rsin B (should all be same)

Percentage Accurate: 75.9% → 99.4%

Time: 19.3s

Alternatives: 13

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: 75.9% 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: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

   1. associate-*r/ 81.3%

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

   2. +-commutative 81.3%

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

3. Simplified 81.3%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in r around 0 99.5%

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

   1. associate-/l* 99.5%

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

   2. +-commutative 99.5%

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

   3. mul-1-neg 99.5%

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

   4. *-commutative 99.5%

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

   5. fma-def 99.5%

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

   6. fma-neg 99.5%

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

9. Simplified 99.5%

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

    1. div-sub 99.5%

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

    2. sub-neg 99.5%

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

    3. associate-/l* 99.5%

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

    4. associate-/l* 99.5%

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

11. Applied egg-rr 99.5%

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

    1. associate-/r/ 99.5%

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

    2. fma-def 99.5%

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

    3. associate-/r/ 99.5%

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

    4. *-inverses 99.5%

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

    5. distribute-lft-neg-in 99.5%

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

    6. metadata-eval 99.5%

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

    7. neg-mul-1 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-6} \lor \neg \left(a \leq 1.25 \cdot 10^{-23}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -2.7e-6) (not (<= a 1.25e-23)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -2.7e-6) || !(a <= 1.25e-23)) {
		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 <= (-2.7d-6)) .or. (.not. (a <= 1.25d-23))) 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 <= -2.7e-6) || !(a <= 1.25e-23)) {
		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 <= -2.7e-6) or not (a <= 1.25e-23):
		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 <= -2.7e-6) || !(a <= 1.25e-23))
		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 <= -2.7e-6) || ~((a <= 1.25e-23)))
		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, -2.7e-6], N[Not[LessEqual[a, 1.25e-23]], $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 < -2.69999999999999998e-6 or 1.2500000000000001e-23 < a

   1. Initial program 58.8%

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

      1. remove-double-neg 58.8%

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

      2. remove-double-neg 58.8%

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

      3. +-commutative 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in b around 0 58.2%

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

   if -2.69999999999999998e-6 < a < 1.2500000000000001e-23

   1. Initial program 99.3%

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

      1. remove-double-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around 0 99.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-6} \lor \neg \left(a \leq 1.25 \cdot 10^{-23}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -3.8e-5)
   (* (sin b) (/ r (cos a)))
   (if (<= a 1.25e-23) (* r (/ (sin b) (cos b))) (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -3.8e-5) {
		tmp = sin(b) * (r / cos(a));
	} else if (a <= 1.25e-23) {
		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 <= (-3.8d-5)) then
        tmp = sin(b) * (r / cos(a))
    else if (a <= 1.25d-23) 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 <= -3.8e-5) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else if (a <= 1.25e-23) {
		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 <= -3.8e-5:
		tmp = math.sin(b) * (r / math.cos(a))
	elif a <= 1.25e-23:
		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 <= -3.8e-5)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	elseif (a <= 1.25e-23)
		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 <= -3.8e-5)
		tmp = sin(b) * (r / cos(a));
	elseif (a <= 1.25e-23)
		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, -3.8e-5], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.25e-23], 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 < -3.8000000000000002e-5

   1. Initial program 57.4%

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

      1. associate-*r/ 57.2%

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

      2. +-commutative 57.2%

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

   3. Simplified 57.2%

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

      1. associate-/l* 57.5%

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

      2. associate-/r/ 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in b around 0 57.2%

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

   if -3.8000000000000002e-5 < a < 1.2500000000000001e-23

   1. Initial program 99.3%

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

      1. remove-double-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around 0 99.3%

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

   if 1.2500000000000001e-23 < a

   1. Initial program 59.8%

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

      1. remove-double-neg 59.8%

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

      2. remove-double-neg 59.8%

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

      3. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in b around 0 58.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00215:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.00215)
   (* (sin b) (/ r (cos a)))
   (if (<= a 1.25e-23) (* (sin b) (/ r (cos b))) (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.00215) {
		tmp = sin(b) * (r / cos(a));
	} else if (a <= 1.25e-23) {
		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 <= (-0.00215d0)) then
        tmp = sin(b) * (r / cos(a))
    else if (a <= 1.25d-23) 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 <= -0.00215) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else if (a <= 1.25e-23) {
		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 <= -0.00215:
		tmp = math.sin(b) * (r / math.cos(a))
	elif a <= 1.25e-23:
		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 <= -0.00215)
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	elseif (a <= 1.25e-23)
		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 <= -0.00215)
		tmp = sin(b) * (r / cos(a));
	elseif (a <= 1.25e-23)
		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, -0.00215], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.25e-23], 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 < -0.00215

