rsin B (should all be same)

Percentage Accurate: 77.3% → 99.5%

Time: 16.8s

Alternatives: 13

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. +-commutative 76.4%

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

3. Simplified 76.4%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in r around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. *-commutative 99.5%

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

   4. sub-neg 99.5%

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

   5. *-commutative 99.5%

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

8. Simplified 99.5%

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

   1. sub-neg 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

10. Applied egg-rr 99.5%

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

    1. +-commutative 99.5%

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

    2. fma-def 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. +-commutative 76.4%

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

3. Simplified 76.4%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

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 76.4%

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

   1. +-commutative 76.4%

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

3. Simplified 76.4%

   \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
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: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(N[(N[Sin[b], $MachinePrecision] * r), $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 76.4%

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

   1. associate-*r/ 76.4%

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

   2. +-commutative 76.4%

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

3. Simplified 76.4%

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

   1. cos-sum 99.4%

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

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%

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

Alternative 5: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0146 \lor \neg \left(a \leq 8.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.0146) (not (<= a 8.5e-5)))
   (/ (sin b) (/ (cos a) r))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.0146) || !(a <= 8.5e-5)) {
		tmp = sin(b) / (cos(a) / r);
	} 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 <= (-0.0146d0)) .or. (.not. (a <= 8.5d-5))) then
        tmp = sin(b) / (cos(a) / r)
    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 <= -0.0146) || !(a <= 8.5e-5)) {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.0146) or not (a <= 8.5e-5):
		tmp = math.sin(b) / (math.cos(a) / r)
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.0146) || !(a <= 8.5e-5))
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	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 <= -0.0146) || ~((a <= 8.5e-5)))
		tmp = sin(b) / (cos(a) / r);
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.0146], N[Not[LessEqual[a, 8.5e-5]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $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 < -0.0146000000000000001 or 8.500000000000001e-5 < a

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      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. Taylor expanded in b around 0 55.5%

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

   if -0.0146000000000000001 < a < 8.500000000000001e-5

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in a around 0 97.9%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0146 \lor \neg \left(a \leq 8.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 6: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.012:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.012)
   (* r (tan b))
   (if (<= b 9.2e-12) (/ (* b r) (cos (+ b a))) (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.012) {
		tmp = r * tan(b);
	} else if (b <= 9.2e-12) {
		tmp = (b * r) / cos((b + 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 (b <= (-0.012d0)) then
        tmp = r * tan(b)
    else if (b <= 9.2d-12) then
        tmp = (b * r) / cos((b + 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 (b <= -0.012) {
		tmp = r * Math.tan(b);
	} else if (b <= 9.2e-12) {
		tmp = (b * r) / Math.cos((b + a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -0.012:
		tmp = r * math.tan(b)
	elif b <= 9.2e-12:
		tmp = (b * r) / math.cos((b + a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.012)
		tmp = Float64(r * tan(b));
	elseif (b <= 9.2e-12)
		tmp = Float64(Float64(b * r) / cos(Float64(b + 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 (b <= -0.012)
		tmp = r * tan(b);
	elseif (b <= 9.2e-12)
		tmp = (b * r) / cos((b + a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.012], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2e-12], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.012

   1. Initial program 56.4%

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

      1. +-commutative 56.4%

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

   3. Simplified 56.4%

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

      1. cos-sum 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. fma-def 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. distribute-lft-neg-out 99.3%

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

      2. fma-neg 99.2%

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

      3. cos-sum 56.4%

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

      4. add-log-exp 56.0%

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

      5. add-cube-cbrt 54.7%

         \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left(\left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      6. log-prod 54.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      7. pow2 54.8%

         \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

   7. Applied egg-rr 54.8%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]
   8. Step-by-step derivation

      1. log-pow 54.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

      2. distribute-lft1-in 54.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      3. metadata-eval 54.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

   9. Simplified 54.8%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

   10. Taylor expanded in a around 0 55.6%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\log \left({\left(e^{\cos b}\right)}^{0.3333333333333333}\right)}} \]
   11. Step-by-step derivation

       1. log-pow 56.2%

          \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos b}\right)\right)}} \]

       2. rem-log-exp 56.6%

          \[\leadsto r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\cos b}\right)} \]

   12. Simplified 56.6%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \cos b\right)}} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 39.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)\right)} \]

       2. expm1-udef 15.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)} - 1} \]

