rsin A (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 17.5s

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: 76.3% 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: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 75.1%

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

   2. remove-double-neg 75.1%

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

   3. sin-neg 75.1%

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

   4. neg-mul-1 75.1%

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

   5. associate-/r* 75.1%

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

   6. associate-/l* 75.1%

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

   7. *-commutative 75.1%

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

   8. associate-*l/ 75.2%

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

   9. associate-/l* 75.2%

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

   10. sin-neg 75.2%

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

   11. distribute-lft-neg-in 75.2%

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

   12. distribute-rgt-neg-in 75.2%

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

   13. associate-/l* 75.2%

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

   14. metadata-eval 75.2%

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

   15. /-rgt-identity 75.2%

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

   16. +-commutative 75.2%

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

3. Simplified 75.2%

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

   1. cos-sum 99.4%

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

   2. sub-neg 99.4%

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

5. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

   2. distribute-lft-neg-in 99.4%

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

   3. *-commutative 99.4%

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

   4. fma-def 99.5%

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

   5. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 75.1%

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

   2. remove-double-neg 75.1%

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

   3. sin-neg 75.1%

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

   4. neg-mul-1 75.1%

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

   5. associate-/r* 75.1%

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

   6. associate-/l* 75.1%

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

   7. *-commutative 75.1%

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

   8. associate-*l/ 75.2%

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

   9. associate-/l* 75.2%

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

   10. sin-neg 75.2%

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

   11. distribute-lft-neg-in 75.2%

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

   12. distribute-rgt-neg-in 75.2%

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

   13. associate-/l* 75.2%

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

   14. metadata-eval 75.2%

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

   15. /-rgt-identity 75.2%

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

   16. +-commutative 75.2%

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

3. Simplified 75.2%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 77.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

   1. sin-mult 75.8%

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

   2. div-sub 75.8%

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

   3. cos-diff 76.5%

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

   4. *-commutative 76.5%

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

   5. *-commutative 76.5%

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

   6. cos-diff 75.8%

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

7. Applied egg-rr 75.7%

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

   1. +-inverses 75.7%

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

9. Simplified 75.7%

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

10. Final simplification 75.7%

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

Alternative 4: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \lor \neg \left(b \leq 5.4 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.3) (not (<= b 5.4e-5)))
   (* r (/ (sin b) (cos b)))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.3) || !(b <= 5.4e-5)) {
		tmp = r * (sin(b) / 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 <= (-1.3d0)) .or. (.not. (b <= 5.4d-5))) then
        tmp = r * (sin(b) / 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 <= -1.3) || !(b <= 5.4e-5)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.3) or not (b <= 5.4e-5):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.3) || !(b <= 5.4e-5))
		tmp = Float64(r * Float64(sin(b) / 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 <= -1.3) || ~((b <= 5.4e-5)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.3], N[Not[LessEqual[b, 5.4e-5]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / 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 < -1.30000000000000004 or 5.3999999999999998e-5 < b

   1. Initial program 55.9%

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

      1. associate-/l* 56.0%

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

      2. remove-double-neg 56.0%

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

      3. sin-neg 56.0%

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

      4. neg-mul-1 56.0%

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

      5. associate-/r* 56.0%

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

      6. associate-/l* 55.9%

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

      7. *-commutative 55.9%

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

      8. associate-*l/ 56.0%

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

      9. associate-/l* 56.0%

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

      10. sin-neg 56.0%

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

      11. distribute-lft-neg-in 56.0%

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

      12. distribute-rgt-neg-in 56.0%

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

      13. associate-/l* 56.0%

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

      14. metadata-eval 56.0%

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

      15. /-rgt-identity 56.0%

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

      16. +-commutative 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in a around 0 55.1%

