rsin B (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 16.1s

Alternatives: 21

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: 76.6% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. associate-*r/ 73.9%

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

   2. +-commutative 73.9%

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

3. Simplified 73.9%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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. *-un-lft-identity 99.5%

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

   3. *-un-lft-identity 99.5%

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

   4. prod-diff 99.5%

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

5. Applied egg-rr 99.5%

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

   1. *-rgt-identity 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-lft-neg-in 99.5%

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

   4. *-commutative 99.5%

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

   5. fma-udef 99.5%

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

   6. *-rgt-identity 99.5%

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

   7. distribute-lft-neg-in 99.5%

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

   8. *-rgt-identity 99.5%

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

   9. fma-udef 99.6%

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

   10. *-commutative 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

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(Float64(r * sin(b)) / fma(cos(b), cos(a), Float64(sin(b) * Float64(-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 73.9%

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

   1. associate-*r/ 73.9%

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

   2. +-commutative 73.9%

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

3. Simplified 73.9%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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. cancel-sign-sub-inv 99.5%

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

   3. fma-def 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. +-commutative 73.9%

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

3. Simplified 73.9%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. *-commutative 73.9%

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

   2. associate-/r/ 73.9%

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

   3. +-commutative 73.9%

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

3. Simplified 73.9%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. associate-*r/ 73.9%

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

   2. +-commutative 73.9%

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

3. Simplified 73.9%

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

   1. cos-sum 99.4%

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

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. Final simplification 99.5%

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

Alternative 6: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-5} \lor \neg \left(a \leq 16500000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -1.45e-5) (not (<= a 16500000000.0)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.45e-5) || !(a <= 16500000000.0)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.45d-5)) .or. (.not. (a <= 16500000000.0d0))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.45e-5) || !(a <= 16500000000.0)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -1.45e-5) or not (a <= 16500000000.0):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -1.45e-5) || !(a <= 16500000000.0))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -1.45e-5) || ~((a <= 16500000000.0)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -1.45e-5], N[Not[LessEqual[a, 16500000000.0]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e-5 or 1.65e10 < a

   1. Initial program 49.1%

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

      1. +-commutative 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in b around 0 49.0%

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

   if -1.45e-5 < a < 1.65e10

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 98.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-5} \lor \neg \left(a \leq 16500000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 7: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-6} \lor \neg \left(a \leq 16500000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -1.9e-6) (not (<= a 16500000000.0)))
   (* r (/ (sin b) (cos a)))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.9e-6) || !(a <= 16500000000.0)) {
		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 <= (-1.9d-6)) .or. (.not. (a <= 16500000000.0d0))) 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 <= -1.9e-6) || !(a <= 16500000000.0)) {
		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 <= -1.9e-6) or not (a <= 16500000000.0):
		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 <= -1.9e-6) || !(a <= 16500000000.0))
		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 <= -1.9e-6) || ~((a <= 16500000000.0)))
		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, -1.9e-6], N[Not[LessEqual[a, 16500000000.0]], $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 < -1.9e-6 or 1.65e10 < a

   1. Initial program 49.1%

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

      1. +-commutative 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in b around 0 49.0%

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

   if -1.9e-6 < a < 1.65e10

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/r/ 99.0%

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

      3. +-commutative 99.0%

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

   3. Simplified 99.0%

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

      1. clear-num 98.9%

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

      2. associate-/r/ 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. div-inv 99.0%

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

      2. associate-*l/ 99.0%

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

      3. *-un-lft-identity 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in a around 0 99.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-6} \lor \neg \left(a \leq 16500000000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 8: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-5} \lor \neg \left(a \leq 16500000000\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -1.2e-5) (not (<= a 16500000000.0)))
   (/ r (/ (cos a) (sin b)))
   (* (sin b) (/ r (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -1.2e-5) || !(a <= 16500000000.0)) {
		tmp = r / (cos(a) / sin(b));
	} 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 <= (-1.2d-5)) .or. (.not. (a <= 16500000000.0d0))) then
        tmp = r / (cos(a) / sin(b))
    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 <= -1.2e-5) || !(a <= 16500000000.0)) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -1.2e-5) or not (a <= 16500000000.0):
		tmp = r / (math.cos(a) / math.sin(b))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -1.2e-5) || !(a <= 16500000000.0))
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	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 <= -1.2e-5) || ~((a <= 16500000000.0)))
		tmp = r / (cos(a) / sin(b));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -1.2e-5], N[Not[LessEqual[a, 16500000000.0]], $MachinePrecision]], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $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 < -1.2e-5 or 1.65e10 < a