   1. Initial program 57.4%

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

      1. associate-*r/ 57.2%

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

      2. +-commutative 57.2%

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

   3. Simplified 57.2%

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

      1. associate-/l* 57.5%

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

      2. associate-/r/ 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in b around 0 57.2%

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

   if -0.00215 < a < 1.2500000000000001e-23

   1. Initial program 99.3%

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

      1. remove-double-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around 0 99.3%

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

      1. associate-/l* 99.2%

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

      2. associate-/r/ 99.3%

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

   7. Simplified 99.3%

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

   if 1.2500000000000001e-23 < a

   1. Initial program 59.8%

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

      1. remove-double-neg 59.8%

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

      2. remove-double-neg 59.8%

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

      3. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in b around 0 58.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00215:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -1.4e-5)
   (/ r (/ (cos a) (sin b)))
   (if (<= a 1.25e-23) (* (sin b) (/ r (cos b))) (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -1.4e-5) {
		tmp = r / (cos(a) / sin(b));
	} else if (a <= 1.25e-23) {
		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 <= (-1.4d-5)) then
        tmp = r / (cos(a) / sin(b))
    else if (a <= 1.25d-23) 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 <= -1.4e-5) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else if (a <= 1.25e-23) {
		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 <= -1.4e-5:
		tmp = r / (math.cos(a) / math.sin(b))
	elif a <= 1.25e-23:
		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 <= -1.4e-5)
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	elseif (a <= 1.25e-23)
		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 <= -1.4e-5)
		tmp = r / (cos(a) / sin(b));
	elseif (a <= 1.25e-23)
		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, -1.4e-5], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.25e-23], 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 < -1.39999999999999998e-5

   1. Initial program 57.4%

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

      1. associate-*r/ 57.2%

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

      2. +-commutative 57.2%

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

   3. Simplified 57.2%

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

      1. cos-sum 99.0%

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

      2. cancel-sign-sub-inv 99.0%

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

      3. fma-def 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in r around 0 99.0%

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

      1. associate-/l* 99.2%

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

      2. +-commutative 99.2%

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

      3. mul-1-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. fma-def 99.2%

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

      6. fma-neg 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in b around 0 57.2%

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

   if -1.39999999999999998e-5 < a < 1.2500000000000001e-23

   1. Initial program 99.3%

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

      1. remove-double-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around 0 99.3%

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

      1. associate-/l* 99.2%

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

      2. associate-/r/ 99.3%

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

   7. Simplified 99.3%

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

   if 1.2500000000000001e-23 < a

   1. Initial program 59.8%

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

      1. remove-double-neg 59.8%

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

      2. remove-double-neg 59.8%

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

      3. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in b around 0 58.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -5.3e-5)
   (/ r (/ (cos a) (sin b)))
   (if (<= a 5.5e-11) (* (sin b) (/ r (cos b))) (/ (sin b) (/ (cos a) r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -5.3e-5) {
		tmp = r / (cos(a) / sin(b));
	} else if (a <= 5.5e-11) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = sin(b) / (cos(a) / r);
	}
	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 <= (-5.3d-5)) then
        tmp = r / (cos(a) / sin(b))
    else if (a <= 5.5d-11) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = sin(b) / (cos(a) / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -5.3e-5) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else if (a <= 5.5e-11) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -5.3e-5:
		tmp = r / (math.cos(a) / math.sin(b))
	elif a <= 5.5e-11:
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = math.sin(b) / (math.cos(a) / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -5.3e-5)
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	elseif (a <= 5.5e-11)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -5.3e-5)
		tmp = r / (cos(a) / sin(b));
	elseif (a <= 5.5e-11)
		tmp = sin(b) * (r / cos(b));
	else
		tmp = sin(b) / (cos(a) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -5.3e-5], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.5e-11], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.3000000000000001e-5

   1. Initial program 57.4%

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

      1. associate-*r/ 57.2%

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

      2. +-commutative 57.2%

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

   3. Simplified 57.2%

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

      1. cos-sum 99.0%

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

      2. cancel-sign-sub-inv 99.0%

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

      3. fma-def 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in r around 0 99.0%

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

      1. associate-/l* 99.2%

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

      2. +-commutative 99.2%

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

      3. mul-1-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. fma-def 99.2%

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

      6. fma-neg 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in b around 0 57.2%

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

   if -5.3000000000000001e-5 < a < 5.49999999999999975e-11

   1. Initial program 99.3%

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

      1. remove-double-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around 0 99.3%

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

      1. associate-/l* 99.2%

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

      2. associate-/r/ 99.3%

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

   7. Simplified 99.3%

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

   if 5.49999999999999975e-11 < a

   1. Initial program 57.8%

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

      1. remove-double-neg 57.8%

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

      2. remove-double-neg 57.8%

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

      3. +-commutative 57.8%

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

   3. Simplified 57.8%

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

      1. associate-*r/ 57.7%

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

      2. *-commutative 57.7%

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

      3. associate-/l* 57.9%

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

   6. Applied egg-rr 57.9%

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

   7. Taylor expanded in b around 0 57.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% 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 81.4%

