rsin A (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 23.0s

Alternatives: 12

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $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 76.3%

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

   1. +-commutative 76.3%

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

3. Simplified 76.3%

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

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

5. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. fma-def 99.5%

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

   4. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. *-commutative 76.3%

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

   2. associate-/l* 76.2%

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

   3. +-commutative 76.2%

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

3. Simplified 76.2%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

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 76.3%

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

   1. +-commutative 76.3%

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

3. Simplified 76.3%

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

   1. cos-sum 99.4%

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

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: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00013 \lor \neg \left(a \leq 5.6 \cdot 10^{+19}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.00013) (not (<= a 5.6e+19)))
   (* (sin b) (/ r (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00013) || !(a <= 5.6e+19)) {
		tmp = sin(b) * (r / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-0.00013d0)) .or. (.not. (a <= 5.6d+19))) then
        tmp = sin(b) * (r / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.00013) || !(a <= 5.6e+19)) {
		tmp = Math.sin(b) * (r / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.00013) or not (a <= 5.6e+19):
		tmp = math.sin(b) * (r / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.00013) || !(a <= 5.6e+19))
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.00013) || ~((a <= 5.6e+19)))
		tmp = sin(b) * (r / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.00013], N[Not[LessEqual[a, 5.6e+19]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.29999999999999989e-4 or 5.6e19 < a

   1. Initial program 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-/l* 53.9%

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

      3. +-commutative 53.9%

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

   3. Simplified 53.9%

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

      1. add-cube-cbrt 53.4%

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

      2. pow3 53.4%

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

   5. Applied egg-rr 53.4%

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

   6. Taylor expanded in b around 0 23.2%

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

   7. Taylor expanded in b around inf 53.4%

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

      1. associate-*r/ 53.4%

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

      2. *-commutative 53.4%

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

   9. Simplified 53.4%

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

   if -1.29999999999999989e-4 < a < 5.6e19

   1. Initial program 96.0%

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

      1. *-commutative 96.0%

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

      2. associate-/l* 95.9%

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

      3. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in a around 0 96.0%

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

      1. associate-/l* 95.9%

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

      2. associate-/r/ 96.1%

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

   6. Simplified 96.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00013 \lor \neg \left(a \leq 5.6 \cdot 10^{+19}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

Alternative 5: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+19}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -2.05e-5)
   (/ (sin b) (/ (cos a) r))
   (if (<= a 5.6e+19) (* r (/ (sin b) (cos b))) (* (sin b) (/ r (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -2.05e-5) {
		tmp = sin(b) / (cos(a) / r);
	} else if (a <= 5.6e+19) {
		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 <= (-2.05d-5)) then
        tmp = sin(b) / (cos(a) / r)
    else if (a <= 5.6d+19) 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 <= -2.05e-5) {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	} else if (a <= 5.6e+19) {
		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 <= -2.05e-5:
		tmp = math.sin(b) / (math.cos(a) / r)
	elif a <= 5.6e+19:
		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 <= -2.05e-5)
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	elseif (a <= 5.6e+19)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -2.05e-5)
		tmp = sin(b) / (cos(a) / r);
	elseif (a <= 5.6e+19)
		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, -2.05e-5], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.6e+19], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.05000000000000002e-5

   1. Initial program 56.4%

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

      1. *-commutative 56.4%

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

      2. associate-/l* 56.4%

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

      3. +-commutative 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in b around 0 55.8%

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

   if -2.05000000000000002e-5 < a < 5.6e19

   1. Initial program 96.0%

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

      1. *-commutative 96.0%

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

      2. associate-/l* 95.9%

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

      3. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in a around 0 96.0%

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

      1. associate-/l* 95.9%

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

      2. associate-/r/ 96.1%

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

   6. Simplified 96.1%

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

   if 5.6e19 < a

   1. Initial program 51.4%

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

      1. *-commutative 51.4%

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

      2. associate-/l* 51.4%

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

      3. +-commutative 51.4%

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

   3. Simplified 51.4%

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

      1. add-cube-cbrt 50.8%

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

      2. pow3 50.7%

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

   5. Applied egg-rr 50.7%

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

   6. Taylor expanded in b around 0 24.0%

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

   7. Taylor expanded in b around inf 51.1%

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

      1. associate-*r/ 51.1%

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

      2. *-commutative 51.1%

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

   9. Simplified 51.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+19}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 6: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -11:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.55:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\left|t_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -11.0) t_0 (if (<= b 1.55) (* r (/ b (cos a))) (fabs t_0)))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -11.0) {
		tmp = t_0;
	} else if (b <= 1.55) {
		tmp = r * (b / cos(a));
	} else {
		tmp = fabs(t_0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (b <= (-11.0d0)) then
        tmp = t_0
    else if (b <= 1.55d0) then
        tmp = r * (b / cos(a))
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -11.0) {
		tmp = t_0;
	} else if (b <= 1.55) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -11.0:
		tmp = t_0
	elif b <= 1.55:
		tmp = r * (b / math.cos(a))
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -11.0)
		tmp = t_0;
	elseif (b <= 1.55)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -11.0)
		tmp = t_0;
	elseif (b <= 1.55)
		tmp = r * (b / cos(a));
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -11.0], t$95$0, If[LessEqual[b, 1.55], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -11