   1. Initial program 49.1%

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

      1. +-commutative 49.1%

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

   3. Simplified 49.1%

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

      1. expm1-log1p-u 47.1%

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

   5. Applied egg-rr 47.1%

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

   6. Taylor expanded in b around 0 47.5%

      \[\leadsto r \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b}{\color{blue}{\cos a}}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 49.0%

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

      2. clear-num 49.0%

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

      3. un-div-inv 49.0%

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

   8. Applied egg-rr 49.0%

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

   if -1.2e-5 < a < 1.65e10

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/r/ 99.0%

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

      3. +-commutative 99.0%

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

   3. Simplified 99.0%

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

      1. clear-num 98.9%

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

      2. associate-/r/ 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. div-inv 99.0%

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

      2. associate-*l/ 99.0%

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

      3. *-un-lft-identity 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in a around 0 99.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-5} \lor \neg \left(a \leq 16500000000\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 9: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 16500000000:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -1.45e-5)
   (/ r (/ (cos a) (sin b)))
   (if (<= a 16500000000.0)
     (* (sin b) (/ r (cos b)))
     (/ (sin b) (/ (cos a) r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -1.45e-5) {
		tmp = r / (cos(a) / sin(b));
	} else if (a <= 16500000000.0) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = sin(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 (a <= (-1.45d-5)) then
        tmp = r / (cos(a) / sin(b))
    else if (a <= 16500000000.0d0) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = sin(b) / (cos(a) / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -1.45e-5) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else if (a <= 16500000000.0) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -1.45e-5:
		tmp = r / (math.cos(a) / math.sin(b))
	elif a <= 16500000000.0:
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = math.sin(b) / (math.cos(a) / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -1.45e-5)
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	elseif (a <= 16500000000.0)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -1.45e-5)
		tmp = r / (cos(a) / sin(b));
	elseif (a <= 16500000000.0)
		tmp = sin(b) * (r / cos(b));
	else
		tmp = sin(b) / (cos(a) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -1.45e-5], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 16500000000.0], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.45e-5

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

   3. Simplified 46.3%

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

      1. expm1-log1p-u 43.3%

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

   5. Applied egg-rr 43.3%

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

   6. Taylor expanded in b around 0 44.1%

      \[\leadsto r \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b}{\color{blue}{\cos a}}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 45.6%

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

      2. clear-num 45.6%

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

      3. un-div-inv 45.6%

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

   8. Applied egg-rr 45.6%

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

   if -1.45e-5 < a < 1.65e10

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/r/ 99.0%

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

      3. +-commutative 99.0%

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

   3. Simplified 99.0%

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

      1. clear-num 98.9%

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

      2. associate-/r/ 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. div-inv 99.0%

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

      2. associate-*l/ 99.0%

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

      3. *-un-lft-identity 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in a around 0 99.0%

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

   if 1.65e10 < a

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

      2. associate-/r/ 51.3%

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

      3. +-commutative 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in b around 0 51.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 16500000000:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 10: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 16500000000:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -1.42e-5)
   (/ r (/ (cos a) (sin b)))
   (if (<= a 16500000000.0)
     (/ (* r (sin b)) (cos b))
     (/ (sin b) (/ (cos a) r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -1.42e-5) {
		tmp = r / (cos(a) / sin(b));
	} else if (a <= 16500000000.0) {
		tmp = (r * sin(b)) / cos(b);
	} else {
		tmp = sin(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 (a <= (-1.42d-5)) then
        tmp = r / (cos(a) / sin(b))
    else if (a <= 16500000000.0d0) then
        tmp = (r * sin(b)) / cos(b)
    else
        tmp = sin(b) / (cos(a) / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -1.42e-5) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else if (a <= 16500000000.0) {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	} else {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -1.42e-5:
		tmp = r / (math.cos(a) / math.sin(b))
	elif a <= 16500000000.0:
		tmp = (r * math.sin(b)) / math.cos(b)
	else:
		tmp = math.sin(b) / (math.cos(a) / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -1.42e-5)
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	elseif (a <= 16500000000.0)
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	else
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -1.42e-5)
		tmp = r / (cos(a) / sin(b));
	elseif (a <= 16500000000.0)
		tmp = (r * sin(b)) / cos(b);
	else
		tmp = sin(b) / (cos(a) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -1.42e-5], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 16500000000.0], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.42e-5

