rsin B (should all be same)

Percentage Accurate: 77.1% → 99.5%

Time: 13.1s

Alternatives: 11

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: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. +-commutative 76.3%

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

3. Simplified 76.3%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 2: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -0.00023:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 66000000:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos a)))))
   (if (<= a -0.00023) t_0 (if (<= a 66000000.0) (* (tan b) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(a));
	double tmp;
	if (a <= -0.00023) {
		tmp = t_0;
	} else if (a <= 66000000.0) {
		tmp = tan(b) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * (sin(b) / cos(a))
    if (a <= (-0.00023d0)) then
        tmp = t_0
    else if (a <= 66000000.0d0) then
        tmp = tan(b) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * (Math.sin(b) / Math.cos(a));
	double tmp;
	if (a <= -0.00023) {
		tmp = t_0;
	} else if (a <= 66000000.0) {
		tmp = Math.tan(b) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * (math.sin(b) / math.cos(a))
	tmp = 0
	if a <= -0.00023:
		tmp = t_0
	elif a <= 66000000.0:
		tmp = math.tan(b) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(a)))
	tmp = 0.0
	if (a <= -0.00023)
		tmp = t_0;
	elseif (a <= 66000000.0)
		tmp = Float64(tan(b) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * (sin(b) / cos(a));
	tmp = 0.0;
	if (a <= -0.00023)
		tmp = t_0;
	elseif (a <= 66000000.0)
		tmp = tan(b) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00023], t$95$0, If[LessEqual[a, 66000000.0], N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3000000000000001e-4 or 6.6e7 < a

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in b around 0 55.7%

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

   if -2.3000000000000001e-4 < a < 6.6e7

   1. Initial program 97.4%

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

      1. associate-*r/ 97.4%

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

      2. +-commutative 97.4%

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

   3. Simplified 97.4%

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

      1. add-log-exp 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. rem-log-exp 97.4%

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

      2. expm1-log1p-u 97.4%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(b + a\right)\right)\right)}} \]

      3. expm1-define 97.2%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(b + a\right)\right)} - 1}} \]

      4. log1p-undefine 97.1%

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

      5. rem-exp-log 97.1%

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

      6. +-commutative 97.1%

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

   8. Applied egg-rr 97.1%

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

   9. Taylor expanded in a around 0 97.1%

      \[\leadsto \frac{r \cdot \sin b}{\left(\color{blue}{\cos b} + 1\right) - 1} \]
   10. Step-by-step derivation

       1. associate--l+ 97.4%

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

       2. metadata-eval 97.4%

          \[\leadsto \frac{r \cdot \sin b}{\cos b + \color{blue}{0}} \]

       3. +-rgt-identity 97.4%

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

       4. associate-/l* 97.4%

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

       5. *-commutative 97.4%

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

       6. quot-tan 97.6%

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

   11. Applied egg-rr 97.6%

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

Alternative 3: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -0.00024:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 66000000:\\ \;\;\;\;\tan b \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos a)))))
   (if (<= a -0.00024) t_0 (if (<= a 66000000.0) (* (tan b) r) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(a));
	double tmp;
	if (a <= -0.00024) {
		tmp = t_0;
	} else if (a <= 66000000.0) {
		tmp = tan(b) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * (r / cos(a))
    if (a <= (-0.00024d0)) then
        tmp = t_0
    else if (a <= 66000000.0d0) then
        tmp = tan(b) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * (r / Math.cos(a));
	double tmp;
	if (a <= -0.00024) {
		tmp = t_0;
	} else if (a <= 66000000.0) {
		tmp = Math.tan(b) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * (r / math.cos(a))
	tmp = 0
	if a <= -0.00024:
		tmp = t_0
	elif a <= 66000000.0:
		tmp = math.tan(b) * r
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(a)))
	tmp = 0.0
	if (a <= -0.00024)
		tmp = t_0;
	elseif (a <= 66000000.0)
		tmp = Float64(tan(b) * r);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * (r / cos(a));
	tmp = 0.0;
	if (a <= -0.00024)
		tmp = t_0;
	elseif (a <= 66000000.0)
		tmp = tan(b) * r;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00024], t$95$0, If[LessEqual[a, 66000000.0], N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.40000000000000006e-4 or 6.6e7 < a