   1. Initial program 56.7%

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

      1. *-commutative 56.7%

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

      2. associate-/l* 56.7%

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

      3. +-commutative 56.7%

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

   3. Simplified 56.7%

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

      1. add-cube-cbrt 55.8%

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

      2. pow3 55.9%

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

   5. Applied egg-rr 55.9%

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

   6. Taylor expanded in b around 0 7.1%

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

   7. Taylor expanded in a around 0 13.0%

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

   if -11 < b < 1.55000000000000004

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/l* 99.0%

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

      3. +-commutative 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in b around 0 97.7%

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

      1. associate-/l* 97.5%

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

   6. Simplified 97.5%

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

      1. associate-/r/ 97.7%

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

   8. Applied egg-rr 97.7%

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

   if 1.55000000000000004 < b

   1. Initial program 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/l* 57.0%

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

      3. +-commutative 57.0%

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

   3. Simplified 57.0%

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

      1. associate-/l* 56.8%

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

      2. *-commutative 56.8%

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

      3. add-sqr-sqrt 29.6%

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

      4. sqrt-unprod 19.9%

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

      5. pow2 19.9%

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

      6. associate-/l* 19.9%

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

      7. associate-/r/ 19.9%

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

      8. *-commutative 19.9%

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

   5. Applied egg-rr 19.9%

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

      1. unpow2 19.9%

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

      2. rem-sqrt-square 32.4%

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

      3. +-commutative 32.4%

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

   7. Simplified 32.4%

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

   8. Taylor expanded in b around 0 11.7%

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

   9. Taylor expanded in a around 0 12.6%

      \[\leadsto \left|\color{blue}{\sin b \cdot r}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -11:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 1.55:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\left|r \cdot \sin b\right|\\ \end{array} \]

Alternative 7: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. *-commutative 76.3%

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

   2. associate-/l* 76.2%

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

   3. +-commutative 76.2%

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

3. Simplified 76.2%

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

   1. clear-num 75.8%

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

   2. associate-/r/ 76.2%

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

   3. clear-num 76.3%

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

5. Applied egg-rr 76.3%

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

6. Final simplification 76.3%

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

Alternative 8: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. *-commutative 76.3%

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

   2. associate-/l* 76.2%

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

   3. +-commutative 76.2%

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

3. Simplified 76.2%

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

   1. associate-/r/ 76.4%

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

5. Applied egg-rr 76.4%

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

6. Final simplification 76.4%

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

Alternative 9: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. *-commutative 76.3%

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

   2. associate-/l* 76.2%

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

   3. +-commutative 76.2%

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

3. Simplified 76.2%

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

   1. add-cube-cbrt 75.2%

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

   2. pow3 75.2%

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

5. Applied egg-rr 75.2%

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

6. Taylor expanded in b around 0 22.4%

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

7. Taylor expanded in b around inf 51.4%

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

   1. associate-*r/ 51.4%

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

   2. *-commutative 51.4%

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

9. Simplified 51.4%

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

10. Final simplification 51.4%

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

Alternative 10: 54.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \lor \neg \left(b \leq 1.5\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.8) (not (<= b 1.5))) (* r (sin b)) (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8) || !(b <= 1.5)) {
		tmp = r * sin(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-4.8d0)) .or. (.not. (b <= 1.5d0))) then
        tmp = r * sin(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8) || !(b <= 1.5)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.8) or not (b <= 1.5):
		tmp = r * math.sin(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.8) || !(b <= 1.5))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.8) || ~((b <= 1.5)))
		tmp = r * sin(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.8], N[Not[LessEqual[b, 1.5]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.79999999999999982 or 1.5 < b

   1. Initial program 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/l* 56.8%

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

      3. +-commutative 56.8%

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

   3. Simplified 56.8%

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

      1. add-cube-cbrt 56.0%

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

      2. pow3 56.0%

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

   5. Applied egg-rr 56.0%

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

   6. Taylor expanded in b around 0 6.7%

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

   7. Taylor expanded in a around 0 12.4%

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

   if -4.79999999999999982 < b < 1.5

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/l* 99.0%

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

      3. +-commutative 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in b around 0 97.7%

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

      1. associate-/l* 97.5%

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

   6. Simplified 97.5%

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

      1. associate-/r/ 97.7%

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

   8. Applied egg-rr 97.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \lor \neg \left(b \leq 1.5\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11: 38.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. *-commutative 76.3%

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

   2. associate-/l* 76.2%

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

   3. +-commutative 76.2%

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

3. Simplified 76.2%

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

   1. add-cube-cbrt 75.2%

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

   2. pow3 75.2%

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

5. Applied egg-rr 75.2%

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

6. Taylor expanded in b around 0 22.4%

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

7. Taylor expanded in a around 0 37.3%

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

8. Final simplification 37.3%

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

Alternative 12: 34.8% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. *-commutative 76.3%

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

   2. associate-/l* 76.2%

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

   3. +-commutative 76.2%

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

3. Simplified 76.2%

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

4. Taylor expanded in b around 0 46.8%

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

   1. *-commutative 46.8%

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

   2. associate-/l* 46.7%

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

   3. associate-/r/ 46.8%

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

6. Simplified 46.8%

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

7. Taylor expanded in a around 0 32.6%

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

8. Final simplification 32.6%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023274 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))