cos((b + a))
    else
        tmp = 1.0d0 / (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 5000000000.0) {
		tmp = (r * (b * (1.0 + (b * (b * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))))) / Math.cos((b + a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -0.58:
		tmp = t_0
	elif b <= 5000000000.0:
		tmp = (r * (b * (1.0 + (b * (b * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))))) / math.cos((b + a))
	else:
		tmp = 1.0 / (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 5000000000.0)
		tmp = Float64(Float64(r * 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(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 5000000000.0)
		tmp = (r * (b * (1.0 + (b * (b * (-0.16666666666666666 + ((b * b) * 0.008333333333333333))))))) / cos((b + a));
	else
		tmp = 1.0 / (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.58], t$95$0, If[LessEqual[b, 5000000000.0], N[(N[(r * 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]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.57999999999999996

   1. Initial program 53.6%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 11.6%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in a around 0

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

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

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

      2. sin-lowering-sin.f64 13.4%

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

   10. Simplified 13.4%

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

   if -0.57999999999999996 < b < 5e9

   1. Initial program 99.3%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in b around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 97.8%

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

   7. Simplified 97.8%

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

   if 5e9 < b

   1. Initial program 62.8%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 63.0%

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

   3. Simplified 63.0%

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

      1. +-commutative N/A

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

      2. cos-sum N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 67.8%

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

       1. clear-num N/A

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

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

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

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

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

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

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

       5. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

       11. sin-lowering-sin.f64 67.6%

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

   11. Applied egg-rr 67.6%

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

   12. Taylor expanded in b around 0

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

   14. Simplified 13.7%

       \[\leadsto \frac{1}{\frac{\color{blue}{1}}{r \cdot \sin b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 55.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -0.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 920000000000:\\ \;\;\;\;\frac{\frac{r \cdot b}{\cos \left(b + a\right)}}{1 + \left(b \cdot b\right) \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -0.58)
     t_0
     (if (<= b 920000000000.0)
       (/ (/ (* r b) (cos (+ b a))) (+ 1.0 (* (* b b) 0.16666666666666666)))
       (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 920000000000.0) {
		tmp = ((r * b) / cos((b + a))) / (1.0 + ((b * b) * 0.16666666666666666));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-0.58d0)) then
        tmp = t_0
    else if (b <= 920000000000.0d0) then
        tmp = ((r * b) / cos((b + a))) / (1.0d0 + ((b * b) * 0.16666666666666666d0))
    else
        tmp = 1.0d0 / (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 920000000000.0) {
		tmp = ((r * b) / Math.cos((b + a))) / (1.0 + ((b * b) * 0.16666666666666666));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -0.58:
		tmp = t_0
	elif b <= 920000000000.0:
		tmp = ((r * b) / math.cos((b + a))) / (1.0 + ((b * b) * 0.16666666666666666))
	else:
		tmp = 1.0 / (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 920000000000.0)
		tmp = Float64(Float64(Float64(r * b) / cos(Float64(b + a))) / Float64(1.0 + Float64(Float64(b * b) * 0.16666666666666666)));
	else
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 920000000000.0)
		tmp = ((r * b) / cos((b + a))) / (1.0 + ((b * b) * 0.16666666666666666));
	else
		tmp = 1.0 / (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.58], t$95$0, If[LessEqual[b, 920000000000.0], N[(N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(b * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.57999999999999996

   1. Initial program 53.6%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 11.6%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in a around 0

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

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

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

      2. sin-lowering-sin.f64 13.4%

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

   10. Simplified 13.4%

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

   if -0.57999999999999996 < b < 9.2e11

   1. Initial program 99.3%

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

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 97.6%

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in b around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. /-lowering-/.f64 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in r around 0

