rsin B (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 18.4s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. associate-*r/ 72.0%

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

   2. /-rgt-identity 72.0%

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

   3. metadata-eval 72.0%

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

   4. associate-/l/ 72.0%

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

   5. associate-*l/ 72.1%

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

   6. associate-/l/ 72.1%

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

   7. metadata-eval 72.1%

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

   8. metadata-eval 72.1%

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

   9. metadata-eval 72.1%

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

   10. /-rgt-identity 72.1%

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

   11. +-commutative 72.1%

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

3. Simplified 72.1%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

3. Simplified 72.0%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return sin(b) * (r / ((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 = sin(b) * (r / ((cos(b) * cos(a)) - (sin(b) * sin(a))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(N[Sin[b], $MachinePrecision] * N[(r / 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 72.0%

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

   1. associate-*r/ 72.0%

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

   2. /-rgt-identity 72.0%

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

   3. metadata-eval 72.0%

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

   4. associate-/l/ 72.0%

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

   5. associate-*l/ 72.1%

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

   6. associate-/l/ 72.1%

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

   7. metadata-eval 72.1%

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

   8. metadata-eval 72.1%

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

   9. metadata-eval 72.1%

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

   10. /-rgt-identity 72.1%

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

   11. +-commutative 72.1%

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

3. Simplified 72.1%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+16} \lor \neg \left(a \leq 0.08\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -2.9e+16) (not (<= a 0.08)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -2.9e+16) || !(a <= 0.08)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-2.9d+16)) .or. (.not. (a <= 0.08d0))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -2.9e+16) || !(a <= 0.08)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -2.9e+16) or not (a <= 0.08):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -2.9e+16) || !(a <= 0.08))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -2.9e+16) || ~((a <= 0.08)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -2.9e+16], N[Not[LessEqual[a, 0.08]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.9e16 or 0.0800000000000000017 < a

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in b around 0 48.8%

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

   if -2.9e16 < a < 0.0800000000000000017

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in a around 0 97.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+16} \lor \neg \left(a \leq 0.08\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 5: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+16}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 0.075:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -2.9e+16)
   (* (sin b) (/ r (cos a)))
   (if (<= a 0.075) (* r (/ (sin b) (cos b))) (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -2.9e+16) {
		tmp = sin(b) * (r / cos(a));
	} else if (a <= 0.075) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = r * (sin(b) / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.9d+16)) then
        tmp = sin(b) * (r / cos(a))
    else if (a <= 0.075d0) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = r * (sin(b) / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -2.9e+16) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else if (a <= 0.075) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -2.9e+16:
		tmp = math.sin(b) * (r / math.cos(a))
	elif a <= 0.075:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -2.9e+16)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	elseif (a <= 0.075)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -2.9e+16)
		tmp = sin(b) * (r / cos(a));
	elseif (a <= 0.075)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -2.9e+16], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.075], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.9e16

   1. Initial program 46.0%

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

      1. associate-*r/ 46.0%

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

      2. /-rgt-identity 46.0%

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

      3. metadata-eval 46.0%

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

      4. associate-/l/ 46.0%

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

      5. associate-*l/ 46.1%

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

      6. associate-/l/ 46.1%

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

      7. metadata-eval 46.1%

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

      8. metadata-eval 46.1%

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

      9. metadata-eval 46.1%

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

      10. /-rgt-identity 46.1%

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

      11. +-commutative 46.1%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in b around 0 46.1%

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

   if -2.9e16 < a < 0.0749999999999999972

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in a around 0 97.3%

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

   if 0.0749999999999999972 < a

   1. Initial program 51.5%

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

      1. +-commutative 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in b around 0 52.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+16}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 0.075:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 6: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+16}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 0.075:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -2.9e+16)
   (* (sin b) (/ r (cos a)))
   (if (<= a 0.075) (* (sin b) (/ r (cos b))) (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -2.9e+16) {
		tmp = sin(b) * (r / cos(a));
	} else if (a <= 0.075) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = r * (sin(b) / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.9d+16)) then
        tmp = sin(b) * (r / cos(a))
    else if (a <= 0.075d0) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = r * (sin(b) / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -2.9e+16) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else if (a <= 0.075) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -2.9e+16:
		tmp = math.sin(b) * (r / math.cos(a))
	elif a <= 0.075:
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -2.9e+16)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	elseif (a <= 0.075)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -2.9e+16)
		tmp = sin(b) * (r / cos(a));
	elseif (a <= 0.075)
		tmp = sin(b) * (r / cos(b));
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -2.9e+16], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.075], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.9e16

