rsin B (should all be same)

Percentage Accurate: 77.0% → 99.5%

Time: 12.4s

Alternatives: 9

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: 77.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin b\\ \frac{r \cdot \sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot t_0\right) + \mathsf{fma}\left(t_0, \sin a, \sin b \cdot \sin a\right)} \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (sin b))))
   (/
    (* r (sin b))
    (+
     (fma 1.0 (* (cos a) (cos b)) (* (sin a) t_0))
     (fma t_0 (sin a) (* (sin b) (sin a)))))))

C

double code(double r, double a, double b) {
	double t_0 = -sin(b);
	return (r * sin(b)) / (fma(1.0, (cos(a) * cos(b)), (sin(a) * t_0)) + fma(t_0, sin(a), (sin(b) * sin(a))));
}

Julia

function code(r, a, b)
	t_0 = Float64(-sin(b))
	return Float64(Float64(r * sin(b)) / Float64(fma(1.0, Float64(cos(a) * cos(b)), Float64(sin(a) * t_0)) + fma(t_0, sin(a), Float64(sin(b) * sin(a)))))
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = (-N[Sin[b], $MachinePrecision])}, 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[a], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Sin[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

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

   1. associate-*r/ 75.2%

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

   2. +-commutative 75.2%

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

3. Simplified 75.2%

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

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

   2. *-rgt-identity 99.5%

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

   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 99.5%

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

   10. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. +-commutative 75.2%

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

3. Simplified 75.2%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 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 75.2%

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

   1. associate-*r/ 75.2%

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

   2. +-commutative 75.2%

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

3. Simplified 75.2%

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

   1. cos-sum 99.5%

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

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 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-6} \lor \neg \left(b \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.32e-6) (not (<= b 1.75e-6)))
   (* r (tan b))
   (* r (/ (sin b) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.32e-6) || !(b <= 1.75e-6)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (sin(b) / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.32d-6)) .or. (.not. (b <= 1.75d-6))) then
        tmp = r * tan(b)
    else
        tmp = r * (sin(b) / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.32e-6) || !(b <= 1.75e-6)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.32e-6) or not (b <= 1.75e-6):
		tmp = r * math.tan(b)
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.32e-6) || !(b <= 1.75e-6))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.32e-6) || ~((b <= 1.75e-6)))
		tmp = r * tan(b);
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.32e-6], N[Not[LessEqual[b, 1.75e-6]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3200000000000001e-6 or 1.74999999999999997e-6 < b

   1. Initial program 49.3%

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

      1. associate-*r/ 49.3%

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

      2. +-commutative 49.3%

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

   3. Simplified 49.3%

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

      1. add-exp-log 28.0%

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

   5. Applied egg-rr 28.0%

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

   6. Taylor expanded in a around 0 28.0%

      \[\leadsto \frac{e^{\log \left(r \cdot \sin b\right)}}{\color{blue}{\cos b}} \]
   7. Step-by-step derivation

      1. add-exp-log 16.8%

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

      2. add-exp-log 24.1%

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

      3. *-un-lft-identity 24.1%

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

      4. times-frac 24.1%

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

      5. /-rgt-identity 24.1%

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

   8. Applied egg-rr 24.1%

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

      1. add-exp-log 49.8%

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

      2. *-commutative 49.8%

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

      3. quot-tan 49.8%

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

   10. Applied egg-rr 49.8%

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

   if -1.3200000000000001e-6 < b < 1.74999999999999997e-6

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in b around 0 99.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-6} \lor \neg \left(b \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 5: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -8e-7)
   (* r (tan b))
   (if (<= b 1.75e-6) (* r (/ (sin b) (cos a))) (* (sin b) (/ r (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -8e-7) {
		tmp = r * tan(b);
	} else if (b <= 1.75e-6) {
		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 (b <= (-8d-7)) then
        tmp = r * tan(b)
    else if (b <= 1.75d-6) 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 (b <= -8e-7) {
		tmp = r * Math.tan(b);
	} else if (b <= 1.75e-6) {
		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 b <= -8e-7:
		tmp = r * math.tan(b)
	elif b <= 1.75e-6:
		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 (b <= -8e-7)
		tmp = Float64(r * tan(b));
	elseif (b <= 1.75e-6)
		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 (b <= -8e-7)
		tmp = r * tan(b);
	elseif (b <= 1.75e-6)
		tmp = r * (sin(b) / cos(a));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -8e-7], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.75e-6], 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 3 regimes

2. if b < -7.9999999999999996e-7

   1. Initial program 55.8%

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

      1. associate-*r/ 55.9%

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

      2. +-commutative 55.9%

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

   3. Simplified 55.9%

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

      1. add-exp-log 35.4%

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

   5. Applied egg-rr 35.4%

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

   6. Taylor expanded in a around 0 35.0%

      \[\leadsto \frac{e^{\log \left(r \cdot \sin b\right)}}{\color{blue}{\cos b}} \]
   7. Step-by-step derivation

