rsin A (should all be same)

Percentage Accurate: 76.9% → 99.4%

Time: 16.5s

Alternatives: 12

Speedup: 1.0×

Specification

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]

Initial Program: 76.9% 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]

Alternative 1: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 74.5%

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

   2. +-commutative 74.5%

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

3. Simplified 74.5%

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

   1. cos-sum 99.4%

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

   2. fma-neg 99.5%

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

5. Applied egg-rr 99.5%

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

   1. fma-neg 99.4%

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

   2. div-sub 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/l* 99.4%

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

   5. quot-tan 99.5%

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

   6. associate-/l* 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 74.5%

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

   2. +-commutative 74.5%

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

3. Simplified 74.5%

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

   1. cos-sum 99.4%

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

   2. fma-neg 99.5%

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

5. Applied egg-rr 99.5%

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

   1. fma-neg 99.4%

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

   2. div-sub 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/l* 99.4%

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

   5. quot-tan 99.5%

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

   6. associate-/l* 99.5%

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

7. Applied egg-rr 99.5%

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

8. Taylor expanded in b around 0 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-5} \lor \neg \left(b \leq 390000\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -5.8e-5) (not (<= b 390000.0)))
   (* (sin b) (/ r (cos b)))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -5.8e-5) || !(b <= 390000.0)) {
		tmp = sin(b) * (r / cos(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 <= (-5.8d-5)) .or. (.not. (b <= 390000.0d0))) then
        tmp = sin(b) * (r / cos(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 <= -5.8e-5) || !(b <= 390000.0)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -5.8e-5) or not (b <= 390000.0):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -5.8e-5) || !(b <= 390000.0))
		tmp = Float64(sin(b) * Float64(r / cos(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 <= -5.8e-5) || ~((b <= 390000.0)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -5.8e-5], N[Not[LessEqual[b, 390000.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.8e-5 or 3.9e5 < b

   1. Initial program 49.8%

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

      1. associate-/l* 49.9%

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

      2. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in a around 0 50.0%

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

      1. associate-/l* 50.1%

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

      2. associate-/r/ 50.0%

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

   6. Simplified 50.0%

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

   if -5.8e-5 < b < 3.9e5

   1. Initial program 98.0%

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

      1. associate-/l* 97.9%

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

      2. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 98.1%

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

      1. associate-/l* 98.0%

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

      2. associate-/r/ 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-5} \lor \neg \left(b \leq 390000\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 4: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00068 \lor \neg \left(b \leq 390000\right):\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00068) (not (<= b 390000.0)))
   (/ r (/ (cos b) (sin b)))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00068) || !(b <= 390000.0)) {
		tmp = r / (cos(b) / 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 <= (-0.00068d0)) .or. (.not. (b <= 390000.0d0))) then
        tmp = r / (cos(b) / 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 <= -0.00068) || !(b <= 390000.0)) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00068) or not (b <= 390000.0):
		tmp = r / (math.cos(b) / math.sin(b))
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.00068) || !(b <= 390000.0))
		tmp = Float64(r / Float64(cos(b) / 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 <= -0.00068) || ~((b <= 390000.0)))
		tmp = r / (cos(b) / sin(b));
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00068], N[Not[LessEqual[b, 390000.0]], $MachinePrecision]], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.8e-4 or 3.9e5 < b