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

   if -1.30000000000000004 < b < 5.3999999999999998e-5

   1. Initial program 97.9%

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

      1. associate-/l* 97.9%

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

      2. remove-double-neg 97.9%

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

      3. sin-neg 97.9%

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

      4. neg-mul-1 97.9%

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

      5. associate-/r* 97.9%

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

      6. associate-/l* 97.9%

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

      7. *-commutative 97.9%

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

      8. associate-*l/ 98.0%

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

      9. associate-/l* 98.0%

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

      10. sin-neg 98.0%

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

      11. distribute-lft-neg-in 98.0%

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

      12. distribute-rgt-neg-in 98.0%

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

      13. associate-/l* 98.0%

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

      14. metadata-eval 98.0%

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

      15. /-rgt-identity 98.0%

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

      16. +-commutative 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in b around 0 98.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \lor \neg \left(b \leq 5.4 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 5: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.3)
   (/ (sin b) (/ (cos b) r))
   (if (<= b 2.5e-5) (* r (/ b (cos a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.3) {
		tmp = sin(b) / (cos(b) / r);
	} else if (b <= 2.5e-5) {
		tmp = r * (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 (b <= (-1.3d0)) then
        tmp = sin(b) / (cos(b) / r)
    else if (b <= 2.5d-5) then
        tmp = r * (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 (b <= -1.3) {
		tmp = Math.sin(b) / (Math.cos(b) / r);
	} else if (b <= 2.5e-5) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.3:
		tmp = math.sin(b) / (math.cos(b) / r)
	elif b <= 2.5e-5:
		tmp = r * (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 (b <= -1.3)
		tmp = Float64(sin(b) / Float64(cos(b) / r));
	elseif (b <= 2.5e-5)
		tmp = Float64(r * Float64(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 (b <= -1.3)
		tmp = sin(b) / (cos(b) / r);
	elseif (b <= 2.5e-5)
		tmp = r * (b / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.3], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-5], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.30000000000000004

   1. Initial program 49.1%

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

      1. associate-/l* 49.1%

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

      2. remove-double-neg 49.1%

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

      3. sin-neg 49.1%

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

      4. neg-mul-1 49.1%

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

      5. associate-/r* 49.1%

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

      6. associate-/l* 49.1%

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

      7. *-commutative 49.1%

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

      8. associate-*l/ 49.2%

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

      9. associate-/l* 49.2%

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

      10. sin-neg 49.2%

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

      11. distribute-lft-neg-in 49.2%

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

      12. distribute-rgt-neg-in 49.2%

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

      13. associate-/l* 49.2%

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

      14. metadata-eval 49.2%

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

      15. /-rgt-identity 49.2%

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

      16. +-commutative 49.2%

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

   3. Simplified 49.2%

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

      1. associate-*l/ 49.1%

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

      2. associate-/l* 49.3%

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

   5. Applied egg-rr 49.3%

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

   6. Taylor expanded in a around 0 49.5%

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

   if -1.30000000000000004 < b < 2.50000000000000012e-5

   1. Initial program 97.9%

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

      1. associate-/l* 97.9%

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

      2. remove-double-neg 97.9%

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

      3. sin-neg 97.9%

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

      4. neg-mul-1 97.9%

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

      5. associate-/r* 97.9%

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

      6. associate-/l* 97.9%

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

      7. *-commutative 97.9%

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

      8. associate-*l/ 98.0%

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

      9. associate-/l* 98.0%

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

      10. sin-neg 98.0%

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

      11. distribute-lft-neg-in 98.0%

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

      12. distribute-rgt-neg-in 98.0%

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

      13. associate-/l* 98.0%

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

      14. metadata-eval 98.0%

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

      15. /-rgt-identity 98.0%

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

      16. +-commutative 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in b around 0 98.0%

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

   if 2.50000000000000012e-5 < b

   1. Initial program 63.7%

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

      1. associate-/l* 63.7%

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

      2. remove-double-neg 63.7%

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

      3. sin-neg 63.7%

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

      4. neg-mul-1 63.7%

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

      5. associate-/r* 63.7%

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

      6. associate-/l* 63.7%

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

      7. *-commutative 63.7%

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

      8. associate-*l/ 63.7%

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

      9. associate-/l* 63.7%

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

      10. sin-neg 63.7%

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

      11. distribute-lft-neg-in 63.7%

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

      12. distribute-rgt-neg-in 63.7%

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

      13. associate-/l* 63.7%

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

      14. metadata-eval 63.7%

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

      15. /-rgt-identity 63.7%

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

      16. +-commutative 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in a around 0 61.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 6: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 75.1%