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

      1. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 97.6%

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

   10. Simplified 97.6%

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

   if 9.2e11 < b

   1. Initial program 62.8%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 63.0%

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

   3. Simplified 63.0%

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

      1. +-commutative N/A

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

      2. cos-sum N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 67.8%

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

       1. clear-num N/A

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

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

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

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

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

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

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

       5. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

       11. sin-lowering-sin.f64 67.6%

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

   11. Applied egg-rr 67.6%

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

   12. Taylor expanded in b around 0

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

   14. Simplified 13.7%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.58:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 920000000000:\\ \;\;\;\;\frac{\frac{r \cdot b}{\cos \left(b + a\right)}}{1 + \left(b \cdot b\right) \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -1850000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3100:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + \left(b \cdot b\right) \cdot -0.16666666666666666\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -1850000000.0)
     t_0
     (if (<= b 3100.0)
       (/ (* r (* b (+ 1.0 (* (* b b) -0.16666666666666666)))) (cos (+ b a)))
       (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -1850000000.0) {
		tmp = t_0;
	} else if (b <= 3100.0) {
		tmp = (r * (b * (1.0 + ((b * b) * -0.16666666666666666)))) / cos((b + a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-1850000000.0d0)) then
        tmp = t_0
    else if (b <= 3100.0d0) then
        tmp = (r * (b * (1.0d0 + ((b * b) * (-0.16666666666666666d0))))) / cos((b + a))
    else
        tmp = 1.0d0 / (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -1850000000.0) {
		tmp = t_0;
	} else if (b <= 3100.0) {
		tmp = (r * (b * (1.0 + ((b * b) * -0.16666666666666666)))) / Math.cos((b + a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -1850000000.0:
		tmp = t_0
	elif b <= 3100.0:
		tmp = (r * (b * (1.0 + ((b * b) * -0.16666666666666666)))) / math.cos((b + a))
	else:
		tmp = 1.0 / (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -1850000000.0)
		tmp = t_0;
	elseif (b <= 3100.0)
		tmp = Float64(Float64(r * Float64(b * Float64(1.0 + Float64(Float64(b * b) * -0.16666666666666666)))) / cos(Float64(b + a)));
	else
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -1850000000.0)
		tmp = t_0;
	elseif (b <= 3100.0)
		tmp = (r * (b * (1.0 + ((b * b) * -0.16666666666666666)))) / cos((b + a));
	else
		tmp = 1.0 / (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1850000000.0], t$95$0, If[LessEqual[b, 3100.0], N[(N[(r * N[(b * N[(1.0 + N[(N[(b * b), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.85e9

   1. Initial program 55.1%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 11.7%

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

   7. Simplified 11.7%

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

   8. Taylor expanded in a around 0

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

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

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

      2. sin-lowering-sin.f64 13.4%

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

   10. Simplified 13.4%

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

   if -1.85e9 < b < 3100

   1. Initial program 98.0%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 97.0%

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

   7. Simplified 97.0%

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

   if 3100 < b

   1. Initial program 63.4%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 63.6%

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

   3. Simplified 63.6%

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

      1. +-commutative N/A

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

      2. cos-sum N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 68.3%

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

       1. clear-num N/A

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

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

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

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

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

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

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

       5. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

       11. sin-lowering-sin.f64 68.2%

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

   11. Applied egg-rr 68.2%

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

   12. Taylor expanded in b around 0

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

   14. Simplified 13.5%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1850000000:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 3100:\\ \;\;\;\;\frac{r \cdot \left(b \cdot \left(1 + \left(b \cdot b\right) \cdot -0.16666666666666666\right)\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -1950000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 55:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + \left(b \cdot b\right) \cdot -0.16666666666666666\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -1950000000.0)
     t_0
     (if (<= b 55.0)
       (* r (/ (* b (+ 1.0 (* (* b b) -0.16666666666666666))) (cos (+ b a))))
       (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -1950000000.0) {
		tmp = t_0;
	} else if (b <= 55.0) {
		tmp = r * ((b * (1.0 + ((b * b) * -0.16666666666666666))) / cos((b + a)));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-1950000000.0d0)) then
        tmp = t_0
    else if (b <= 55.0d0) then
        tmp = r * ((b * (1.0d0 + ((b * b) * (-0.16666666666666666d0)))) / cos((b + a)))
    else
        tmp = 1.0d0 / (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -1950000000.0) {
		tmp = t_0;
	} else if (b <= 55.0) {
		tmp = r * ((b * (1.0 + ((b * b) * -0.16666666666666666))) / Math.cos((b + a)));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -1950000000.0:
		tmp = t_0
	elif b <= 55.0:
		tmp = r * ((b * (1.0 + ((b * b) * -0.16666666666666666))) / math.cos((b + a)))
	else:
		tmp = 1.0 / (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -1950000000.0)
		tmp = t_0;
	elseif (b <= 55.0)
		tmp = Float64(r * Float64(Float64(b * Float64(1.0 + Float64(Float64(b * b) * -0.16666666666666666))) / cos(Float64(b + a))));
	else
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -1950000000.0)
		tmp = t_0;
	elseif (b <= 55.0)
		tmp = r * ((b * (1.0 + ((b * b) * -0.16666666666666666))) / cos((b + a)));
	else
		tmp = 1.0 / (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1950000000.0], t$95$0, If[LessEqual[b, 55.0], N[(r * N[(N[(b * N[(1.0 + N[(N[(b * b), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.95e9

