rsin A (should all be same)

Percentage Accurate: 77.5% → 99.5%

Time: 15.1s

Alternatives: 14

Speedup: 0.7×

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: 77.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. associate-/l* 78.6%

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

   2. remove-double-neg 78.6%

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

   3. sin-neg 78.6%

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

   4. neg-mul-1 78.6%

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

   5. associate-/r* 78.6%

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

   6. associate-/l* 78.6%

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

   7. *-commutative 78.6%

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

   8. associate-*l/ 78.6%

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

   9. associate-/l* 78.6%

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

   10. sin-neg 78.6%

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

   11. distribute-lft-neg-in 78.6%

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

   12. distribute-rgt-neg-in 78.6%

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

   13. associate-/l* 78.6%

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

   14. metadata-eval 78.6%

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

   15. /-rgt-identity 78.6%

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

   16. +-commutative 78.6%

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

3. Simplified 78.6%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
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. fma-neg 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{r}{\mathsf{fma}\left(\frac{1}{\tan b}, \cos a, -\sin a\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ r (fma (/ 1.0 (tan b)) (cos a) (- (sin a)))))

C

double code(double r, double a, double b) {
	return r / fma((1.0 / tan(b)), cos(a), -sin(a));
}

Julia

function code(r, a, b)
	return Float64(r / fma(Float64(1.0 / tan(b)), cos(a), Float64(-sin(a))))
end

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. associate-/l* 78.6%

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

   2. +-commutative 78.6%

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

3. Simplified 78.6%

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

   1. cos-sum 99.3%

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

5. Applied egg-rr 99.3%

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

   1. div-sub 99.3%

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

   2. sub-neg 99.3%

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

   3. associate-/l* 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-/l* 99.3%

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

7. Applied egg-rr 99.3%

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

   1. sub-neg 99.3%

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

   2. associate-/r/ 99.3%

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

   3. *-inverses 99.3%

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

9. Simplified 99.3%

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

    1. fma-neg 99.3%

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

    2. clear-num 99.3%

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

    3. quot-tan 99.4%

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

    4. /-rgt-identity 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 99.4% 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 78.6%

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

   1. associate-/l* 78.6%

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

   2. +-commutative 78.6%

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

3. Simplified 78.6%

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

   1. cos-sum 99.3%

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

5. Applied egg-rr 99.3%

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

   1. div-sub 99.3%

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

   2. sub-neg 99.3%

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

   3. associate-/l* 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-/l* 99.3%

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

7. Applied egg-rr 99.3%

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

   1. sub-neg 99.3%

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

   2. associate-/r/ 99.3%

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

   3. *-inverses 99.3%

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

9. Simplified 99.3%

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

    1. sub-neg 99.3%

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

    2. /-rgt-identity 99.3%

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

    3. *-commutative 99.3%

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

    4. clear-num 99.3%

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

    5. un-div-inv 99.3%

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

    6. quot-tan 99.4%

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

11. Applied egg-rr 99.4%

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

    1. unsub-neg 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 4: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-6} \lor \neg \left(b \leq 2.4 \cdot 10^{-14}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -7.6e-6) (not (<= b 2.4e-14)))
   (* r (/ (sin b) (cos b)))
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.6e-6) || !(b <= 2.4e-14)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = (b * r) / 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 <= (-7.6d-6)) .or. (.not. (b <= 2.4d-14))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.6e-6) || !(b <= 2.4e-14)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -7.6e-6) or not (b <= 2.4e-14):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -7.6e-6) || !(b <= 2.4e-14))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -7.6e-6) || ~((b <= 2.4e-14)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -7.6e-6], N[Not[LessEqual[b, 2.4e-14]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.6000000000000001e-6 or 2.4e-14 < b