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

   2. +-commutative 75.1%

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

3. Simplified 75.1%

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

   1. associate-/r/ 75.2%

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

5. Applied egg-rr 75.2%

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

6. Final simplification 75.2%

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

Alternative 7: 54.6% 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 75.1%

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

   1. associate-/l* 75.1%

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

   2. remove-double-neg 75.1%

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

   3. sin-neg 75.1%

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

   4. neg-mul-1 75.1%

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

   5. associate-/r* 75.1%

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

   6. associate-/l* 75.1%

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

   7. *-commutative 75.1%

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

   8. associate-*l/ 75.2%

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

   9. associate-/l* 75.2%

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

   10. sin-neg 75.2%

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

   11. distribute-lft-neg-in 75.2%

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

   12. distribute-rgt-neg-in 75.2%

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

   13. associate-/l* 75.2%

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

   14. metadata-eval 75.2%

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

   15. /-rgt-identity 75.2%

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

   16. +-commutative 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in b around 0 51.3%

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

5. Final simplification 51.3%

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

Alternative 8: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -420000:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{elif}\;b \leq 1.12:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -420000.0)
   (/ (- r) (sin a))
   (if (<= b 1.12) (* r (/ b (cos a))) (/ r (- (/ 1.0 b) (sin a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -420000.0) {
		tmp = -r / sin(a);
	} else if (b <= 1.12) {
		tmp = r * (b / cos(a));
	} else {
		tmp = r / ((1.0 / b) - sin(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 <= (-420000.0d0)) then
        tmp = -r / sin(a)
    else if (b <= 1.12d0) then
        tmp = r * (b / cos(a))
    else
        tmp = r / ((1.0d0 / b) - sin(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -420000.0) {
		tmp = -r / Math.sin(a);
	} else if (b <= 1.12) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = r / ((1.0 / b) - Math.sin(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -420000.0:
		tmp = -r / math.sin(a)
	elif b <= 1.12:
		tmp = r * (b / math.cos(a))
	else:
		tmp = r / ((1.0 / b) - math.sin(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -420000.0)
		tmp = Float64(Float64(-r) / sin(a));
	elseif (b <= 1.12)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(r / Float64(Float64(1.0 / b) - sin(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -420000.0)
		tmp = -r / sin(a);
	elseif (b <= 1.12)
		tmp = r * (b / cos(a));
	else
		tmp = r / ((1.0 / b) - sin(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -420000.0], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.12], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(N[(1.0 / b), $MachinePrecision] - N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.2e5

   1. Initial program 49.1%

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

      1. associate-/l* 49.1%

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

      2. +-commutative 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in b around 0 10.5%

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

      1. +-commutative 10.5%

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

      2. neg-mul-1 10.5%

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

      3. unsub-neg 10.5%

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

   6. Simplified 10.5%

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

   7. Taylor expanded in b around inf 10.6%

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

      1. associate-*r/ 10.6%

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

      2. neg-mul-1 10.6%

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

   9. Simplified 10.6%

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

   if -4.2e5 < b < 1.1200000000000001

   1. Initial program 97.1%

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

      1. associate-/l* 97.0%

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

      2. remove-double-neg 97.0%

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

      3. sin-neg 97.0%

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

      4. neg-mul-1 97.0%

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

      5. associate-/r* 97.0%

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

      6. associate-/l* 97.1%

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

      7. *-commutative 97.1%

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

      8. associate-*l/ 97.2%

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

      9. associate-/l* 97.2%

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

      10. sin-neg 97.2%

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

      11. distribute-lft-neg-in 97.2%

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

      12. distribute-rgt-neg-in 97.2%

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

      13. associate-/l* 97.2%

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

      14. metadata-eval 97.2%

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

      15. /-rgt-identity 97.2%

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

      16. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in b around 0 96.6%

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

   if 1.1200000000000001 < b

   1. Initial program 63.7%

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

      1. associate-/l* 63.7%

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

      2. +-commutative 63.7%

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

   3. Simplified 63.7%

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

   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 a around 0 11.2%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -420000:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{elif}\;b \leq 1.12:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \end{array} \]

