rsin A (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 14.6s

Alternatives: 9

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: 76.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. +-commutative 76.4%

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

3. Simplified 76.4%

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

   1. cos-sum 99.5%

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

   2. fma-neg 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{1}{\frac{\cos a}{\tan b} - \sin a} \end{array} \]

FPCore

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

C

double code(double r, double a, double b) {
	return r * (1.0 / ((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 * (1.0d0 / ((cos(a) / tan(b)) - sin(a)))
end function

Java

public static double code(double r, double a, double b) {
	return r * (1.0 / ((Math.cos(a) / Math.tan(b)) - Math.sin(a)));
}

Python

def code(r, a, b):
	return r * (1.0 / ((math.cos(a) / math.tan(b)) - math.sin(a)))

Julia

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

MATLAB

function tmp = code(r, a, b)
	tmp = r * (1.0 / ((cos(a) / tan(b)) - sin(a)));
end

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate-/l* 76.3%

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

   2. +-commutative 76.3%

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

3. Simplified 76.3%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. sub-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. *-un-lft-identity 99.4%

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

   5. times-frac 99.3%

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

   6. clear-num 99.3%

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

   7. quot-tan 99.4%

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

7. Applied egg-rr 99.4%

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

   1. sub-neg 99.4%

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

   2. /-rgt-identity 99.4%

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

   3. associate-*r/ 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. associate-/l* 99.4%

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

9. Simplified 99.4%

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

    1. div-inv 99.5%

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

    2. associate-/r/ 99.5%

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

    3. pow1 99.5%

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

    4. pow1 99.5%

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

    5. pow-div 99.5%

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

    6. metadata-eval 99.5%

       \[\leadsto r \cdot \frac{1}{\frac{\cos a}{\tan b} - {\sin b}^{\color{blue}{0}} \cdot \sin a} \]

    7. metadata-eval 99.5%

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

    8. *-un-lft-identity 99.5%

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate-/l* 76.3%

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

   2. +-commutative 76.3%

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

3. Simplified 76.3%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. sub-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. *-un-lft-identity 99.4%

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

   5. times-frac 99.3%

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

   6. clear-num 99.3%

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

   7. quot-tan 99.4%

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

7. Applied egg-rr 99.4%

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

   1. sub-neg 99.4%

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

   2. /-rgt-identity 99.4%

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

   3. associate-*r/ 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. associate-/l* 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in b around 0 99.5%

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

11. Final simplification 99.5%

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

Alternative 4: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. +-commutative 76.4%

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

3. Simplified 76.4%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

   1. sin-mult 77.5%

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

   2. cos-sum 78.0%

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

   3. div-sub 78.0%

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

   4. cos-diff 99.4%

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

   5. add-sqr-sqrt 55.6%

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

   6. sqrt-unprod 87.8%

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

   7. sqr-neg 87.8%

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

   8. sqrt-unprod 48.2%

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

   9. add-sqr-sqrt 77.5%

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

   10. sub-neg 77.5%

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

   11. cos-sum 78.2%

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

   12. cos-sum 77.5%

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

7. Applied egg-rr 77.5%

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

   1. +-inverses 77.5%

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

9. Simplified 77.5%

   \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{0}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 67.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r \cdot \sin b}{\cos b \cdot \cos a - 0}\right)\right)} \]

    2. expm1-udef 27.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{r \cdot \sin b}{\cos b \cdot \cos a - 0}\right)} - 1} \]

    3. *-commutative 27.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin b \cdot r}}{\cos b \cdot \cos a - 0}\right)} - 1 \]

    4. --rgt-identity 27.2%

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

    5. times-frac 27.2%

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

    6. tan-quot 27.2%

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

11. Applied egg-rr 27.2%

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

    1. expm1-def 67.4%

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

    2. expm1-log1p 77.6%

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

    3. *-commutative 77.6%

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

13. Simplified 77.6%

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

14. Final simplification 77.6%

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

Alternative 5: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-6} \lor \neg \left(b \leq 1.65 \cdot 10^{-38}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.8e-6) (not (<= b 1.65e-38)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8e-6) || !(b <= 1.65e-38)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-4.8d-6)) .or. (.not. (b <= 1.65d-38))) then
        tmp = r * tan(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8e-6) || !(b <= 1.65e-38)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.8e-6) or not (b <= 1.65e-38):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.8e-6) || !(b <= 1.65e-38))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.8e-6) || ~((b <= 1.65e-38)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.8e-6], N[Not[LessEqual[b, 1.65e-38]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.7999999999999998e-6 or 1.6500000000000001e-38 < b