       3. associate-*r* 15.7%

          \[\leadsto e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{\color{blue}{\left(3 \cdot 0.3333333333333333\right) \cdot \cos b}}\right)} - 1 \]

       4. metadata-eval 15.7%

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

       5. *-un-lft-identity 15.7%

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

       6. quot-tan 15.7%

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

   14. Applied egg-rr 15.7%

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

       1. expm1-def 39.8%

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

       2. expm1-log1p 56.7%

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

   16. Simplified 56.7%

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

   if -0.012 < b < 9.19999999999999957e-12

   1. Initial program 98.9%

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

      1. associate-*r/ 98.9%

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

      2. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in b around 0 98.9%

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

   if 9.19999999999999957e-12 < b

   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. Taylor expanded in a around 0 51.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.012:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 7: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.018:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.018)
   (/ (sin b) (/ (cos a) r))
   (if (<= a 2.1e-5) (* r (/ (sin b) (cos b))) (/ (* (sin b) r) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.018) {
		tmp = sin(b) / (cos(a) / r);
	} else if (a <= 2.1e-5) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = (sin(b) * r) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.018d0)) then
        tmp = sin(b) / (cos(a) / r)
    else if (a <= 2.1d-5) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = (sin(b) * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.018) {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	} else if (a <= 2.1e-5) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = (Math.sin(b) * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.018:
		tmp = math.sin(b) / (math.cos(a) / r)
	elif a <= 2.1e-5:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = (math.sin(b) * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.018)
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	elseif (a <= 2.1e-5)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(Float64(sin(b) * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.018)
		tmp = sin(b) / (cos(a) / r);
	elseif (a <= 2.1e-5)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = (sin(b) * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.018], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.1e-5], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0179999999999999986

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

      2. associate-/r/ 48.4%

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

      3. +-commutative 48.4%

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

   3. Simplified 48.4%

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

   4. Taylor expanded in b around 0 48.9%

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

   if -0.0179999999999999986 < a < 2.09999999999999988e-5

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in a around 0 97.9%

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

   if 2.09999999999999988e-5 < a

   1. Initial program 61.5%

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

      1. associate-*r/ 61.6%

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

      2. +-commutative 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in b around 0 61.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.018:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \end{array} \]

Alternative 8: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

2. Final simplification 76.4%

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

Alternative 9: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.012 \lor \neg \left(b \leq 9.2 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.012) (not (<= b 9.2e-12)))
   (* r (tan b))
   (/ (* b r) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.012) || !(b <= 9.2e-12)) {
		tmp = r * tan(b);
	} else {
		tmp = (b * r) / cos((b + a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.012d0)) .or. (.not. (b <= 9.2d-12))) then
        tmp = r * tan(b)
    else
        tmp = (b * r) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.012) || !(b <= 9.2e-12)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (b * r) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.012) or not (b <= 9.2e-12):
		tmp = r * math.tan(b)
	else:
		tmp = (b * r) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.012) || !(b <= 9.2e-12))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.012) || ~((b <= 9.2e-12)))
		tmp = r * tan(b);
	else
		tmp = (b * r) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.012], N[Not[LessEqual[b, 9.2e-12]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.012 or 9.19999999999999957e-12 < b

   1. Initial program 54.3%

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

      1. +-commutative 54.3%

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

   3. Simplified 54.3%

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

      1. cos-sum 99.1%

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

      2. cancel-sign-sub-inv 99.1%

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

      3. fma-def 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. distribute-lft-neg-out 99.2%

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

      2. fma-neg 99.1%

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

      3. cos-sum 54.3%

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

      4. add-log-exp 54.0%

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

      5. add-cube-cbrt 52.9%

         \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left(\left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      6. log-prod 52.9%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      7. pow2 52.9%

         \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

   7. Applied egg-rr 52.9%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]
   8. Step-by-step derivation

      1. log-pow 52.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

      2. distribute-lft1-in 52.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      3. metadata-eval 52.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

   9. Simplified 52.8%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

   10. Taylor expanded in a around 0 53.0%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\log \left({\left(e^{\cos b}\right)}^{0.3333333333333333}\right)}} \]
   11. Step-by-step derivation

       1. log-pow 53.6%

          \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos b}\right)\right)}} \]

       2. rem-log-exp 53.9%

          \[\leadsto r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\cos b}\right)} \]

   12. Simplified 53.9%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \cos b\right)}} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 37.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)\right)} \]