   1. Initial program 55.5%

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

      1. associate-*r/ 55.5%

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

      2. +-commutative 55.5%

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

   3. Simplified 55.5%

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

      1. associate-*r/ 55.5%

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

      2. *-commutative 55.5%

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

      3. div-inv 55.4%

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

      4. associate-*l* 55.5%

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

   6. Applied egg-rr 55.5%

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

   7. Taylor expanded in b around 0 55.8%

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

   if -2.40000000000000006e-4 < a < 6.6e7

   1. Initial program 97.4%

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

      1. associate-*r/ 97.4%

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

      2. +-commutative 97.4%

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

   3. Simplified 97.4%

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

      1. add-log-exp 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. rem-log-exp 97.4%

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

      2. expm1-log1p-u 97.4%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(b + a\right)\right)\right)}} \]

      3. expm1-define 97.2%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(b + a\right)\right)} - 1}} \]

      4. log1p-undefine 97.1%

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

      5. rem-exp-log 97.1%

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

      6. +-commutative 97.1%

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

   8. Applied egg-rr 97.1%

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

   9. Taylor expanded in a around 0 97.1%

      \[\leadsto \frac{r \cdot \sin b}{\left(\color{blue}{\cos b} + 1\right) - 1} \]
   10. Step-by-step derivation

       1. associate--l+ 97.4%

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

       2. metadata-eval 97.4%

          \[\leadsto \frac{r \cdot \sin b}{\cos b + \color{blue}{0}} \]

       3. +-rgt-identity 97.4%

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

       4. associate-/l* 97.4%

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

       5. *-commutative 97.4%

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

       6. quot-tan 97.6%

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

   11. Applied egg-rr 97.6%

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

Alternative 4: 77.1% 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]

Derivation

1. Initial program 76.3%

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

Alternative 5: 77.1% 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 76.3%

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

   1. associate-*r/ 76.3%

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

   2. +-commutative 76.3%

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

3. Simplified 76.3%

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

   1. *-commutative 76.3%

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

   2. associate-/l* 76.3%

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

6. Applied egg-rr 76.3%

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

Alternative 6: 77.0% 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 76.3%

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

   1. associate-*r/ 76.3%

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

   2. +-commutative 76.3%

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

3. Simplified 76.3%

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

   1. add-log-exp 76.1%

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

6. Applied egg-rr 76.1%

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

   1. cos-sum 99.3%

      \[\leadsto \frac{r \cdot \sin b}{\log \left(e^{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}}\right)} \]

   2. add-log-exp 99.5%

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

   3. fma-neg 99.5%

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

   4. distribute-rgt-neg-in 99.5%

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

8. Applied egg-rr 99.5%

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

   1. fma-undefine 99.5%

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

   2. add-sqr-sqrt 48.6%

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

   3. sqrt-unprod 88.6%

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

   4. sqr-neg 88.6%

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

   5. sqrt-unprod 40.0%

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

   6. add-sqr-sqrt 76.5%

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

   7. cos-diff 76.4%

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

10. Applied egg-rr 76.4%

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

Alternative 7: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.00045:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -1.75e-6) t_0 (if (<= b 0.00045) (/ (* b r) (cos (+ b a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -1.75e-6) {
		tmp = t_0;
	} else if (b <= 0.00045) {
		tmp = (b * r) / 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 = tan(b) * r
    if (b <= (-1.75d-6)) then
        tmp = t_0
    else if (b <= 0.00045d0) then
        tmp = (b * r) / 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 = Math.tan(b) * r;
	double tmp;
	if (b <= -1.75e-6) {
		tmp = t_0;
	} else if (b <= 0.00045) {
		tmp = (b * r) / Math.cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -1.75e-6:
		tmp = t_0
	elif b <= 0.00045:
		tmp = (b * r) / math.cos((b + a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -1.75e-6)
		tmp = t_0;
	elseif (b <= 0.00045)
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -1.75e-6)
		tmp = t_0;
	elseif (b <= 0.00045)
		tmp = (b * r) / cos((b + a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -1.75e-6], t$95$0, If[LessEqual[b, 0.00045], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.74999999999999997e-6 or 4.4999999999999999e-4 < b