   1. Initial program 61.1%

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

      1. associate-/l* 61.1%

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

      2. remove-double-neg 61.1%

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

      3. sin-neg 61.1%

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

      4. neg-mul-1 61.1%

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

      5. associate-/r* 61.1%

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

      6. associate-/l* 61.1%

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

      7. *-commutative 61.1%

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

      8. associate-*l/ 61.1%

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

      9. associate-/l* 61.1%

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

      10. sin-neg 61.1%

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

      11. distribute-lft-neg-in 61.1%

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

      12. distribute-rgt-neg-in 61.1%

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

      13. associate-/l* 61.1%

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

      14. metadata-eval 61.1%

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

      15. /-rgt-identity 61.1%

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

      16. +-commutative 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in a around 0 61.1%

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

   if -7.6000000000000001e-6 < b < 2.4e-14

   1. Initial program 99.8%

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

      1. associate-/l* 99.6%

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

      2. remove-double-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. neg-mul-1 99.6%

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

      5. associate-/r* 99.6%

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

      6. associate-/l* 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-/l* 99.7%

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

      10. sin-neg 99.7%

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

      11. distribute-lft-neg-in 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. associate-/l* 99.7%

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

      14. metadata-eval 99.7%

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

      15. /-rgt-identity 99.7%

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

      16. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in b around 0 99.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-6} \lor \neg \left(b \leq 2.4 \cdot 10^{-14}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 5: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-6} \lor \neg \left(b \leq 4.2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -2.05e-6) (not (<= b 4.2e-6)))
   (/ r (/ (cos b) (sin b)))
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.05e-6) || !(b <= 4.2e-6)) {
		tmp = r / (cos(b) / sin(b));
	} else {
		tmp = (b * r) / 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 <= (-2.05d-6)) .or. (.not. (b <= 4.2d-6))) then
        tmp = r / (cos(b) / sin(b))
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -2.05e-6) || !(b <= 4.2e-6)) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -2.05e-6) or not (b <= 4.2e-6):
		tmp = r / (math.cos(b) / math.sin(b))
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -2.05e-6) || !(b <= 4.2e-6))
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -2.05e-6) || ~((b <= 4.2e-6)))
		tmp = r / (cos(b) / sin(b));
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -2.05e-6], N[Not[LessEqual[b, 4.2e-6]], $MachinePrecision]], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.0499999999999999e-6 or 4.1999999999999996e-6 < b

   1. Initial program 60.8%

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

      1. associate-/l* 60.9%

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

      2. +-commutative 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in a around 0 60.9%

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

   if -2.0499999999999999e-6 < b < 4.1999999999999996e-6

   1. Initial program 99.8%

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

      1. associate-/l* 99.6%

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

      2. remove-double-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. neg-mul-1 99.6%

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

      5. associate-/r* 99.6%

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

      6. associate-/l* 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-/l* 99.7%

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

      10. sin-neg 99.7%

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

      11. distribute-lft-neg-in 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. associate-/l* 99.7%

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

      14. metadata-eval 99.7%

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

      15. /-rgt-identity 99.7%

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

      16. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in b around 0 99.8%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-6} \lor \neg \left(b \leq 4.2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 6: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -9.5e-7)
   (/ (sin b) (/ (cos b) r))
   (if (<= b 5e-5) (/ (* b r) (cos a)) (/ r (/ (cos b) (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -9.5e-7) {
		tmp = sin(b) / (cos(b) / r);
	} else if (b <= 5e-5) {
		tmp = (b * r) / 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 <= (-9.5d-7)) then
        tmp = sin(b) / (cos(b) / r)
    else if (b <= 5d-5) then
        tmp = (b * r) / 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 <= -9.5e-7) {
		tmp = Math.sin(b) / (Math.cos(b) / r);
	} else if (b <= 5e-5) {
		tmp = (b * r) / Math.cos(a);
	} else {
		tmp = r / (Math.cos(b) / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -9.5e-7:
		tmp = math.sin(b) / (math.cos(b) / r)
	elif b <= 5e-5:
		tmp = (b * r) / 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 <= -9.5e-7)
		tmp = Float64(sin(b) / Float64(cos(b) / r));
	elseif (b <= 5e-5)
		tmp = Float64(Float64(b * r) / 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 <= -9.5e-7)
		tmp = sin(b) / (cos(b) / r);
	elseif (b <= 5e-5)
		tmp = (b * r) / cos(a);
	else
		tmp = r / (cos(b) / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -9.5e-7], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-5], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $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 < -9.5000000000000001e-7