Alternative 9: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -340000:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -340000.0)
   (/ (- r) (sin a))
   (if (<= b 2.7e+21) (/ (* b r) (cos (+ b a))) (/ r (- (/ 1.0 b) (sin a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -340000.0) {
		tmp = -r / sin(a);
	} else if (b <= 2.7e+21) {
		tmp = (b * r) / cos((b + a));
	} else {
		tmp = r / ((1.0 / b) - sin(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 <= (-340000.0d0)) then
        tmp = -r / sin(a)
    else if (b <= 2.7d+21) then
        tmp = (b * r) / cos((b + a))
    else
        tmp = r / ((1.0d0 / b) - sin(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -340000.0) {
		tmp = -r / Math.sin(a);
	} else if (b <= 2.7e+21) {
		tmp = (b * r) / Math.cos((b + a));
	} else {
		tmp = r / ((1.0 / b) - Math.sin(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -340000.0:
		tmp = -r / math.sin(a)
	elif b <= 2.7e+21:
		tmp = (b * r) / math.cos((b + a))
	else:
		tmp = r / ((1.0 / b) - math.sin(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -340000.0)
		tmp = Float64(Float64(-r) / sin(a));
	elseif (b <= 2.7e+21)
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	else
		tmp = Float64(r / Float64(Float64(1.0 / b) - sin(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -340000.0)
		tmp = -r / sin(a);
	elseif (b <= 2.7e+21)
		tmp = (b * r) / cos((b + a));
	else
		tmp = r / ((1.0 / b) - sin(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -340000.0], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e+21], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / N[(N[(1.0 / b), $MachinePrecision] - N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4e5

   1. Initial program 49.1%

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

      1. associate-/l* 49.1%

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

      2. +-commutative 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in b around 0 10.5%

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

      1. +-commutative 10.5%

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

      2. neg-mul-1 10.5%

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

      3. unsub-neg 10.5%

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

   6. Simplified 10.5%

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

   7. Taylor expanded in b around inf 10.6%

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

      1. associate-*r/ 10.6%

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

      2. neg-mul-1 10.6%

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

   9. Simplified 10.6%

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

   if -3.4e5 < b < 2.7e21

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in b around 0 93.6%

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

   if 2.7e21 < b

   1. Initial program 61.4%

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

      1. associate-/l* 61.4%

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

      2. +-commutative 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in b around 0 11.6%

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

      1. +-commutative 11.6%

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

      2. neg-mul-1 11.6%

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

      3. unsub-neg 11.6%

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

   6. Simplified 11.6%

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

   7. Taylor expanded in a around 0 11.6%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -340000:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \end{array} \]

Alternative 10: 53.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -430000 \lor \neg \left(b \leq 42000000000000\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -430000.0) (not (<= b 42000000000000.0)))
   (/ (- r) (sin a))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -430000.0) || !(b <= 42000000000000.0)) {
		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 <= (-430000.0d0)) .or. (.not. (b <= 42000000000000.0d0))) 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 <= -430000.0) || !(b <= 42000000000000.0)) {
		tmp = -r / Math.sin(a);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -430000.0) or not (b <= 42000000000000.0):
		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 <= -430000.0) || !(b <= 42000000000000.0))
		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 <= -430000.0) || ~((b <= 42000000000000.0)))
		tmp = -r / sin(a);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -430000.0], N[Not[LessEqual[b, 42000000000000.0]], $MachinePrecision]], 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 < -4.3e5 or 4.2e13 < b

   1. Initial program 55.4%

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

      1. associate-/l* 55.4%

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

      2. +-commutative 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in b around 0 10.9%