   1. Initial program 54.4%

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

      1. associate-/l* 54.3%

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

      2. +-commutative 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in a around 0 54.9%

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

      1. associate-/l* 54.8%

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

      2. associate-/r/ 54.9%

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

   6. Simplified 54.9%

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

      1. expm1-log1p-u 41.4%

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

      2. expm1-udef 19.0%

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

      3. associate-*l/ 19.0%

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

      4. *-un-lft-identity 19.0%

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

      5. times-frac 19.0%

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

      6. tan-quot 19.0%

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

   8. Applied egg-rr 19.0%

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

      1. expm1-def 41.4%

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

      2. expm1-log1p 54.9%

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

      3. /-rgt-identity 54.9%

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

   10. Simplified 54.9%

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

   if -4.7999999999999998e-6 < b < 1.6500000000000001e-38

   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. Step-by-step derivation

      1. cos-sum 99.8%

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

      2. fma-neg 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in b around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-6} \lor \neg \left(b \leq 1.65 \cdot 10^{-38}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 6: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-7} \lor \neg \left(b \leq 1.65 \cdot 10^{-38}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -9.5e-7) (not (<= b 1.65e-38)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -9.5e-7) || !(b <= 1.65e-38)) {
		tmp = r * tan(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 <= (-9.5d-7)) .or. (.not. (b <= 1.65d-38))) then
        tmp = r * tan(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 <= -9.5e-7) || !(b <= 1.65e-38)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -9.5e-7) or not (b <= 1.65e-38):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -9.5e-7) || !(b <= 1.65e-38))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -9.5e-7) || ~((b <= 1.65e-38)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -9.5e-7], N[Not[LessEqual[b, 1.65e-38]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.5000000000000001e-7 or 1.6500000000000001e-38 < b

   1. Initial program 54.4%

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

      1. associate-/l* 54.3%

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

      2. +-commutative 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in a around 0 54.9%

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

      1. associate-/l* 54.8%

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

      2. associate-/r/ 54.9%

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

   6. Simplified 54.9%

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

      1. expm1-log1p-u 41.4%

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

      2. expm1-udef 19.0%

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

      3. associate-*l/ 19.0%

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

      4. *-un-lft-identity 19.0%

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

      5. times-frac 19.0%

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

      6. tan-quot 19.0%

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

   8. Applied egg-rr 19.0%

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

      1. expm1-def 41.4%

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

      2. expm1-log1p 54.9%

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

      3. /-rgt-identity 54.9%

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

   10. Simplified 54.9%

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

   if -9.5000000000000001e-7 < b < 1.6500000000000001e-38

   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. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in b around 0 99.6%

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

      1. associate-/r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-7} \lor \neg \left(b \leq 1.65 \cdot 10^{-38}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 7: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate-/l* 76.3%

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

   2. +-commutative 76.3%

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

3. Simplified 76.3%

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

4. Taylor expanded in a around 0 62.0%

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

   1. associate-/l* 61.9%

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

   2. associate-/r/ 62.0%

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

6. Simplified 62.0%

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

7. Taylor expanded in b around 0 41.1%

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

8. Final simplification 41.1%

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

Alternative 8: 60.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate-/l* 76.3%

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

   2. +-commutative 76.3%

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

3. Simplified 76.3%

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

4. Taylor expanded in a around 0 62.0%

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

   1. associate-/l* 61.9%

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

   2. associate-/r/ 62.0%

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

6. Simplified 62.0%

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

   1. expm1-log1p-u 53.3%

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

   2. expm1-udef 24.5%

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

   3. associate-*l/ 24.5%

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

   4. *-un-lft-identity 24.5%

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

   5. times-frac 24.5%

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

   6. tan-quot 24.5%

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

8. Applied egg-rr 24.5%

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

   1. expm1-def 53.3%

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

   2. expm1-log1p 62.1%

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

   3. /-rgt-identity 62.1%

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

10. Simplified 62.1%

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

11. Final simplification 62.1%

    \[\leadsto r \cdot \tan b \]

Alternative 9: 34.6% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate-/l* 76.3%

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

   2. +-commutative 76.3%

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

3. Simplified 76.3%

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

4. Taylor expanded in b around 0 51.8%

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

5. Taylor expanded in a around 0 37.6%

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

   1. *-commutative 37.6%

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

7. Simplified 37.6%

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

8. Final simplification 37.6%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023310 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))