   1. Initial program 49.8%

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

      1. associate-/l* 49.9%

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

      2. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in a around 0 50.1%

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

   if -6.8e-4 < b < 3.9e5

   1. Initial program 98.0%

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

      1. associate-/l* 97.9%

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

      2. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 98.1%

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

      1. associate-/l* 98.0%

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

      2. associate-/r/ 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00068 \lor \neg \left(b \leq 390000\right):\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 390000:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -5.2e-5)
   (/ (sin b) (/ (cos b) r))
   (if (<= b 390000.0) (* r (/ b (cos a))) (/ r (/ (cos b) (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -5.2e-5) {
		tmp = sin(b) / (cos(b) / r);
	} else if (b <= 390000.0) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r / (cos(b) / 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 <= (-5.2d-5)) then
        tmp = sin(b) / (cos(b) / r)
    else if (b <= 390000.0d0) then
        tmp = r * (b / cos(a))
    else
        tmp = r / (cos(b) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -5.2e-5) {
		tmp = Math.sin(b) / (Math.cos(b) / r);
	} else if (b <= 390000.0) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r / (Math.cos(b) / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -5.2e-5:
		tmp = math.sin(b) / (math.cos(b) / r)
	elif b <= 390000.0:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r / (math.cos(b) / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -5.2e-5)
		tmp = Float64(sin(b) / Float64(cos(b) / r));
	elseif (b <= 390000.0)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -5.2e-5)
		tmp = sin(b) / (cos(b) / r);
	elseif (b <= 390000.0)
		tmp = r * (b / cos(a));
	else
		tmp = r / (cos(b) / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -5.2e-5], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 390000.0], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.19999999999999968e-5

   1. Initial program 51.7%

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

      1. associate-/l* 51.8%

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

      2. +-commutative 51.8%

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

   3. Simplified 51.8%

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

      1. associate-/l* 51.7%

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

      2. add-exp-log 27.9%

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

      3. div-inv 27.9%

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

      4. associate-*l* 27.9%

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

      5. div-inv 27.9%

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

   5. Applied egg-rr 27.9%

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

   6. Taylor expanded in r around 0 51.7%

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

      1. *-commutative 51.7%

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

      2. +-commutative 51.7%

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

      3. associate-/l* 51.8%

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

      4. +-commutative 51.8%

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

   8. Simplified 51.8%

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

   9. Taylor expanded in a around 0 51.5%

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

   if -5.19999999999999968e-5 < b < 3.9e5

   1. Initial program 98.0%

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

      1. associate-/l* 97.9%

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

      2. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 98.1%

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

      1. associate-/l* 98.0%

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

      2. associate-/r/ 98.2%

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

   6. Simplified 98.2%

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

   if 3.9e5 < b

   1. Initial program 47.5%

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

      1. associate-/l* 47.6%

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

      2. +-commutative 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in a around 0 48.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 390000:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 6: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 74.5%

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

   2. +-commutative 74.5%

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

3. Simplified 74.5%

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

   1. associate-/r/ 74.5%

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

5. Applied egg-rr 74.5%

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

6. Final simplification 74.5%

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

Alternative 7: 76.9% 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 74.5%

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

   1. associate-/l* 74.5%

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

   2. +-commutative 74.5%

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

3. Simplified 74.5%

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

   1. clear-num 73.9%

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

   2. associate-/r/ 74.5%

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

   3. clear-num 74.6%

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

5. Applied egg-rr 74.6%

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

6. Final simplification 74.6%

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

Alternative 8: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-5} \lor \neg \left(b \leq 390000\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -8.5e-5) (not (<= b 390000.0)))
   (/ r (- (/ 1.0 (tan b)) a))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -8.5e-5) || !(b <= 390000.0)) {
		tmp = r / ((1.0 / tan(b)) - a);
	} 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 <= (-8.5d-5)) .or. (.not. (b <= 390000.0d0))) then
        tmp = r / ((1.0d0 / tan(b)) - a)
    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 <= -8.5e-5) || !(b <= 390000.0)) {
		tmp = r / ((1.0 / Math.tan(b)) - a);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -8.5e-5) or not (b <= 390000.0):
		tmp = r / ((1.0 / math.tan(b)) - a)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -8.5e-5) || !(b <= 390000.0))
		tmp = Float64(r / Float64(Float64(1.0 / tan(b)) - a));
	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 <= -8.5e-5) || ~((b <= 390000.0)))
		tmp = r / ((1.0 / tan(b)) - a);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -8.5e-5], N[Not[LessEqual[b, 390000.0]], $MachinePrecision]], N[(r / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.500000000000001e-5 or 3.9e5 < b

   1. Initial program 49.8%

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

      1. associate-/l* 49.9%

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

      2. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in a around 0 46.2%

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

      1. +-commutative 46.2%

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

      2. mul-1-neg 46.2%

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

      3. unsub-neg 46.2%

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

   6. Simplified 46.2%

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

      1. expm1-log1p-u 31.3%

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

      2. expm1-udef 31.1%

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

      3. clear-num 31.1%

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

      4. quot-tan 31.1%

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

   8. Applied egg-rr 31.1%

      \[\leadsto \frac{r}{\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\tan b}\right)} - 1\right)} - a} \]
   9. Step-by-step derivation

