rsin B (should all be same)

Percentage Accurate: 76.4% → 99.5%

Time: 13.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. +-commutative 73.4%

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

   2. cos-sum 99.4%

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

   3. fma-neg 99.4%

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

4. Applied egg-rr 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. +-commutative 73.4%

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

   2. cos-sum 99.4%

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

   3. sub-neg 99.4%

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

4. Applied egg-rr 99.4%

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

   1. sub-neg 99.4%

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

6. Simplified 99.4%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. +-commutative 73.4%

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

   2. associate-/l* 73.4%

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

   3. clear-num 72.2%

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

   4. +-commutative 72.2%

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

   5. *-commutative 72.2%

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

4. Applied egg-rr 72.2%

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

   1. +-commutative 72.2%

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

   2. cos-sum 98.3%

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

   3. div-sub 92.3%

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

6. Applied egg-rr 92.3%

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

7. Taylor expanded in r around 0 99.3%

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

   1. associate-/l* 99.4%

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

   2. fma-neg 99.4%

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

9. Simplified 99.4%

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

Alternative 4: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. +-commutative 73.4%

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

   2. associate-/l* 73.4%

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

   3. clear-num 72.2%

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

   4. +-commutative 72.2%

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

   5. *-commutative 72.2%

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

4. Applied egg-rr 72.2%

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

   1. +-commutative 72.2%

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

   2. cos-sum 98.3%

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

   3. div-sub 92.3%

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

6. Applied egg-rr 92.3%

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

7. Taylor expanded in r around 0 99.3%

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

8. Final simplification 99.3%

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

Alternative 5: 77.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. +-commutative 73.4%

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

   2. cos-sum 99.4%

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

   3. sub-neg 99.4%

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

4. Applied egg-rr 99.4%

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

   1. sub-neg 99.4%

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

6. Simplified 99.4%

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

   1. sin-mult 74.7%

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

   2. +-commutative 74.7%

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

   3. div-sub 74.7%

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

   4. cos-diff 74.6%

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

   5. add-sqr-sqrt 40.6%

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

   6. sqrt-unprod 73.3%

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

   7. sqr-neg 73.3%

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

   8. sqrt-unprod 44.4%

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

   9. add-sqr-sqrt 72.6%

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

   10. sub-neg 72.6%

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

   11. cos-sum 74.7%

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

   12. +-commutative 74.7%

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

8. Applied egg-rr 74.7%

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

   1. +-inverses 74.7%

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

10. Simplified 74.7%

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

11. Final simplification 74.7%

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

Alternative 6: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5400000000000 \lor \neg \left(a \leq 3.1 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -5400000000000.0) (not (<= a 3.1e-8)))
   (* r (/ (sin b) (cos a)))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -5400000000000.0) || !(a <= 3.1e-8)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = sin(b) * (r / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-5400000000000.0d0)) .or. (.not. (a <= 3.1d-8))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -5400000000000.0) || !(a <= 3.1e-8)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -5400000000000.0) or not (a <= 3.1e-8):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -5400000000000.0) || !(a <= 3.1e-8))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -5400000000000.0) || ~((a <= 3.1e-8)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -5400000000000.0], N[Not[LessEqual[a, 3.1e-8]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.4e12 or 3.1e-8 < a

   1. Initial program 47.4%

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

   3. Taylor expanded in b around 0 48.2%

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

   if -5.4e12 < a < 3.1e-8

   1. Initial program 98.1%

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

      1. cos-sum 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. prod-diff 99.6%

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

      4. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5400000000000 \lor \neg \left(a \leq 3.1 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. +-commutative 73.4%

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

   2. cos-sum 99.4%

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

   3. fma-neg 99.4%

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

4. Applied egg-rr 99.4%

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

   1. fma-undefine 99.4%

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

   2. add-sqr-sqrt 61.3%

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

   3. sqrt-unprod 90.0%

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

   4. sqr-neg 90.0%

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

   5. sqrt-unprod 40.3%

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

   6. add-sqr-sqrt 73.6%

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

   7. cos-diff 73.5%

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

6. Applied egg-rr 73.5%

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

Alternative 8: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

3. Final simplification 73.4%

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

Alternative 9: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

3. Taylor expanded in b around 0 51.7%

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

Alternative 10: 50.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

3. Taylor expanded in b around 0 47.7%

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

   1. associate-/l* 47.7%

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

5. Simplified 47.7%

   \[\leadsto \color{blue}{b \cdot \frac{r}{\cos a}} \]
6. Add Preprocessing

Alternative 11: 34.9% 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 73.4%

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

3. Taylor expanded in b around 0 47.7%

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

   1. associate-/l* 47.7%

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

5. Simplified 47.7%

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

6. Taylor expanded in a around 0 33.5%

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

7. Final simplification 33.5%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))