   1. Initial program 46.0%

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

      1. associate-*r/ 46.0%

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

      2. /-rgt-identity 46.0%

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

      3. metadata-eval 46.0%

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

      4. associate-/l/ 46.0%

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

      5. associate-*l/ 46.1%

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

      6. associate-/l/ 46.1%

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

      7. metadata-eval 46.1%

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

      8. metadata-eval 46.1%

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

      9. metadata-eval 46.1%

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

      10. /-rgt-identity 46.1%

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

      11. +-commutative 46.1%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in b around 0 46.1%

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

   if -2.9e16 < a < 0.0749999999999999972

   1. Initial program 97.1%

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

      1. associate-*r/ 97.1%

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

      2. /-rgt-identity 97.1%

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

      3. metadata-eval 97.1%

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

      4. associate-/l/ 97.1%

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

      5. associate-*l/ 97.2%

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

      6. associate-/l/ 97.2%

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

      7. metadata-eval 97.2%

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

      8. metadata-eval 97.2%

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

      9. metadata-eval 97.2%

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

      10. /-rgt-identity 97.2%

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

      11. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in a around 0 97.4%

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

   if 0.0749999999999999972 < a

   1. Initial program 51.5%

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

      1. +-commutative 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in b around 0 52.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+16}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 0.075:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 7: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

2. Final simplification 72.0%

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

Alternative 8: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. associate-*r/ 72.0%

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

   2. /-rgt-identity 72.0%

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

   3. metadata-eval 72.0%

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

   4. associate-/l/ 72.0%

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

   5. associate-*l/ 72.1%

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

   6. associate-/l/ 72.1%

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

   7. metadata-eval 72.1%

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

   8. metadata-eval 72.1%

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

   9. metadata-eval 72.1%

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

   10. /-rgt-identity 72.1%

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

   11. +-commutative 72.1%

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

3. Simplified 72.1%

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

4. Final simplification 72.1%

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

Alternative 9: 55.5% 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 72.0%

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

   1. +-commutative 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in b around 0 52.2%

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

5. Final simplification 52.2%

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

Alternative 10: 55.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -960:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 18000:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -960.0)
   (* r (sin b))
   (if (<= b 18000.0) (* r (/ b (cos a))) (* r (- (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -960.0) {
		tmp = r * sin(b);
	} else if (b <= 18000.0) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r * -sin(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-960.0d0)) then
        tmp = r * sin(b)
    else if (b <= 18000.0d0) then
        tmp = r * (b / cos(a))
    else
        tmp = r * -sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -960.0) {
		tmp = r * Math.sin(b);
	} else if (b <= 18000.0) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r * -Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -960.0:
		tmp = r * math.sin(b)
	elif b <= 18000.0:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r * -math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -960.0)
		tmp = Float64(r * sin(b));
	elseif (b <= 18000.0)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r * Float64(-sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -960.0)
		tmp = r * sin(b);
	elseif (b <= 18000.0)
		tmp = r * (b / cos(a));
	else
		tmp = r * -sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -960.0], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 18000.0], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -960

   1. Initial program 55.1%

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

      1. associate-*r/ 55.0%

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

      2. /-rgt-identity 55.0%

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

      3. metadata-eval 55.0%

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

      4. associate-/l/ 55.0%

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

      5. associate-*l/ 55.2%

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

      6. associate-/l/ 55.2%

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

      7. metadata-eval 55.2%

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

      8. metadata-eval 55.2%

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

      9. metadata-eval 55.2%

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

      10. /-rgt-identity 55.2%

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

      11. +-commutative 55.2%

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

   3. Simplified 55.2%

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

      1. cos-sum 99.3%

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

      2. cancel-sign-sub-inv 99.3%

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

      3. fma-def 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in b around 0 7.6%