   1. Initial program 55.1%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 11.7%

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

   7. Simplified 11.7%

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

   8. Taylor expanded in a around 0

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

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

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

      2. sin-lowering-sin.f64 13.4%

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

   10. Simplified 13.4%

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

   if -1.95e9 < b < 55

   1. Initial program 98.0%

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

   3. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 97.0%

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

   5. Simplified 97.0%

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

   if 55 < b

   1. Initial program 63.4%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 63.6%

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

   3. Simplified 63.6%

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

      1. +-commutative N/A

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

      2. cos-sum N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 68.3%

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

       1. clear-num N/A

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

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

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

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

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

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

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

       5. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

       11. sin-lowering-sin.f64 68.2%

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

   11. Applied egg-rr 68.2%

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

   12. Taylor expanded in b around 0

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

   14. Simplified 13.5%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1950000000:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 55:\\ \;\;\;\;r \cdot \frac{b \cdot \left(1 + \left(b \cdot b\right) \cdot -0.16666666666666666\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -0.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 530000000000:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -0.58)
     t_0
     (if (<= b 530000000000.0)
       (/ (* r b) (cos (+ b a)))
       (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 530000000000.0) {
		tmp = (r * b) / cos((b + a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-0.58d0)) then
        tmp = t_0
    else if (b <= 530000000000.0d0) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = 1.0d0 / (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 530000000000.0) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -0.58:
		tmp = t_0
	elif b <= 530000000000.0:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = 1.0 / (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 530000000000.0)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 530000000000.0)
		tmp = (r * b) / cos((b + a));
	else
		tmp = 1.0 / (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.58], t$95$0, If[LessEqual[b, 530000000000.0], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.57999999999999996

   1. Initial program 53.6%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 11.6%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in a around 0

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

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

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

      2. sin-lowering-sin.f64 13.4%

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

   10. Simplified 13.4%

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

   if -0.57999999999999996 < b < 5.3e11

   1. Initial program 99.3%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 97.5%

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

   if 5.3e11 < b

   1. Initial program 62.8%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 63.0%

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

   3. Simplified 63.0%

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

      1. +-commutative N/A

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

      2. cos-sum N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 67.8%

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

       1. clear-num N/A

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

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

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

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

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

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

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

       5. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

       11. sin-lowering-sin.f64 67.6%

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

   11. Applied egg-rr 67.6%

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

   12. Taylor expanded in b around 0

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

   14. Simplified 13.7%

       \[\leadsto \frac{1}{\frac{\color{blue}{1}}{r \cdot \sin b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -0.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 320000000000:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -0.58)
     t_0
     (if (<= b 320000000000.0)
       (* r (/ b (cos (+ b a))))
       (/ 1.0 (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 320000000000.0) {
		tmp = r * (b / cos((b + a)));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-0.58d0)) then
        tmp = t_0
    else if (b <= 320000000000.0d0) then
        tmp = r * (b / cos((b + a)))
    else
        tmp = 1.0d0 / (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 320000000000.0) {
		tmp = r * (b / Math.cos((b + a)));
	} else {
		tmp = 1.0 / (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -0.58:
		tmp = t_0
	elif b <= 320000000000.0:
		tmp = r * (b / math.cos((b + a)))
	else:
		tmp = 1.0 / (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 320000000000.0)
		tmp = Float64(r * Float64(b / cos(Float64(b + a))));
	else
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 320000000000.0)
		tmp = r * (b / cos((b + a)));
	else
		tmp = 1.0 / (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.58], t$95$0, If[LessEqual[b, 320000000000.0], N[(r * N[(b / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.57999999999999996