      1. expm1-def 31.3%

         \[\leadsto \frac{r}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tan b}\right)\right)} - a} \]

      2. expm1-log1p 46.3%

         \[\leadsto \frac{r}{\color{blue}{\frac{1}{\tan b}} - a} \]

   10. Simplified 46.3%

       \[\leadsto \frac{r}{\color{blue}{\frac{1}{\tan b}} - a} \]

   if -8.500000000000001e-5 < b < 3.9e5

   1. Initial program 98.0%

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

      1. associate-/l* 97.9%

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

      2. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 98.1%

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

      1. associate-/l* 98.0%

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

      2. associate-/r/ 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-5} \lor \neg \left(b \leq 390000\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 9: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\tan b} - a\\ \mathbf{if}\;b \leq -0.00025:\\ \;\;\;\;\frac{1}{\frac{t_0}{r}}\\ \mathbf{elif}\;b \leq 390000:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 (tan b)) a)))
   (if (<= b -0.00025)
     (/ 1.0 (/ t_0 r))
     (if (<= b 390000.0) (* r (/ b (cos a))) (/ r t_0)))))

C

double code(double r, double a, double b) {
	double t_0 = (1.0 / tan(b)) - a;
	double tmp;
	if (b <= -0.00025) {
		tmp = 1.0 / (t_0 / r);
	} else if (b <= 390000.0) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / tan(b)) - a
    if (b <= (-0.00025d0)) then
        tmp = 1.0d0 / (t_0 / r)
    else if (b <= 390000.0d0) then
        tmp = r * (b / cos(a))
    else
        tmp = r / t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (1.0 / Math.tan(b)) - a;
	double tmp;
	if (b <= -0.00025) {
		tmp = 1.0 / (t_0 / r);
	} else if (b <= 390000.0) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r / t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (1.0 / math.tan(b)) - a
	tmp = 0
	if b <= -0.00025:
		tmp = 1.0 / (t_0 / r)
	elif b <= 390000.0:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r / t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(1.0 / tan(b)) - a)
	tmp = 0.0
	if (b <= -0.00025)
		tmp = Float64(1.0 / Float64(t_0 / r));
	elseif (b <= 390000.0)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (1.0 / tan(b)) - a;
	tmp = 0.0;
	if (b <= -0.00025)
		tmp = 1.0 / (t_0 / r);
	elseif (b <= 390000.0)
		tmp = r * (b / cos(a));
	else
		tmp = r / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]}, If[LessEqual[b, -0.00025], N[(1.0 / N[(t$95$0 / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 390000.0], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.5000000000000001e-4

   1. Initial program 51.7%

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

      1. associate-/l* 51.8%

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

      2. +-commutative 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in a around 0 48.5%

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

      1. +-commutative 48.5%

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

      2. mul-1-neg 48.5%

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

      3. unsub-neg 48.5%

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

   6. Simplified 48.5%

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

      1. clear-num 48.5%

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

      2. inv-pow 48.5%

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

      3. clear-num 48.4%

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

      4. quot-tan 48.6%

         \[\leadsto {\left(\frac{\frac{1}{\color{blue}{\tan b}} - a}{r}\right)}^{-1} \]

   8. Applied egg-rr 48.6%

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

      1. unpow-1 48.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\tan b} - a}{r}}} \]

   10. Simplified 48.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\tan b} - a}{r}}} \]

   if -2.5000000000000001e-4 < b < 3.9e5

   1. Initial program 98.0%

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

      1. associate-/l* 97.9%

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

      2. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 98.1%

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

      1. associate-/l* 98.0%

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

      2. associate-/r/ 98.2%

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

   6. Simplified 98.2%

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

   if 3.9e5 < b

   1. Initial program 47.5%

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

      1. associate-/l* 47.6%

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

      2. +-commutative 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in a around 0 43.4%