   1. Initial program 60.8%

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

      1. associate-/l* 60.8%

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

      2. remove-double-neg 60.8%

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

      3. sin-neg 60.8%

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

      4. neg-mul-1 60.8%

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

      5. associate-/r* 60.8%

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

      6. associate-/l* 60.8%

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

      7. *-commutative 60.8%

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

      8. associate-*l/ 60.7%

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

      9. associate-/l* 60.7%

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

      10. sin-neg 60.7%

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

      11. distribute-lft-neg-in 60.7%

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

      12. distribute-rgt-neg-in 60.7%

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

      13. associate-/l* 60.7%

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

      14. metadata-eval 60.7%

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

      15. /-rgt-identity 60.7%

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

      16. +-commutative 60.7%

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

   3. Simplified 60.7%

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

      1. associate-*l/ 60.8%

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

      2. associate-/l* 60.8%

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

   5. Applied egg-rr 60.8%

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

   6. Taylor expanded in a around 0 61.0%

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

   if -9.5000000000000001e-7 < b < 5.00000000000000024e-5

   1. Initial program 99.8%

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

      1. associate-/l* 99.6%

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

      2. remove-double-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. neg-mul-1 99.6%

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

      5. associate-/r* 99.6%

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

      6. associate-/l* 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-/l* 99.7%

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

      10. sin-neg 99.7%

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

      11. distribute-lft-neg-in 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. associate-/l* 99.7%

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

      14. metadata-eval 99.7%

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

      15. /-rgt-identity 99.7%

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

      16. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in b around 0 99.8%

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

   if 5.00000000000000024e-5 < b

   1. Initial program 60.8%

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

      1. associate-/l* 61.0%

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

      2. +-commutative 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in a around 0 60.9%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 7: 55.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{+15} \lor \neg \left(b \leq 1.86\right):\\ \;\;\;\;\left|\sin b \cdot r\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -7.4e+15) (not (<= b 1.86)))
   (fabs (* (sin b) r))
   (/ (* b r) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.4e+15) || !(b <= 1.86)) {
		tmp = fabs((sin(b) * r));
	} else {
		tmp = (b * r) / cos((b + 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 <= (-7.4d+15)) .or. (.not. (b <= 1.86d0))) then
        tmp = abs((sin(b) * r))
    else
        tmp = (b * r) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.4e+15) || !(b <= 1.86)) {
		tmp = Math.abs((Math.sin(b) * r));
	} else {
		tmp = (b * r) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -7.4e+15) or not (b <= 1.86):
		tmp = math.fabs((math.sin(b) * r))
	else:
		tmp = (b * r) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -7.4e+15) || !(b <= 1.86))
		tmp = abs(Float64(sin(b) * r));
	else
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -7.4e+15) || ~((b <= 1.86)))
		tmp = abs((sin(b) * r));
	else
		tmp = (b * r) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -7.4e+15], N[Not[LessEqual[b, 1.86]], $MachinePrecision]], N[Abs[N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]], $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.4e15 or 1.8600000000000001 < b

   1. Initial program 59.3%

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

      1. associate-/l* 59.4%

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

      2. remove-double-neg 59.4%

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

      3. sin-neg 59.4%

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

      4. neg-mul-1 59.4%

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

      5. associate-/r* 59.4%

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

      6. associate-/l* 59.3%

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

      7. *-commutative 59.3%

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

      8. associate-*l/ 59.4%

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

      9. associate-/l* 59.4%

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

      10. sin-neg 59.4%

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

      11. distribute-lft-neg-in 59.4%

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

      12. distribute-rgt-neg-in 59.4%

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

      13. associate-/l* 59.4%

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

      14. metadata-eval 59.4%

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

      15. /-rgt-identity 59.4%

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

      16. +-commutative 59.4%

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

   3. Simplified 59.4%

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

      1. associate-*l/ 59.3%

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

      2. associate-/l* 59.4%

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

   5. Applied egg-rr 59.4%

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

   6. Taylor expanded in b around 0 11.7%

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

      1. add-sqr-sqrt 6.5%

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

      2. sqrt-unprod 9.1%

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

      3. pow2 9.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{\sin b}{\frac{\cos a}{r}}\right)}^{2}}} \]