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

   3. Simplified 46.3%

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

      1. expm1-log1p-u 43.3%

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

   5. Applied egg-rr 43.3%

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

   6. Taylor expanded in b around 0 44.1%

      \[\leadsto r \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b}{\color{blue}{\cos a}}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 45.6%

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

      2. clear-num 45.6%

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

      3. un-div-inv 45.6%

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

   8. Applied egg-rr 45.6%

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

   if -1.42e-5 < a < 1.65e10

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 99.0%

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

   if 1.65e10 < a

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

      2. associate-/r/ 51.3%

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

      3. +-commutative 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in b around 0 51.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{-5}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 16500000000:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 11: 76.6% 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 73.9%

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

2. Final simplification 73.9%

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

Alternative 12: 76.6% 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 73.9%

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

   1. *-commutative 73.9%

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

   2. associate-/r/ 73.9%

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

   3. +-commutative 73.9%

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

3. Simplified 73.9%

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

   1. div-inv 73.8%

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

   2. clear-num 73.9%

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

   3. *-commutative 73.9%

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

5. Applied egg-rr 73.9%

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

6. Final simplification 73.9%

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

Alternative 13: 76.6% accurate, 1.0× speedup

Math

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

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(Float64(r * 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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.9%

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

   1. associate-*r/ 73.9%

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

   2. +-commutative 73.9%

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

3. Simplified 73.9%

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

4. Final simplification 73.9%

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

Alternative 14: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. +-commutative 73.9%

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

3. Simplified 73.9%

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

4. Taylor expanded in b around 0 50.1%

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

5. Final simplification 50.1%

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

Alternative 15: 54.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -105000000:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 9.2:\\ \;\;\;\;\frac{b}{\frac{\cos \left(b + a\right)}{r}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -105000000.0)
   (* r (sin b))
   (if (<= b 9.2) (/ b (/ (cos (+ b a)) r)) (/ (sin b) (/ 1.0 r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -105000000.0) {
		tmp = r * sin(b);
	} else if (b <= 9.2) {
		tmp = b / (cos((b + a)) / r);
	} else {
		tmp = sin(b) / (1.0 / 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 <= (-105000000.0d0)) then
        tmp = r * sin(b)
    else if (b <= 9.2d0) then
        tmp = b / (cos((b + a)) / r)
    else
        tmp = sin(b) / (1.0d0 / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -105000000.0) {
		tmp = r * Math.sin(b);
	} else if (b <= 9.2) {
		tmp = b / (Math.cos((b + a)) / r);
	} else {
		tmp = Math.sin(b) / (1.0 / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -105000000.0:
		tmp = r * math.sin(b)
	elif b <= 9.2:
		tmp = b / (math.cos((b + a)) / r)
	else:
		tmp = math.sin(b) / (1.0 / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -105000000.0)
		tmp = Float64(r * sin(b));
	elseif (b <= 9.2)
		tmp = Float64(b / Float64(cos(Float64(b + a)) / r));
	else
		tmp = Float64(sin(b) / Float64(1.0 / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -105000000.0)
		tmp = r * sin(b);
	elseif (b <= 9.2)
		tmp = b / (cos((b + a)) / r);
	else
		tmp = sin(b) / (1.0 / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -105000000.0], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2], N[(b / N[(N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e8

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

   3. Simplified 52.5%

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

      1. expm1-log1p-u 33.1%

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

   5. Applied egg-rr 33.1%

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

   6. Taylor expanded in b around 0 11.5%

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

   7. Taylor expanded in a around 0 11.3%

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

   if -1.05e8 < b < 9.1999999999999993

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

   3. Simplified 97.7%

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

      1. expm1-log1p-u 97.5%

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

   5. Applied egg-rr 97.5%

      \[\leadsto r \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b}{\cos \left(b + a\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 77.8%

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

      2. sqrt-unprod 82.5%

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

      3. pow2 82.5%

         \[\leadsto r \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b}{\sqrt{\color{blue}{{\cos \left(b + a\right)}^{2}}}}\right)\right) \]