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

      1. +-commutative 43.4%

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

      2. mul-1-neg 43.4%

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

      3. unsub-neg 43.4%

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

   6. Simplified 43.4%

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

      1. expm1-log1p-u 28.6%

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

      2. expm1-udef 28.4%

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

      3. clear-num 28.4%

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

      4. quot-tan 28.4%

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

   8. Applied egg-rr 28.4%

      \[\leadsto \frac{r}{\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\tan b}\right)} - 1\right)} - a} \]
   9. Step-by-step derivation

      1. expm1-def 28.6%

         \[\leadsto \frac{r}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tan b}\right)\right)} - a} \]

      2. expm1-log1p 43.4%

         \[\leadsto \frac{r}{\color{blue}{\frac{1}{\tan b}} - a} \]

   10. Simplified 43.4%

       \[\leadsto \frac{r}{\color{blue}{\frac{1}{\tan b}} - a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00025:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\tan b} - a}{r}}\\ \mathbf{elif}\;b \leq 390000:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 10: 52.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1950000000:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1950000000.0) (/ (- r) (sin a)) (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1950000000.0) {
		tmp = -r / sin(a);
	} 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 <= (-1950000000.0d0)) then
        tmp = -r / sin(a)
    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 <= -1950000000.0) {
		tmp = -r / Math.sin(a);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

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

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1950000000.0)
		tmp = Float64(Float64(-r) / sin(a));
	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 <= -1950000000.0)
		tmp = -r / sin(a);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1950000000.0], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.95e9

   1. Initial program 50.1%

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

      1. associate-/l* 50.1%

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

      2. +-commutative 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in b around 0 11.2%

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

      1. +-commutative 11.2%

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

      2. neg-mul-1 11.2%

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

      3. unsub-neg 11.2%

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

   6. Simplified 11.2%

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

   7. Taylor expanded in b around inf 10.8%

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

      1. associate-*r/ 10.8%

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

      2. neg-mul-1 10.8%

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

   9. Simplified 10.8%

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

   if -1.95e9 < b

   1. Initial program 82.8%

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

      1. associate-/l* 82.7%

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

      2. +-commutative 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in b around 0 68.8%

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

      1. associate-/l* 68.7%

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

      2. associate-/r/ 68.9%

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

   6. Simplified 68.9%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1950000000:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11: 35.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (if (<= b -4.5) (/ (- r) (sin a)) (* r b)))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -4.5) {
		tmp = -r / sin(a);
	} else {
		tmp = r * 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 <= (-4.5d0)) then
        tmp = -r / sin(a)
    else
        tmp = r * b
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -4.5) {
		tmp = -r / Math.sin(a);
	} else {
		tmp = r * b;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -4.5:
		tmp = -r / math.sin(a)
	else:
		tmp = r * b
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -4.5)
		tmp = Float64(Float64(-r) / sin(a));
	else
		tmp = Float64(r * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -4.5)
		tmp = -r / sin(a);
	else
		tmp = r * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -4.5], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision], N[(r * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5

   1. Initial program 50.3%

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

      1. associate-/l* 50.4%

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

      2. +-commutative 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in b around 0 11.3%

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

      1. +-commutative 11.3%

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

      2. neg-mul-1 11.3%

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

      3. unsub-neg 11.3%

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

   6. Simplified 11.3%

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

   7. Taylor expanded in b around inf 10.7%

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

      1. associate-*r/ 10.7%

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

      2. neg-mul-1 10.7%

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

   9. Simplified 10.7%

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

   if -4.5 < b

   1. Initial program 83.0%

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

      1. associate-/l* 83.0%

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

      2. +-commutative 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in b around 0 69.4%

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

      1. associate-/l* 69.3%

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

      2. associate-/r/ 69.4%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in a around 0 43.2%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot b\\ \end{array} \]

Alternative 12: 34.3% 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 74.5%

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

   1. associate-/l* 74.5%

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

   2. +-commutative 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in b around 0 52.1%

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

   1. associate-/l* 52.1%

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

   2. associate-/r/ 52.1%

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

6. Simplified 52.1%

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

7. Taylor expanded in a around 0 33.0%

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

8. Final simplification 33.0%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023299 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))