      4. div-inv 9.1%

         \[\leadsto \sqrt{{\color{blue}{\left(\sin b \cdot \frac{1}{\frac{\cos a}{r}}\right)}}^{2}} \]

      5. clear-num 9.1%

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

   8. Applied egg-rr 9.1%

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

      1. unpow2 9.1%

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

      2. rem-sqrt-square 12.2%

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

   10. Simplified 12.2%

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

   11. Taylor expanded in a around 0 12.7%

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

   if -7.4e15 < b < 1.8600000000000001

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 97.5%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{+15} \lor \neg \left(b \leq 1.86\right):\\ \;\;\;\;\left|\sin b \cdot r\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 8: 77.5% 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 78.6%

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

   1. associate-/l* 78.6%

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

   2. +-commutative 78.6%

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

3. Simplified 78.6%

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

   1. associate-/r/ 78.6%

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

5. Applied egg-rr 78.6%

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

6. Final simplification 78.6%

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

Alternative 9: 77.3% 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 78.6%

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

   1. associate-/l* 78.6%

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

   2. remove-double-neg 78.6%

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

   3. sin-neg 78.6%

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

   4. neg-mul-1 78.6%

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

   5. associate-/r* 78.6%

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

   6. associate-/l* 78.6%

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

   7. *-commutative 78.6%

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

   8. associate-*l/ 78.6%

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

   9. associate-/l* 78.6%

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

   10. sin-neg 78.6%

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

   11. distribute-lft-neg-in 78.6%

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

   12. distribute-rgt-neg-in 78.6%

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

   13. associate-/l* 78.6%

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

   14. metadata-eval 78.6%

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

   15. /-rgt-identity 78.6%

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

   16. +-commutative 78.6%

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

3. Simplified 78.6%

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

   1. associate-*l/ 78.6%

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

   2. associate-/l* 78.6%

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

5. Applied egg-rr 78.6%

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

   1. clear-num 78.0%

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

   2. cos-sum 98.8%

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

   3. associate-/r/ 99.3%

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

   4. clear-num 99.4%

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

   5. sub-neg 99.4%

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

   6. add-sqr-sqrt 51.2%

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

   7. sqrt-unprod 89.1%

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

   8. sqr-neg 89.1%

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

   9. sqrt-unprod 48.0%

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

   10. add-sqr-sqrt 78.5%

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

   11. cos-diff 78.7%

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

7. Applied egg-rr 78.7%

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

8. Final simplification 78.7%

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

Alternative 10: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. associate-/l* 78.6%