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

      1. +-commutative 10.9%

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

      2. neg-mul-1 10.9%

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

      3. unsub-neg 10.9%

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

   6. Simplified 10.9%

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

   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 -4.3e5 < b < 4.2e13

   1. Initial program 97.1%

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

      1. associate-/l* 97.1%

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

      2. remove-double-neg 97.1%

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

      3. sin-neg 97.1%

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

      4. neg-mul-1 97.1%

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

      5. associate-/r* 97.1%

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

      6. associate-/l* 97.1%

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

      7. *-commutative 97.1%

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

      8. associate-*l/ 97.2%

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

      9. associate-/l* 97.2%

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

      10. sin-neg 97.2%

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

      11. distribute-lft-neg-in 97.2%

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

      12. distribute-rgt-neg-in 97.2%

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

      13. associate-/l* 97.2%

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

      14. metadata-eval 97.2%

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

      15. /-rgt-identity 97.2%

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

      16. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in b around 0 95.1%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -430000 \lor \neg \left(b \leq 42000000000000\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11: 37.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -360000 \lor \neg \left(b \leq 1.95 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -360000.0) (not (<= b 1.95e-11))) (/ (- r) (sin a)) (* b r)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -360000.0) || !(b <= 1.95e-11)) {
		tmp = -r / sin(a);
	} else {
		tmp = b * 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 ((b <= (-360000.0d0)) .or. (.not. (b <= 1.95d-11))) then
        tmp = -r / sin(a)
    else
        tmp = b * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -360000.0) || !(b <= 1.95e-11)) {
		tmp = -r / Math.sin(a);
	} else {
		tmp = b * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -360000.0) or not (b <= 1.95e-11):
		tmp = -r / math.sin(a)
	else:
		tmp = b * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -360000.0) || !(b <= 1.95e-11))
		tmp = Float64(Float64(-r) / sin(a));
	else
		tmp = Float64(b * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -360000.0) || ~((b <= 1.95e-11)))
		tmp = -r / sin(a);
	else
		tmp = b * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -360000.0], N[Not[LessEqual[b, 1.95e-11]], $MachinePrecision]], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision], N[(b * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.6e5 or 1.95000000000000005e-11 < b

   1. Initial program 56.2%

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

      1. associate-/l* 56.2%

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

      2. +-commutative 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in b around 0 11.5%

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

      1. +-commutative 11.5%

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

      2. neg-mul-1 11.5%

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

      3. unsub-neg 11.5%

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

   6. Simplified 11.5%

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

   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 -3.6e5 < b < 1.95000000000000005e-11

   1. Initial program 97.3%

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

      1. associate-/l* 97.2%

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

      2. remove-double-neg 97.2%

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

      3. sin-neg 97.2%

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

      4. neg-mul-1 97.2%

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

      5. associate-/r* 97.2%

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

      6. associate-/l* 97.3%

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

      7. *-commutative 97.3%

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

      8. associate-*l/ 97.4%

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

      9. associate-/l* 97.4%

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

      10. sin-neg 97.4%

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

      11. distribute-lft-neg-in 97.4%

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

      12. distribute-rgt-neg-in 97.4%

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

      13. associate-/l* 97.4%

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

      14. metadata-eval 97.4%

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

      15. /-rgt-identity 97.4%

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

      16. +-commutative 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in b around 0 96.7%

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

   5. Taylor expanded in a around 0 65.5%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -360000 \lor \neg \left(b \leq 1.95 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot r\\ \end{array} \]

Alternative 12: 34.5% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 75.1%

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

   2. remove-double-neg 75.1%

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

   3. sin-neg 75.1%

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

   4. neg-mul-1 75.1%

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

   5. associate-/r* 75.1%

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

   6. associate-/l* 75.1%

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

   7. *-commutative 75.1%

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

   8. associate-*l/ 75.2%

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

   9. associate-/l* 75.2%

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

   10. sin-neg 75.2%

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

   11. distribute-lft-neg-in 75.2%

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

   12. distribute-rgt-neg-in 75.2%

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

   13. associate-/l* 75.2%

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

   14. metadata-eval 75.2%

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

   15. /-rgt-identity 75.2%

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

   16. +-commutative 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in b around 0 46.9%

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

5. Taylor expanded in a around 0 32.3%

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

6. Final simplification 32.3%

   \[\leadsto b \cdot r \]

Reproduce

herbie shell --seed 2023334 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))