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

3. Final simplification 81.4%

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

Alternative 8: 75.8% accurate, 1.0× speedup

Math

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

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(Float64(r * 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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

   1. associate-*r/ 81.3%

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

   2. +-commutative 81.3%

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

3. Simplified 81.3%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in b around inf 99.5%

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

   1. mul-1-neg 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. *-commutative 99.5%

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

   5. fma-def 99.5%

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

   6. *-commutative 99.5%

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

9. Simplified 99.5%

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

    1. fma-udef 99.5%

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

    2. +-commutative 99.5%

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

    3. *-commutative 99.5%

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

    4. add-sqr-sqrt 52.8%

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

    5. sqrt-unprod 91.1%

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

    6. sqr-neg 91.1%

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

    7. sqrt-unprod 38.2%

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

    8. add-sqr-sqrt 81.2%

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

    9. cos-diff 81.5%

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

11. Applied egg-rr 81.5%

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

12. Final simplification 81.5%

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

Alternative 9: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

   1. remove-double-neg 81.4%

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

   2. remove-double-neg 81.4%

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

   3. +-commutative 81.4%

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

3. Simplified 81.4%

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

5. Taylor expanded in b around 0 55.4%

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

6. Final simplification 55.4%

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

Alternative 10: 54.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.7 \lor \neg \left(b \leq 1.12\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.7) (not (<= b 1.12))) (* r (sin b)) (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.7) || !(b <= 1.12)) {
		tmp = r * sin(b);
	} else {
		tmp = r * (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 ((b <= (-4.7d0)) .or. (.not. (b <= 1.12d0))) then
        tmp = r * sin(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.7) || !(b <= 1.12)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.7) or not (b <= 1.12):
		tmp = r * math.sin(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.7) || !(b <= 1.12))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.7) || ~((b <= 1.12)))
		tmp = r * sin(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.7], N[Not[LessEqual[b, 1.12]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.70000000000000018 or 1.1200000000000001 < b

   1. Initial program 62.4%

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

      1. associate-*r/ 62.4%

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

      2. +-commutative 62.4%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in a around 0 60.0%

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

      1. mul-1-neg 60.0%

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

      2. unsub-neg 60.0%

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

   7. Simplified 60.0%

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

   8. Taylor expanded in b around 0 10.5%

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

   if -4.70000000000000018 < b < 1.1200000000000001

   1. Initial program 99.5%

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

      1. remove-double-neg 99.5%

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

      2. remove-double-neg 99.5%

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

      3. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 98.4%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.7 \lor \neg \left(b \leq 1.12\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -23:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -23.0) (* r (sin b)) (/ (* r b) (cos (+ a b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -23.0) {
		tmp = r * sin(b);
	} else {
		tmp = (r * b) / cos((a + 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 <= (-23.0d0)) then
        tmp = r * sin(b)
    else
        tmp = (r * b) / cos((a + b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -23.0) {
		tmp = r * Math.sin(b);
	} else {
		tmp = (r * b) / Math.cos((a + b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -23.0:
		tmp = r * math.sin(b)
	else:
		tmp = (r * b) / math.cos((a + b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -23.0)
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(a + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -23.0)
		tmp = r * sin(b);
	else
		tmp = (r * b) / cos((a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -23.0], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -23

   1. Initial program 65.8%

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

      1. associate-*r/ 65.8%

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

      2. +-commutative 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in a around 0 62.5%

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

      1. mul-1-neg 62.5%

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

      2. unsub-neg 62.5%

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

   7. Simplified 62.5%

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

   8. Taylor expanded in b around 0 11.0%

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

   if -23 < b

   1. Initial program 86.6%

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

      1. associate-*r/ 86.5%

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

      2. +-commutative 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in b around 0 68.3%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -23:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(a + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

   1. remove-double-neg 81.4%

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

   2. remove-double-neg 81.4%

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

   3. +-commutative 81.4%

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

3. Simplified 81.4%

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

5. Taylor expanded in b around 0 52.1%

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

6. Final simplification 52.1%

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

Alternative 13: 34.8% 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 81.4%

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

   1. remove-double-neg 81.4%

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

   2. remove-double-neg 81.4%

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

   3. +-commutative 81.4%

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

3. Simplified 81.4%

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

5. Taylor expanded in b around 0 52.1%

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

6. Taylor expanded in a around 0 37.0%

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

7. Final simplification 37.0%

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

Reproduce

herbie shell --seed 2024026 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))