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

   2. remove-double-neg 78.6%

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

   3. sin-neg 78.6%

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

   4. neg-mul-1 78.6%

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

   5. associate-/r* 78.6%

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

   6. associate-/l* 78.6%

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

   7. *-commutative 78.6%

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

   8. associate-*l/ 78.6%

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

   9. associate-/l* 78.6%

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

   10. sin-neg 78.6%

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

   11. distribute-lft-neg-in 78.6%

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

   12. distribute-rgt-neg-in 78.6%

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

   13. associate-/l* 78.6%

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

   14. metadata-eval 78.6%

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

   15. /-rgt-identity 78.6%

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

   16. +-commutative 78.6%

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

3. Simplified 78.6%

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

   1. associate-*l/ 78.6%

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

   2. associate-/l* 78.6%

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

5. Applied egg-rr 78.6%

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

6. Taylor expanded in b around 0 52.5%

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

7. Taylor expanded in b around inf 52.5%

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

   1. *-commutative 52.5%

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

   2. associate-*r/ 52.5%

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

9. Simplified 52.5%

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

10. Final simplification 52.5%

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

Alternative 11: 55.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+27} \lor \neg \left(b \leq 9.4 \cdot 10^{+27}\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{\cos a}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -9.8e+27) (not (<= b 9.4e+27)))
   (* (sin b) r)
   (/ b (/ (cos a) r))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -9.8e+27) || !(b <= 9.4e+27)) {
		tmp = sin(b) * r;
	} else {
		tmp = b / (cos(a) / r);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-9.8d+27)) .or. (.not. (b <= 9.4d+27))) then
        tmp = sin(b) * r
    else
        tmp = b / (cos(a) / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -9.8e+27) || !(b <= 9.4e+27)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = b / (Math.cos(a) / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -9.8e+27) or not (b <= 9.4e+27):
		tmp = math.sin(b) * r
	else:
		tmp = b / (math.cos(a) / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -9.8e+27) || !(b <= 9.4e+27))
		tmp = Float64(sin(b) * r);
	else
		tmp = Float64(b / Float64(cos(a) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -9.8e+27) || ~((b <= 9.4e+27)))
		tmp = sin(b) * r;
	else
		tmp = b / (cos(a) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -9.8e+27], N[Not[LessEqual[b, 9.4e+27]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision], N[(b / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.8000000000000003e27 or 9.39999999999999952e27 < b

   1. Initial program 59.2%

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

      1. associate-/l* 59.3%

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

      2. remove-double-neg 59.3%

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

      3. sin-neg 59.3%

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

      4. neg-mul-1 59.3%

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

      5. associate-/r* 59.3%

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

      6. associate-/l* 59.2%

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

      7. *-commutative 59.2%

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

      8. associate-*l/ 59.3%

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

      9. associate-/l* 59.3%

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

      10. sin-neg 59.3%

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

      11. distribute-lft-neg-in 59.3%

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

      12. distribute-rgt-neg-in 59.3%

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

      13. associate-/l* 59.3%

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

      14. metadata-eval 59.3%

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

      15. /-rgt-identity 59.3%

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

      16. +-commutative 59.3%

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

   3. Simplified 59.3%

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

      1. associate-*l/ 59.2%

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

      2. associate-/l* 59.3%

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

   5. Applied egg-rr 59.3%

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

   6. Taylor expanded in b around 0 11.9%

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

   7. Taylor expanded in a around 0 11.7%

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

   if -9.8000000000000003e27 < b < 9.39999999999999952e27

   1. Initial program 97.7%

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

      1. associate-/l* 97.5%

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

      2. remove-double-neg 97.5%

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

      3. sin-neg 97.5%

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

      4. neg-mul-1 97.5%

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

      5. associate-/r* 97.5%

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

      6. associate-/l* 97.7%

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

      7. *-commutative 97.7%

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

      8. associate-*l/ 97.6%

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

      9. associate-/l* 97.6%

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

      10. sin-neg 97.6%

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

      11. distribute-lft-neg-in 97.6%

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

      12. distribute-rgt-neg-in 97.6%

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

      13. associate-/l* 97.6%

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

      14. metadata-eval 97.6%

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

      15. /-rgt-identity 97.6%

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

      16. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in b around 0 92.6%

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

      1. associate-/l* 92.5%

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

   6. Simplified 92.5%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+27} \lor \neg \left(b \leq 9.4 \cdot 10^{+27}\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 12: 55.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+27} \lor \neg \left(b \leq 8.8 \cdot 10^{+27}\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.6e+27) (not (<= b 8.8e+27)))
   (* (sin b) r)
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e+27) || !(b <= 8.8e+27)) {
		tmp = sin(b) * r;
	} else {
		tmp = (b * r) / 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 <= (-1.6d+27)) .or. (.not. (b <= 8.8d+27))) then
        tmp = sin(b) * r
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e+27) || !(b <= 8.8e+27)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.6e+27) or not (b <= 8.8e+27):
		tmp = math.sin(b) * r
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.6e+27) || !(b <= 8.8e+27))
		tmp = Float64(sin(b) * r);
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.6e+27) || ~((b <= 8.8e+27)))
		tmp = sin(b) * r;
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.6e+27], N[Not[LessEqual[b, 8.8e+27]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.60000000000000008e27 or 8.7999999999999995e27 < b