   1. Initial program 52.3%

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

      1. associate-*r/ 52.3%

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

      2. +-commutative 52.3%

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

   3. Simplified 52.3%

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

      1. add-log-exp 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. rem-log-exp 52.3%

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

      2. expm1-log1p-u 52.3%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(b + a\right)\right)\right)}} \]

      3. expm1-define 52.1%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(b + a\right)\right)} - 1}} \]

      4. log1p-undefine 52.0%

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

      5. rem-exp-log 52.0%

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

      6. +-commutative 52.0%

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

   8. Applied egg-rr 52.0%

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

   9. Taylor expanded in a around 0 52.2%

      \[\leadsto \frac{r \cdot \sin b}{\left(\color{blue}{\cos b} + 1\right) - 1} \]
   10. Step-by-step derivation

       1. associate--l+ 52.5%

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

       2. metadata-eval 52.5%

          \[\leadsto \frac{r \cdot \sin b}{\cos b + \color{blue}{0}} \]

       3. +-rgt-identity 52.5%

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

       4. associate-/l* 52.5%

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

       5. *-commutative 52.5%

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

       6. quot-tan 52.6%

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

   11. Applied egg-rr 52.6%

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

   if -1.74999999999999997e-6 < b < 4.4999999999999999e-4

   1. Initial program 99.1%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in b around 0 99.2%

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

Alternative 8: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -1.75e-6) t_0 (if (<= b 2.1e-5) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -1.75e-6) {
		tmp = t_0;
	} else if (b <= 2.1e-5) {
		tmp = r * (b / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(b) * r
    if (b <= (-1.75d-6)) then
        tmp = t_0
    else if (b <= 2.1d-5) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.tan(b) * r;
	double tmp;
	if (b <= -1.75e-6) {
		tmp = t_0;
	} else if (b <= 2.1e-5) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -1.75e-6:
		tmp = t_0
	elif b <= 2.1e-5:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -1.75e-6)
		tmp = t_0;
	elseif (b <= 2.1e-5)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -1.75e-6)
		tmp = t_0;
	elseif (b <= 2.1e-5)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -1.75e-6], t$95$0, If[LessEqual[b, 2.1e-5], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.74999999999999997e-6 or 2.09999999999999988e-5 < b

   1. Initial program 52.3%

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

      1. associate-*r/ 52.3%

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

      2. +-commutative 52.3%

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

   3. Simplified 52.3%

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

      1. add-log-exp 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. rem-log-exp 52.3%

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

      2. expm1-log1p-u 52.3%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(b + a\right)\right)\right)}} \]

      3. expm1-define 52.1%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(b + a\right)\right)} - 1}} \]

      4. log1p-undefine 52.0%

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

      5. rem-exp-log 52.0%

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

      6. +-commutative 52.0%

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

   8. Applied egg-rr 52.0%

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

   9. Taylor expanded in a around 0 52.2%

      \[\leadsto \frac{r \cdot \sin b}{\left(\color{blue}{\cos b} + 1\right) - 1} \]
   10. Step-by-step derivation

       1. associate--l+ 52.5%

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

       2. metadata-eval 52.5%

          \[\leadsto \frac{r \cdot \sin b}{\cos b + \color{blue}{0}} \]