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

      1. mul-1-neg 7.6%

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

      2. unsub-neg 7.6%

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

      3. *-commutative 7.6%

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

   8. Simplified 7.6%

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

   9. Taylor expanded in a around 0 13.6%

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

   11. Simplified 13.6%

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

   if -960 < b < 18000

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in b around 0 96.8%

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

   if 18000 < b

   1. Initial program 41.3%

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

      1. +-commutative 41.3%

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

   3. Simplified 41.3%

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

      1. associate-*r/ 41.4%

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

      2. clear-num 41.4%

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

      3. *-commutative 41.4%

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

   5. Applied egg-rr 41.4%

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

   6. Taylor expanded in a around 0 41.2%

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

      1. associate-/r* 41.3%

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

   8. Simplified 41.3%

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

      1. frac-2neg 41.3%

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

      2. metadata-eval 41.3%

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

      3. div-inv 41.3%

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

      4. distribute-neg-frac 41.3%

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

      5. add-sqr-sqrt 25.1%

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

      6. sqrt-unprod 28.9%

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

      7. sqr-neg 28.9%

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

      8. sqrt-unprod 3.7%

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

      9. add-sqr-sqrt 7.4%

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

      10. frac-2neg 7.4%

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

      11. clear-num 7.4%

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

      12. div-inv 7.4%

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

      13. clear-num 7.4%

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

   10. Applied egg-rr 7.4%

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

   11. Taylor expanded in b around 0 11.6%

       \[\leadsto -1 \cdot \left(\sin b \cdot \color{blue}{r}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -960:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 18000:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \end{array} \]

Alternative 11: 55.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -960:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 18000:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -960.0)
   (* r (sin b))
   (if (<= b 18000.0) (/ (* r b) (cos a)) (* r (- (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -960.0) {
		tmp = r * sin(b);
	} else if (b <= 18000.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = r * -sin(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-960.0d0)) then
        tmp = r * sin(b)
    else if (b <= 18000.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = r * -sin(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -960.0) {
		tmp = r * Math.sin(b);
	} else if (b <= 18000.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = r * -Math.sin(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -960.0:
		tmp = r * math.sin(b)
	elif b <= 18000.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = r * -math.sin(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -960.0)
		tmp = Float64(r * sin(b));
	elseif (b <= 18000.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(r * Float64(-sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -960.0)
		tmp = r * sin(b);
	elseif (b <= 18000.0)
		tmp = (r * b) / cos(a);
	else
		tmp = r * -sin(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -960.0], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 18000.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(r * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -960

   1. Initial program 55.1%

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

      1. associate-*r/ 55.0%

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

      2. /-rgt-identity 55.0%

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

      3. metadata-eval 55.0%

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

      4. associate-/l/ 55.0%

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

      5. associate-*l/ 55.2%

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

      6. associate-/l/ 55.2%

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

      7. metadata-eval 55.2%

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

      8. metadata-eval 55.2%

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

      9. metadata-eval 55.2%

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

      10. /-rgt-identity 55.2%

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

      11. +-commutative 55.2%

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

   3. Simplified 55.2%

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

      1. cos-sum 99.3%

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

      2. cancel-sign-sub-inv 99.3%

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

      3. fma-def 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in b around 0 7.6%