   1. Initial program 59.2%

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

      1. associate-/l* 59.3%

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

      2. remove-double-neg 59.3%

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

      3. sin-neg 59.3%

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

      4. neg-mul-1 59.3%

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

      5. associate-/r* 59.3%

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

      6. associate-/l* 59.2%

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

      7. *-commutative 59.2%

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

      8. associate-*l/ 59.3%

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

      9. associate-/l* 59.3%

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

      10. sin-neg 59.3%

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

      11. distribute-lft-neg-in 59.3%

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

      12. distribute-rgt-neg-in 59.3%

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

      13. associate-/l* 59.3%

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

      14. metadata-eval 59.3%

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

      15. /-rgt-identity 59.3%

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

      16. +-commutative 59.3%

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

   3. Simplified 59.3%

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

      1. associate-*l/ 59.2%

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

      2. associate-/l* 59.3%

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

   5. Applied egg-rr 59.3%

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

   6. Taylor expanded in b around 0 11.9%

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

   7. Taylor expanded in a around 0 11.7%

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

   if -1.60000000000000008e27 < b < 8.7999999999999995e27

   1. Initial program 97.7%

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

      1. associate-/l* 97.5%

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

      2. remove-double-neg 97.5%

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

      3. sin-neg 97.5%

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

      4. neg-mul-1 97.5%

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

      5. associate-/r* 97.5%

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

      6. associate-/l* 97.7%

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

      7. *-commutative 97.7%

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

      8. associate-*l/ 97.6%

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

      9. associate-/l* 97.6%

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

      10. sin-neg 97.6%

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

      11. distribute-lft-neg-in 97.6%

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

      12. distribute-rgt-neg-in 97.6%

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

      13. associate-/l* 97.6%

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

      14. metadata-eval 97.6%

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

      15. /-rgt-identity 97.6%

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

      16. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in b around 0 92.6%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+27} \lor \neg \left(b \leq 8.8 \cdot 10^{+27}\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 13: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. associate-/l* 78.6%

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

   2. remove-double-neg 78.6%

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

   3. sin-neg 78.6%

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

   4. neg-mul-1 78.6%

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

   5. associate-/r* 78.6%

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

   6. associate-/l* 78.6%

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

   7. *-commutative 78.6%

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

   8. associate-*l/ 78.6%

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

   9. associate-/l* 78.6%

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

   10. sin-neg 78.6%

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

   11. distribute-lft-neg-in 78.6%

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

   12. distribute-rgt-neg-in 78.6%

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

   13. associate-/l* 78.6%

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

   14. metadata-eval 78.6%

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

   15. /-rgt-identity 78.6%

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

   16. +-commutative 78.6%

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

3. Simplified 78.6%

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

   1. associate-*l/ 78.6%

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

   2. associate-/l* 78.6%

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

5. Applied egg-rr 78.6%

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

6. Taylor expanded in b around 0 52.5%

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

7. Taylor expanded in a around 0 39.2%

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

8. Final simplification 39.2%

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

Alternative 14: 35.4% 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 78.6%

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

   1. associate-/l* 78.6%

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

   2. remove-double-neg 78.6%

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

   3. sin-neg 78.6%

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

   4. neg-mul-1 78.6%

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

   5. associate-/r* 78.6%

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

   6. associate-/l* 78.6%

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

   7. *-commutative 78.6%

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

   8. associate-*l/ 78.6%

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

   9. associate-/l* 78.6%

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

   10. sin-neg 78.6%

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

   11. distribute-lft-neg-in 78.6%

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

   12. distribute-rgt-neg-in 78.6%

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

   13. associate-/l* 78.6%

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

   14. metadata-eval 78.6%

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

   15. /-rgt-identity 78.6%

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

   16. +-commutative 78.6%

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

3. Simplified 78.6%

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

4. Taylor expanded in b around 0 48.5%

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

   1. associate-/l* 48.4%

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

6. Simplified 48.4%

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

7. Taylor expanded in a around 0 35.3%

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

8. Final simplification 35.3%

   \[\leadsto b \cdot r \]

Reproduce

herbie shell --seed 2023339 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))