   7. Applied egg-rr 82.5%

      \[\leadsto r \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin b}{\color{blue}{\sqrt{{\cos \left(b + a\right)}^{2}}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 82.5%

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

      2. rem-sqrt-square 82.5%

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

      3. +-commutative 82.5%

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

   9. Simplified 82.5%

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

   10. Taylor expanded in b around 0 81.9%

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

       1. associate-/l* 81.8%

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

       2. +-commutative 81.8%

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

       3. rem-square-sqrt 77.0%

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

       4. fabs-sqr 77.0%

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

       5. rem-square-sqrt 96.0%

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

   12. Simplified 96.0%

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

   if 9.1999999999999993 < b

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-/r/ 55.2%

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

      3. +-commutative 55.2%

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

   3. Simplified 55.2%

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

      1. clear-num 55.1%

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

      2. associate-/r/ 55.1%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in a around 0 55.1%

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

   7. Taylor expanded in b around 0 11.8%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -105000000:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 9.2:\\ \;\;\;\;\frac{b}{\frac{\cos \left(b + a\right)}{r}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \]

Alternative 16: 54.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -106000000:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 15.6:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -106000000.0)
   (* r (sin b))
   (if (<= b 15.6) (/ (* r b) (cos (+ b a))) (/ (sin b) (/ 1.0 r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -106000000.0) {
		tmp = r * sin(b);
	} else if (b <= 15.6) {
		tmp = (r * b) / cos((b + a));
	} else {
		tmp = sin(b) / (1.0 / 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 <= (-106000000.0d0)) then
        tmp = r * sin(b)
    else if (b <= 15.6d0) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = sin(b) / (1.0d0 / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -106000000.0) {
		tmp = r * Math.sin(b);
	} else if (b <= 15.6) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = Math.sin(b) / (1.0 / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -106000000.0:
		tmp = r * math.sin(b)
	elif b <= 15.6:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = math.sin(b) / (1.0 / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -106000000.0)
		tmp = Float64(r * sin(b));
	elseif (b <= 15.6)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = Float64(sin(b) / Float64(1.0 / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -106000000.0)
		tmp = r * sin(b);
	elseif (b <= 15.6)
		tmp = (r * b) / cos((b + a));
	else
		tmp = sin(b) / (1.0 / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -106000000.0], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 15.6], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.06e8

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

   3. Simplified 52.5%

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

      1. expm1-log1p-u 33.1%

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

   5. Applied egg-rr 33.1%

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

   6. Taylor expanded in b around 0 11.5%

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

   7. Taylor expanded in a around 0 11.3%

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

   if -1.06e8 < b < 15.5999999999999996

   1. Initial program 97.7%

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

      1. associate-*r/ 97.7%

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

      2. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in b around 0 96.1%

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

   if 15.5999999999999996 < b

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-/r/ 55.2%

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

      3. +-commutative 55.2%

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

   3. Simplified 55.2%

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

      1. clear-num 55.1%

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

      2. associate-/r/ 55.1%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in a around 0 55.1%

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

   7. Taylor expanded in b around 0 11.8%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -106000000:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 15.6:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \]

Alternative 17: 54.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.55 \lor \neg \left(b \leq 30\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.55) (not (<= b 30.0))) (* r (sin b)) (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.55) || !(b <= 30.0)) {
		tmp = r * sin(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.55d0)) .or. (.not. (b <= 30.0d0))) then
        tmp = r * sin(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.55) || !(b <= 30.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.55) or not (b <= 30.0):
		tmp = r * math.sin(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.55) || !(b <= 30.0))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.55) || ~((b <= 30.0)))
		tmp = r * sin(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.55], N[Not[LessEqual[b, 30.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.55000000000000004 or 30 < b