   1. Initial program 53.6%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in b around 0

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

      1. cos-lowering-cos.f64 11.6%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in a around 0

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

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

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

      2. sin-lowering-sin.f64 13.4%

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   10. Simplified 13.4%

       \[\leadsto \color{blue}{r \cdot \sin b} \]

   if -0.57999999999999996 < b < 3.2e11

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{b}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 97.5%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos \left(a + b\right)} \]

   if 3.2e11 < b

   1. Initial program 62.8%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      7. +-lowering-+.f64 63.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 63.0%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + b\right)\right) \]

      2. cos-sum N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos a \cdot \cos b - \color{blue}{\sin a \cdot \sin b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\left(\cos a \cdot \cos b\right), \color{blue}{\left(\sin a \cdot \sin b\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos a, \cos b\right), \left(\color{blue}{\sin a} \cdot \sin b\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \cos b\right), \left(\sin \color{blue}{a} \cdot \sin b\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \left(\sin a \cdot \sin b\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \left(\sin b \cdot \color{blue}{\sin a}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \mathsf{*.f64}\left(\sin b, \color{blue}{\sin a}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin \color{blue}{a}\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin b \cdot \sin a}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{cos.f64}\left(b\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 67.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1} \cdot \cos b - \sin b \cdot \sin a} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot \cos b - \sin b \cdot \sin a}{r \cdot \sin b}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot \cos b - \sin b \cdot \sin a}{r \cdot \sin b}\right)}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 \cdot \cos b - \sin b \cdot \sin a\right), \color{blue}{\left(r \cdot \sin b\right)}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 \cdot \cos b\right), \left(\sin b \cdot \sin a\right)\right), \left(\color{blue}{r} \cdot \sin b\right)\right)\right) \]

       5. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\cos b, \left(\sin b \cdot \sin a\right)\right), \left(r \cdot \sin b\right)\right)\right) \]

       6. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(b\right), \left(\sin b \cdot \sin a\right)\right), \left(r \cdot \sin b\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{*.f64}\left(\sin b, \sin a\right)\right), \left(r \cdot \sin b\right)\right)\right) \]

       8. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin a\right)\right), \left(r \cdot \sin b\right)\right)\right) \]

       9. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right), \left(r \cdot \sin b\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right), \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right)\right)\right) \]

       11. sin-lowering-sin.f64 67.6%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right), \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]

   11. Applied egg-rr 67.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{\cos b - \sin b \cdot \sin a}{r \cdot \sin b}}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 13.7%

       \[\leadsto \frac{1}{\frac{\color{blue}{1}}{r \cdot \sin b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.58:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 320000000000:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{r \cdot \sin b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -0.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 160000000000:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -0.58)
     t_0
     (if (<= b 160000000000.0) (* r (/ b (cos (+ b a)))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 160000000000.0) {
		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 * sin(b)
    if (b <= (-0.58d0)) then
        tmp = t_0
    else if (b <= 160000000000.0d0) 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.sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 160000000000.0) {
		tmp = r * (b / Math.cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -0.58:
		tmp = t_0
	elif b <= 160000000000.0:
		tmp = r * (b / math.cos((b + a)))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 160000000000.0)
		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 * sin(b);
	tmp = 0.0;
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 160000000000.0)
		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[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.58], t$95$0, If[LessEqual[b, 160000000000.0], 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 < -0.57999999999999996 or 1.6e11 < b

   1. Initial program 58.1%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      7. +-lowering-+.f64 58.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \color{blue}{\cos a}\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 12.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 12.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{r \cdot \sin b} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