       2. expm1-udef 13.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)} - 1} \]

       3. associate-*r* 13.3%

          \[\leadsto e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{\color{blue}{\left(3 \cdot 0.3333333333333333\right) \cdot \cos b}}\right)} - 1 \]

       4. metadata-eval 13.3%

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

       5. *-un-lft-identity 13.3%

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

       6. quot-tan 13.3%

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

   14. Applied egg-rr 13.3%

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

       1. expm1-def 37.8%

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

       2. expm1-log1p 54.0%

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

   16. Simplified 54.0%

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

   if -0.012 < b < 9.19999999999999957e-12

   1. Initial program 98.9%

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

      1. associate-*r/ 98.9%

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

      2. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in b around 0 98.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.012 \lor \neg \left(b \leq 9.2 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 10: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.012 \lor \neg \left(b \leq 9.2 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.012) (not (<= b 9.2e-12)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.012) || !(b <= 9.2e-12)) {
		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 <= (-0.012d0)) .or. (.not. (b <= 9.2d-12))) 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 <= -0.012) || !(b <= 9.2e-12)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.012) or not (b <= 9.2e-12):
		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 <= -0.012) || !(b <= 9.2e-12))
		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 <= -0.012) || ~((b <= 9.2e-12)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.012], N[Not[LessEqual[b, 9.2e-12]], $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 < -0.012 or 9.19999999999999957e-12 < b

   1. Initial program 54.3%

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

      1. +-commutative 54.3%

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

   3. Simplified 54.3%

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

      1. cos-sum 99.1%

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

      2. cancel-sign-sub-inv 99.1%

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

      3. fma-def 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. distribute-lft-neg-out 99.2%

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

      2. fma-neg 99.1%

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

      3. cos-sum 54.3%

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

      4. add-log-exp 54.0%

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

      5. add-cube-cbrt 52.9%

         \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left(\left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      6. log-prod 52.9%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      7. pow2 52.9%

         \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

   7. Applied egg-rr 52.9%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]
   8. Step-by-step derivation

      1. log-pow 52.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

      2. distribute-lft1-in 52.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      3. metadata-eval 52.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

   9. Simplified 52.8%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

   10. Taylor expanded in a around 0 53.0%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\log \left({\left(e^{\cos b}\right)}^{0.3333333333333333}\right)}} \]
   11. Step-by-step derivation

       1. log-pow 53.6%

          \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos b}\right)\right)}} \]

       2. rem-log-exp 53.9%

          \[\leadsto r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\cos b}\right)} \]

   12. Simplified 53.9%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \cos b\right)}} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 37.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)\right)} \]

       2. expm1-udef 13.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)} - 1} \]

       3. associate-*r* 13.3%

          \[\leadsto e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{\color{blue}{\left(3 \cdot 0.3333333333333333\right) \cdot \cos b}}\right)} - 1 \]

       4. metadata-eval 13.3%

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

       5. *-un-lft-identity 13.3%

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

       6. quot-tan 13.3%

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

   14. Applied egg-rr 13.3%

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

       1. expm1-def 37.8%

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

       2. expm1-log1p 54.0%

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

   16. Simplified 54.0%

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

   if -0.012 < b < 9.19999999999999957e-12

   1. Initial program 98.9%

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

      1. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in b around 0 98.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.012 \lor \neg \left(b \leq 9.2 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11: 76.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.012 \lor \neg \left(b \leq 9.2 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.012) (not (<= b 9.2e-12)))
   (* r (tan b))
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.012) || !(b <= 9.2e-12)) {
		tmp = r * tan(b);
	} else {
		tmp = (b * r) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.012d0)) .or. (.not. (b <= 9.2d-12))) then
        tmp = r * tan(b)
    else
        tmp = (b * r) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.012) || !(b <= 9.2e-12)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.012) or not (b <= 9.2e-12):
		tmp = r * math.tan(b)
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.012) || !(b <= 9.2e-12))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.012) || ~((b <= 9.2e-12)))
		tmp = r * tan(b);
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.012], N[Not[LessEqual[b, 9.2e-12]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.012 or 9.19999999999999957e-12 < b

   1. Initial program 54.3%

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

      1. +-commutative 54.3%

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

   3. Simplified 54.3%

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

      1. cos-sum 99.1%

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

      2. cancel-sign-sub-inv 99.1%

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

      3. fma-def 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. distribute-lft-neg-out 99.2%