   1. Initial program 53.9%

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

      1. +-commutative 53.9%

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

   3. Simplified 53.9%

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

      1. expm1-log1p-u 35.8%

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

   5. Applied egg-rr 35.8%

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

   6. Taylor expanded in b around 0 10.0%

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

   7. Taylor expanded in a around 0 11.4%

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

   if -0.55000000000000004 < b < 30

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in b around 0 97.6%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.55 \lor \neg \left(b \leq 30\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 18: 54.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.16:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 24:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.16)
   (* r (sin b))
   (if (<= b 24.0) (* r (/ b (cos a))) (/ (sin b) (/ 1.0 r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.16) {
		tmp = r * sin(b);
	} else if (b <= 24.0) {
		tmp = r * (b / cos(a));
	} else {
		tmp = sin(b) / (1.0 / 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 <= (-1.16d0)) then
        tmp = r * sin(b)
    else if (b <= 24.0d0) then
        tmp = r * (b / cos(a))
    else
        tmp = sin(b) / (1.0d0 / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.16) {
		tmp = r * Math.sin(b);
	} else if (b <= 24.0) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = Math.sin(b) / (1.0 / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.16:
		tmp = r * math.sin(b)
	elif b <= 24.0:
		tmp = r * (b / math.cos(a))
	else:
		tmp = math.sin(b) / (1.0 / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.16)
		tmp = Float64(r * sin(b));
	elseif (b <= 24.0)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = Float64(sin(b) / Float64(1.0 / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1.16)
		tmp = r * sin(b);
	elseif (b <= 24.0)
		tmp = r * (b / cos(a));
	else
		tmp = sin(b) / (1.0 / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.16], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 24.0], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15999999999999992

   1. Initial program 52.6%

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

      1. +-commutative 52.6%

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

   3. Simplified 52.6%

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

      1. expm1-log1p-u 33.5%

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

   5. Applied egg-rr 33.5%

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

   6. Taylor expanded in b around 0 11.1%

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

   7. Taylor expanded in a around 0 11.0%

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

   if -1.15999999999999992 < b < 24

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in b around 0 97.6%

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

   if 24 < b

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-/r/ 55.2%

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

      3. +-commutative 55.2%

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

   3. Simplified 55.2%

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

      1. clear-num 55.1%

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

      2. associate-/r/ 55.1%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in a around 0 55.1%

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

   7. Taylor expanded in b around 0 11.8%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.16:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 24:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \]

Alternative 19: 54.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 30:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -1.02)
   (* r (sin b))
   (if (<= b 30.0) (/ (* r b) (cos a)) (/ (sin b) (/ 1.0 r)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.02) {
		tmp = r * sin(b);
	} else if (b <= 30.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = sin(b) / (1.0 / 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 <= (-1.02d0)) then
        tmp = r * sin(b)
    else if (b <= 30.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = sin(b) / (1.0d0 / r)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -1.02) {
		tmp = r * Math.sin(b);
	} else if (b <= 30.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = Math.sin(b) / (1.0 / r);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -1.02:
		tmp = r * math.sin(b)
	elif b <= 30.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = math.sin(b) / (1.0 / r)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -1.02)
		tmp = Float64(r * sin(b));
	elseif (b <= 30.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(sin(b) / Float64(1.0 / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -1.02)
		tmp = r * sin(b);
	elseif (b <= 30.0)
		tmp = (r * b) / cos(a);
	else
		tmp = sin(b) / (1.0 / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -1.02], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 30.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.02

   1. Initial program 52.6%

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

      1. +-commutative 52.6%

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

   3. Simplified 52.6%

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

      1. expm1-log1p-u 33.5%

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

   5. Applied egg-rr 33.5%

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

   6. Taylor expanded in b around 0 11.1%

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

   7. Taylor expanded in a around 0 11.0%

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

   if -1.02 < b < 30

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in b around 0 97.6%

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

   if 30 < b

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-/r/ 55.2%

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

      3. +-commutative 55.2%

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

   3. Simplified 55.2%

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

      1. clear-num 55.1%

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

      2. associate-/r/ 55.1%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in a around 0 55.1%

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

   7. Taylor expanded in b around 0 11.8%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 30:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \end{array} \]

Alternative 20: 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 73.9%

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

   1. +-commutative 73.9%

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

3. Simplified 73.9%

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

   1. expm1-log1p-u 63.9%

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

5. Applied egg-rr 63.9%

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

6. Taylor expanded in b around 0 49.4%

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

7. Taylor expanded in a around 0 36.2%

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

8. Final simplification 36.2%

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

Alternative 21: 34.4% 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 73.9%

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

   1. +-commutative 73.9%

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

3. Simplified 73.9%

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

4. Taylor expanded in b around 0 45.8%

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

5. Taylor expanded in a around 0 31.9%

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

6. Final simplification 31.9%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023194 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))