      2. sin-lowering-sin.f64 13.6%

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   10. Simplified 13.6%

       \[\leadsto \color{blue}{r \cdot \sin b} \]

   if -0.57999999999999996 < b < 1.6e11

   1. Initial program 99.3%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(r, \mathsf{/.f64}\left(\color{blue}{b}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(a, b\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 97.5%

      \[\leadsto r \cdot \frac{\color{blue}{b}}{\cos \left(a + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.58:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 160000000000:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -0.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.75:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -0.58) t_0 (if (<= b 1.75) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 1.75) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-0.58d0)) then
        tmp = t_0
    else if (b <= 1.75d0) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -0.58) {
		tmp = t_0;
	} else if (b <= 1.75) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -0.58:
		tmp = t_0
	elif b <= 1.75:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 1.75)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -0.58)
		tmp = t_0;
	elseif (b <= 1.75)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.58], t$95$0, If[LessEqual[b, 1.75], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.57999999999999996 or 1.75 < b

   1. Initial program 58.4%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      7. +-lowering-+.f64 58.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \color{blue}{\cos a}\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 12.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 12.1%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{r \cdot \sin b} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

      2. sin-lowering-sin.f64 13.4%

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

   10. Simplified 13.4%

       \[\leadsto \color{blue}{r \cdot \sin b} \]

   if -0.57999999999999996 < b < 1.75

   1. Initial program 99.2%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      7. +-lowering-+.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + b\right)\right) \]

      2. cos-sum N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos a \cdot \cos b - \color{blue}{\sin a \cdot \sin b}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\left(\cos a \cdot \cos b\right), \color{blue}{\left(\sin a \cdot \sin b\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos a, \cos b\right), \left(\color{blue}{\sin a} \cdot \sin b\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \cos b\right), \left(\sin \color{blue}{a} \cdot \sin b\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \left(\sin a \cdot \sin b\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \left(\sin b \cdot \color{blue}{\sin a}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \mathsf{*.f64}\left(\sin b, \color{blue}{\sin a}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin \color{blue}{a}\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(a\right), \mathsf{cos.f64}\left(b\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin b \cdot \sin a}} \]

   7. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   8. 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 98.1%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right)\right) \]

   9. Simplified 98.1%

      \[\leadsto \color{blue}{b \cdot \frac{r}{\cos a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 39.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ r \cdot \sin b \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (sin b)))

C

double code(double r, double a, double b) {
	return r * sin(b);
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * sin(b)
end function

Java

public static double code(double r, double a, double b) {
	return r * Math.sin(b);
}

Python

def code(r, a, b):
	return r * math.sin(b)

Julia

function code(r, a, b)
	return Float64(r * sin(b))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * sin(b);
end

Wolfram

code[r_, a_, b_] := N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos \left(a + b\right)}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   7. +-lowering-+.f64 80.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 80.2%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \color{blue}{\cos a}\right) \]
6. Step-by-step derivation

   1. cos-lowering-cos.f64 57.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(a\right)\right) \]

7. Simplified 57.8%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{r \cdot \sin b} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\sin b}\right) \]

   2. sin-lowering-sin.f64 41.6%

      \[\leadsto \mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right) \]

10. Simplified 41.6%

    \[\leadsto \color{blue}{r \cdot \sin b} \]
11. Add Preprocessing

Alternative 19: 34.9% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ r \cdot b \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r b))

C

double code(double r, double a, double b) {
	return r * b;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * b
end function

Java

public static double code(double r, double a, double b) {
	return r * b;
}

Python

def code(r, a, b):
	return r * b

Julia

function code(r, a, b)
	return Float64(r * b)
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * b;
end

Wolfram

code[r_, a_, b_] := N[(r * b), $MachinePrecision]

Derivation

1. Initial program 80.1%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing

3. Taylor expanded in b around 0

   \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{b}{\cos a}\right)}\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(r, \mathsf{/.f64}\left(b, \color{blue}{\cos a}\right)\right) \]

   2. cos-lowering-cos.f64 53.7%

      \[\leadsto \mathsf{*.f64}\left(r, \mathsf{/.f64}\left(b, \mathsf{cos.f64}\left(a\right)\right)\right) \]

5. Simplified 53.7%

   \[\leadsto r \cdot \color{blue}{\frac{b}{\cos a}} \]

6. Taylor expanded in a around 0

   \[\leadsto \color{blue}{b \cdot r} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto r \cdot \color{blue}{b} \]

   2. *-lowering-*.f64 37.0%

      \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{b}\right) \]

8. Simplified 37.0%

   \[\leadsto \color{blue}{r \cdot b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))