      1. add-exp-log 18.1%

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

      2. add-exp-log 24.1%

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

      3. *-un-lft-identity 24.1%

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

      4. times-frac 24.1%

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

      5. /-rgt-identity 24.1%

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

   8. Applied egg-rr 24.1%

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

      1. add-exp-log 55.3%

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

      2. *-commutative 55.3%

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

      3. quot-tan 55.4%

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

   10. Applied egg-rr 55.4%

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

   if -7.9999999999999996e-7 < b < 1.74999999999999997e-6

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in b around 0 99.3%

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

   if 1.74999999999999997e-6 < b

   1. Initial program 43.4%

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

      1. associate-*r/ 43.4%

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

      2. +-commutative 43.4%

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

   3. Simplified 43.4%

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

      1. cos-sum 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. prod-diff 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. *-rgt-identity 99.0%

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

      3. distribute-lft-neg-in 99.0%

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

      4. *-commutative 99.0%

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

      5. fma-udef 99.0%

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

      6. *-rgt-identity 99.0%

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

      7. distribute-lft-neg-in 99.0%

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

      8. *-rgt-identity 99.0%

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

      9. fma-udef 99.0%

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

      10. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in r around 0 98.9%

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

      1. associate-+r+ 99.0%

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

      2. *-commutative 99.0%

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

      3. *-commutative 99.0%

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

      4. distribute-lft1-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. neg-mul-1 99.0%

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

      7. distribute-lft-neg-in 99.0%

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

      8. +-commutative 99.0%

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

      9. cancel-sign-sub-inv 99.0%

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

      10. *-commutative 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in a around 0 44.8%

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

       1. associate-*r/ 44.9%

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

   13. Simplified 44.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 6: 77.0% 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 75.2%

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

2. Final simplification 75.2%

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

Alternative 7: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-6} \lor \neg \left(b \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.32e-6) (not (<= b 1.75e-6)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.32e-6) || !(b <= 1.75e-6)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.32d-6)) .or. (.not. (b <= 1.75d-6))) then
        tmp = r * tan(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.32e-6) || !(b <= 1.75e-6)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.32e-6) or not (b <= 1.75e-6):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.32e-6) || !(b <= 1.75e-6))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.32e-6) || ~((b <= 1.75e-6)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.32e-6], N[Not[LessEqual[b, 1.75e-6]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3200000000000001e-6 or 1.74999999999999997e-6 < b

   1. Initial program 49.3%

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

      1. associate-*r/ 49.3%

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

      2. +-commutative 49.3%

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

   3. Simplified 49.3%

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

      1. add-exp-log 28.0%

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

   5. Applied egg-rr 28.0%

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

   6. Taylor expanded in a around 0 28.0%

      \[\leadsto \frac{e^{\log \left(r \cdot \sin b\right)}}{\color{blue}{\cos b}} \]
   7. Step-by-step derivation

      1. add-exp-log 16.8%

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

      2. add-exp-log 24.1%

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

      3. *-un-lft-identity 24.1%

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

      4. times-frac 24.1%

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

      5. /-rgt-identity 24.1%

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

   8. Applied egg-rr 24.1%

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

      1. add-exp-log 49.8%

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

      2. *-commutative 49.8%

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

      3. quot-tan 49.8%

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

   10. Applied egg-rr 49.8%

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

   if -1.3200000000000001e-6 < b < 1.74999999999999997e-6

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in b around 0 99.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-6} \lor \neg \left(b \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 8: 60.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. associate-*r/ 75.2%

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

   2. +-commutative 75.2%

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

3. Simplified 75.2%

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

   1. add-exp-log 42.7%

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

5. Applied egg-rr 42.7%

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

6. Taylor expanded in a around 0 32.6%

   \[\leadsto \frac{e^{\log \left(r \cdot \sin b\right)}}{\color{blue}{\cos b}} \]
7. Step-by-step derivation

   1. add-exp-log 27.2%

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

   2. add-exp-log 30.7%

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

   3. *-un-lft-identity 30.7%

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

   4. times-frac 30.7%

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

   5. /-rgt-identity 30.7%

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

8. Applied egg-rr 30.7%

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

   1. add-exp-log 58.0%

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

   2. *-commutative 58.0%

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

   3. quot-tan 58.1%

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

10. Applied egg-rr 58.1%

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

11. Final simplification 58.1%

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

Alternative 9: 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 75.2%

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

   1. +-commutative 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in b around 0 54.4%

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

5. Taylor expanded in a around 0 36.8%

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

6. Final simplification 36.8%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023203 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))