       3. +-rgt-identity 52.5%

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

       4. associate-/l* 52.5%

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

       5. *-commutative 52.5%

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

       6. quot-tan 52.6%

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

   11. Applied egg-rr 52.6%

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

   if -1.74999999999999997e-6 < b < 2.09999999999999988e-5

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in b around 0 99.1%

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

Alternative 9: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan b \cdot r\\ \mathbf{if}\;b \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.000112:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (tan b) r)))
   (if (<= b -1.25e-6) t_0 (if (<= b 0.000112) (/ (* b r) (cos a)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = tan(b) * r;
	double tmp;
	if (b <= -1.25e-6) {
		tmp = t_0;
	} else if (b <= 0.000112) {
		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 = tan(b) * r
    if (b <= (-1.25d-6)) then
        tmp = t_0
    else if (b <= 0.000112d0) 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 = Math.tan(b) * r;
	double tmp;
	if (b <= -1.25e-6) {
		tmp = t_0;
	} else if (b <= 0.000112) {
		tmp = (b * r) / Math.cos(a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.tan(b) * r
	tmp = 0
	if b <= -1.25e-6:
		tmp = t_0
	elif b <= 0.000112:
		tmp = (b * r) / math.cos(a)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(tan(b) * r)
	tmp = 0.0
	if (b <= -1.25e-6)
		tmp = t_0;
	elseif (b <= 0.000112)
		tmp = Float64(Float64(b * r) / cos(a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = tan(b) * r;
	tmp = 0.0;
	if (b <= -1.25e-6)
		tmp = t_0;
	elseif (b <= 0.000112)
		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[Tan[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -1.25e-6], t$95$0, If[LessEqual[b, 0.000112], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.2500000000000001e-6 or 1.11999999999999998e-4 < b

   1. Initial program 52.3%

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

      1. associate-*r/ 52.3%

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

      2. +-commutative 52.3%

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

   3. Simplified 52.3%

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

      1. add-log-exp 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. rem-log-exp 52.3%

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

      2. expm1-log1p-u 52.3%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(b + a\right)\right)\right)}} \]

      3. expm1-define 52.1%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(b + a\right)\right)} - 1}} \]

      4. log1p-undefine 52.0%

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

      5. rem-exp-log 52.0%

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

      6. +-commutative 52.0%

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

   8. Applied egg-rr 52.0%

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

   9. Taylor expanded in a around 0 52.2%

      \[\leadsto \frac{r \cdot \sin b}{\left(\color{blue}{\cos b} + 1\right) - 1} \]
   10. Step-by-step derivation

       1. associate--l+ 52.5%

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

       2. metadata-eval 52.5%

          \[\leadsto \frac{r \cdot \sin b}{\cos b + \color{blue}{0}} \]

       3. +-rgt-identity 52.5%

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

       4. associate-/l* 52.5%

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

       5. *-commutative 52.5%

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

       6. quot-tan 52.6%

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

   11. Applied egg-rr 52.6%

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

   if -1.2500000000000001e-6 < b < 1.11999999999999998e-4

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in b around 0 99.2%

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

Alternative 10: 60.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. associate-*r/ 76.3%

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

   2. +-commutative 76.3%

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

3. Simplified 76.3%

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

   1. add-log-exp 76.1%

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

6. Applied egg-rr 76.1%

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

   1. rem-log-exp 76.3%

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

   2. expm1-log1p-u 76.2%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(b + a\right)\right)\right)}} \]

   3. expm1-define 76.0%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(b + a\right)\right)} - 1}} \]

   4. log1p-undefine 76.0%

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

   5. rem-exp-log 76.0%

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

   6. +-commutative 76.0%

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

8. Applied egg-rr 76.0%

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

9. Taylor expanded in a around 0 60.5%

   \[\leadsto \frac{r \cdot \sin b}{\left(\color{blue}{\cos b} + 1\right) - 1} \]
10. Step-by-step derivation

    1. associate--l+ 60.6%

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

    2. metadata-eval 60.6%

       \[\leadsto \frac{r \cdot \sin b}{\cos b + \color{blue}{0}} \]

    3. +-rgt-identity 60.6%

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

    4. associate-/l* 60.6%

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

    5. *-commutative 60.6%

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

    6. quot-tan 60.7%

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

11. Applied egg-rr 60.7%

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

Alternative 11: 34.7% 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 76.3%

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

   1. +-commutative 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in b around 0 53.0%

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

6. Taylor expanded in a around 0 37.3%

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

Reproduce

herbie shell --seed 2024052 -o generate:simplify
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))