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

      1. mul-1-neg 7.6%

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

      2. unsub-neg 7.6%

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

      3. *-commutative 7.6%

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

   8. Simplified 7.6%

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

   9. Taylor expanded in a around 0 13.6%

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

   11. Simplified 13.6%

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

   if -960 < b < 18000

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in b around 0 96.9%

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

   if 18000 < b

   1. Initial program 41.3%

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

      1. +-commutative 41.3%

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

   3. Simplified 41.3%

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

      1. associate-*r/ 41.4%

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

      2. clear-num 41.4%

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

      3. *-commutative 41.4%

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

   5. Applied egg-rr 41.4%

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

   6. Taylor expanded in a around 0 41.2%

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

      1. associate-/r* 41.3%

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

   8. Simplified 41.3%

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

      1. frac-2neg 41.3%

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

      2. metadata-eval 41.3%

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

      3. div-inv 41.3%

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

      4. distribute-neg-frac 41.3%

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

      5. add-sqr-sqrt 25.1%

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

      6. sqrt-unprod 28.9%

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

      7. sqr-neg 28.9%

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

      8. sqrt-unprod 3.7%

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

      9. add-sqr-sqrt 7.4%

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

      10. frac-2neg 7.4%

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

      11. clear-num 7.4%

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

      12. div-inv 7.4%

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

      13. clear-num 7.4%

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

   10. Applied egg-rr 7.4%

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

   11. Taylor expanded in b around 0 11.6%

       \[\leadsto -1 \cdot \left(\sin b \cdot \color{blue}{r}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -960:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 18000:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(-\sin b\right)\\ \end{array} \]

Alternative 12: 39.4% 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 72.0%

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

   1. associate-*r/ 72.0%

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

   2. /-rgt-identity 72.0%

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

   3. metadata-eval 72.0%

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

   4. associate-/l/ 72.0%

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

   5. associate-*l/ 72.1%

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

   6. associate-/l/ 72.1%

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

   7. metadata-eval 72.1%

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

   8. metadata-eval 72.1%

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

   9. metadata-eval 72.1%

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

   10. /-rgt-identity 72.1%

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

   11. +-commutative 72.1%

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

3. Simplified 72.1%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in b around 0 49.9%

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

   1. mul-1-neg 49.9%

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

   2. unsub-neg 49.9%

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

   3. *-commutative 49.9%

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

8. Simplified 49.9%

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

9. Taylor expanded in a around 0 37.1%

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

11. Simplified 37.1%

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

12. Final simplification 37.1%

    \[\leadsto r \cdot \sin b \]

Alternative 13: 35.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-0.3333333333333333 \cdot \frac{b}{r} + \frac{1}{r \cdot b}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ 1.0 (+ (* -0.3333333333333333 (/ b r)) (/ 1.0 (* r b)))))

C

double code(double r, double a, double b) {
	return 1.0 / ((-0.3333333333333333 * (b / r)) + (1.0 / (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 = 1.0d0 / (((-0.3333333333333333d0) * (b / r)) + (1.0d0 / (r * b)))
end function

Java

public static double code(double r, double a, double b) {
	return 1.0 / ((-0.3333333333333333 * (b / r)) + (1.0 / (r * b)));
}

Python

def code(r, a, b):
	return 1.0 / ((-0.3333333333333333 * (b / r)) + (1.0 / (r * b)))

Julia

function code(r, a, b)
	return Float64(1.0 / Float64(Float64(-0.3333333333333333 * Float64(b / r)) + Float64(1.0 / Float64(r * b))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = 1.0 / ((-0.3333333333333333 * (b / r)) + (1.0 / (r * b)));
end

Wolfram

code[r_, a_, b_] := N[(1.0 / N[(N[(-0.3333333333333333 * N[(b / r), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(r * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

3. Simplified 72.0%

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

   1. associate-*r/ 72.0%

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

   2. clear-num 71.5%

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

   3. *-commutative 71.5%

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

5. Applied egg-rr 71.5%

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

6. Taylor expanded in a around 0 55.9%

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

   1. associate-/r* 55.9%

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

8. Simplified 55.9%

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

9. Taylor expanded in b around 0 32.8%

   \[\leadsto \frac{1}{\color{blue}{-0.3333333333333333 \cdot \frac{b}{r} + \frac{1}{b \cdot r}}} \]

10. Final simplification 32.8%

    \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{b}{r} + \frac{1}{r \cdot b}} \]

Alternative 14: 35.3% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(r, a, b):
	return r * b

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in b around 0 48.3%

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

5. Taylor expanded in a around 0 32.8%

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

6. Final simplification 32.8%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023332 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))