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

      2. fma-neg 99.1%

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

      3. cos-sum 54.3%

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

      4. add-log-exp 54.0%

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

      5. add-cube-cbrt 52.9%

         \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left(\left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      6. log-prod 52.9%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      7. pow2 52.9%

         \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

   7. Applied egg-rr 52.9%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]
   8. Step-by-step derivation

      1. log-pow 52.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

      2. distribute-lft1-in 52.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

      3. metadata-eval 52.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

   9. Simplified 52.8%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

   10. Taylor expanded in a around 0 53.0%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\log \left({\left(e^{\cos b}\right)}^{0.3333333333333333}\right)}} \]
   11. Step-by-step derivation

       1. log-pow 53.6%

          \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos b}\right)\right)}} \]

       2. rem-log-exp 53.9%

          \[\leadsto r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\cos b}\right)} \]

   12. Simplified 53.9%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \cos b\right)}} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 37.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)\right)} \]

       2. expm1-udef 13.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)} - 1} \]

       3. associate-*r* 13.3%

          \[\leadsto e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{\color{blue}{\left(3 \cdot 0.3333333333333333\right) \cdot \cos b}}\right)} - 1 \]

       4. metadata-eval 13.3%

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

       5. *-un-lft-identity 13.3%

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

       6. quot-tan 13.3%

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

   14. Applied egg-rr 13.3%

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

       1. expm1-def 37.8%

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

       2. expm1-log1p 54.0%

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

   16. Simplified 54.0%

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

   if -0.012 < b < 9.19999999999999957e-12

   1. Initial program 98.9%

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

      1. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in b around 0 98.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.012 \lor \neg \left(b \leq 9.2 \cdot 10^{-12}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 12: 60.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. +-commutative 76.4%

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

3. Simplified 76.4%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

   1. distribute-lft-neg-out 99.5%

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

   2. fma-neg 99.4%

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

   3. cos-sum 76.4%

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

   4. add-log-exp 76.1%

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

   5. add-cube-cbrt 74.4%

      \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left(\left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

   6. log-prod 74.4%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left(\sqrt[3]{e^{\cos \left(b + a\right)}} \cdot \sqrt[3]{e^{\cos \left(b + a\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

   7. pow2 74.4%

      \[\leadsto r \cdot \frac{\sin b}{\log \color{blue}{\left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

7. Applied egg-rr 74.4%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left({\left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]
8. Step-by-step derivation

   1. log-pow 74.4%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

   2. distribute-lft1-in 74.4%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

   3. metadata-eval 74.4%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)} \]

9. Simplified 74.4%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{3 \cdot \log \left(\sqrt[3]{e^{\cos \left(b + a\right)}}\right)}} \]

10. Taylor expanded in a around 0 58.7%

    \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\log \left({\left(e^{\cos b}\right)}^{0.3333333333333333}\right)}} \]
11. Step-by-step derivation

    1. log-pow 59.0%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos b}\right)\right)}} \]

    2. rem-log-exp 59.2%

       \[\leadsto r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\cos b}\right)} \]

12. Simplified 59.2%

    \[\leadsto r \cdot \frac{\sin b}{3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \cos b\right)}} \]
13. Step-by-step derivation

    1. expm1-log1p-u 46.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)\right)} \]

    2. expm1-udef 21.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{3 \cdot \left(0.3333333333333333 \cdot \cos b\right)}\right)} - 1} \]

    3. associate-*r* 21.7%

       \[\leadsto e^{\mathsf{log1p}\left(r \cdot \frac{\sin b}{\color{blue}{\left(3 \cdot 0.3333333333333333\right) \cdot \cos b}}\right)} - 1 \]

    4. metadata-eval 21.7%

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

    5. *-un-lft-identity 21.7%

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

    6. quot-tan 21.7%

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

14. Applied egg-rr 21.7%

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

    1. expm1-def 46.7%

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

    2. expm1-log1p 59.2%

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

16. Simplified 59.2%

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

17. Final simplification 59.2%

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

Alternative 13: 35.3% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. +-commutative 76.4%

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

3. Simplified 76.4%

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

4. Taylor expanded in b around 0 51.9%

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

5. Taylor expanded in a around 0 34.9%

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

6. Final simplification 34.9%

   \[\leadsto b \cdot r \]

Reproduce

herbie shell --seed 2023187 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))