rsin A (should all be same)

Percentage Accurate: 77.1% → 99.5%

Time: 12.5s

Alternatives: 14

Speedup: 1.0×

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: 77.1% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. +-commutative 77.1%

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

3. Simplified 77.1%

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

   1. cos-sum 99.5%

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

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%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-*r/ 77.0%

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

   2. *-commutative 77.0%

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

   3. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. cos-sum 99.5%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \lor \neg \left(b \leq 2.1 \cdot 10^{-26}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -7.5) (not (<= b 2.1e-26)))
   (* r (/ (sin b) (cos b)))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.5) || !(b <= 2.1e-26)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = (r * b) / 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 <= (-7.5d0)) .or. (.not. (b <= 2.1d-26))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.5) || !(b <= 2.1e-26)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -7.5) or not (b <= 2.1e-26):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -7.5) || !(b <= 2.1e-26))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -7.5) || ~((b <= 2.1e-26)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -7.5], N[Not[LessEqual[b, 2.1e-26]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.5 or 2.10000000000000008e-26 < b

   1. Initial program 57.3%

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

      1. associate-*r/ 57.3%

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

      2. *-commutative 57.3%

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

      3. +-commutative 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in a around 0 56.9%

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

   if -7.5 < b < 2.10000000000000008e-26

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in b around 0 98.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \lor \neg \left(b \leq 2.1 \cdot 10^{-26}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 4: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -7.5)
   (* r (/ (sin b) (cos b)))
   (if (<= b 2.1e-26) (/ (* r b) (cos (+ b a))) (/ r (/ (cos b) (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -7.5) {
		tmp = r * (sin(b) / cos(b));
	} else if (b <= 2.1e-26) {
		tmp = (r * b) / cos((b + a));
	} else {
		tmp = r / (cos(b) / sin(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 <= (-7.5d0)) then
        tmp = r * (sin(b) / cos(b))
    else if (b <= 2.1d-26) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = r / (cos(b) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -7.5) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else if (b <= 2.1e-26) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = r / (Math.cos(b) / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -7.5:
		tmp = r * (math.sin(b) / math.cos(b))
	elif b <= 2.1e-26:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = r / (math.cos(b) / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -7.5)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (b <= 2.1e-26)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -7.5)
		tmp = r * (sin(b) / cos(b));
	elseif (b <= 2.1e-26)
		tmp = (r * b) / cos((b + a));
	else
		tmp = r / (cos(b) / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -7.5], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-26], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5

   1. Initial program 54.5%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. +-commutative 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in a around 0 54.2%

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

   if -7.5 < b < 2.10000000000000008e-26

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in b around 0 98.5%

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

   if 2.10000000000000008e-26 < b

   1. Initial program 59.7%

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

      1. associate-/l* 59.6%

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

      2. +-commutative 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in a around 0 59.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -7.5)
   (* r (/ (sin b) (cos b)))
   (if (<= b 2.1e-26) (/ (* r b) (cos (+ b a))) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -7.5) {
		tmp = r * (sin(b) / cos(b));
	} else if (b <= 2.1e-26) {
		tmp = (r * b) / 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 <= (-7.5d0)) then
        tmp = r * (sin(b) / cos(b))
    else if (b <= 2.1d-26) then
        tmp = (r * b) / 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 <= -7.5) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else if (b <= 2.1e-26) {
		tmp = (r * b) / 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 <= -7.5:
		tmp = r * (math.sin(b) / math.cos(b))
	elif b <= 2.1e-26:
		tmp = (r * b) / 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 <= -7.5)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (b <= 2.1e-26)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -7.5)
		tmp = r * (sin(b) / cos(b));
	elseif (b <= 2.1e-26)
		tmp = (r * b) / cos((b + a));
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -7.5], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-26], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5

   1. Initial program 54.5%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. +-commutative 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in a around 0 54.2%

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

   if -7.5 < b < 2.10000000000000008e-26

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in b around 0 98.5%

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

   if 2.10000000000000008e-26 < b

   1. Initial program 59.7%

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

      1. associate-*r/ 59.6%

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

      2. *-commutative 59.6%

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

      3. +-commutative 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in a around 0 59.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]

Alternative 6: 77.1% 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 77.1%

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

   1. associate-/l* 77.0%

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

   2. +-commutative 77.0%

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

3. Simplified 77.0%

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

   1. associate-/r/ 77.1%

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

5. Applied egg-rr 77.1%

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

6. Final simplification 77.1%

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

Alternative 7: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

2. Final simplification 77.1%

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

Alternative 8: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-*r/ 77.0%

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

   2. *-commutative 77.0%

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

   3. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in b around 0 53.6%

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

5. Final simplification 53.6%

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

Alternative 9: 54.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -29000000 \lor \neg \left(b \leq 1.9 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -29000000.0) (not (<= b 1.9e+17)))
   (/ r (- (/ 1.0 b) (sin a)))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -29000000.0) || !(b <= 1.9e+17)) {
		tmp = r / ((1.0 / b) - sin(a));
	} 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 <= (-29000000.0d0)) .or. (.not. (b <= 1.9d+17))) then
        tmp = r / ((1.0d0 / b) - sin(a))
    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 <= -29000000.0) || !(b <= 1.9e+17)) {
		tmp = r / ((1.0 / b) - Math.sin(a));
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -29000000.0) or not (b <= 1.9e+17):
		tmp = r / ((1.0 / b) - math.sin(a))
	else:
		tmp = (r * b) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -29000000.0) || !(b <= 1.9e+17))
		tmp = Float64(r / Float64(Float64(1.0 / b) - sin(a)));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -29000000.0) || ~((b <= 1.9e+17)))
		tmp = r / ((1.0 / b) - sin(a));
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -29000000.0], N[Not[LessEqual[b, 1.9e+17]], $MachinePrecision]], N[(r / N[(N[(1.0 / b), $MachinePrecision] - N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.9e7 or 1.9e17 < b

   1. Initial program 56.0%

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

      1. associate-/l* 55.9%

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

      2. +-commutative 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in b around 0 10.4%

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

      1. neg-mul-1 10.4%

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

      2. +-commutative 10.4%

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

      3. unsub-neg 10.4%

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

   6. Simplified 10.4%

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

   7. Taylor expanded in a around 0 10.4%

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

   if -2.9e7 < b < 1.9e17

   1. Initial program 98.5%

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

      1. associate-*r/ 98.5%

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

      2. *-commutative 98.5%

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

      3. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in b around 0 96.5%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -29000000 \lor \neg \left(b \leq 1.9 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 10: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \lor \neg \left(b \leq 6.2 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -9.5) (not (<= b 6.2e+19)))
   (/ r (- (/ 1.0 b) (sin a)))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -9.5) || !(b <= 6.2e+19)) {
		tmp = r / ((1.0 / b) - sin(a));
	} else {
		tmp = (r * b) / 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 <= (-9.5d0)) .or. (.not. (b <= 6.2d+19))) then
        tmp = r / ((1.0d0 / b) - sin(a))
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -9.5) || !(b <= 6.2e+19)) {
		tmp = r / ((1.0 / b) - Math.sin(a));
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -9.5) or not (b <= 6.2e+19):
		tmp = r / ((1.0 / b) - math.sin(a))
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -9.5) || !(b <= 6.2e+19))
		tmp = Float64(r / Float64(Float64(1.0 / b) - sin(a)));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -9.5) || ~((b <= 6.2e+19)))
		tmp = r / ((1.0 / b) - sin(a));
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -9.5], N[Not[LessEqual[b, 6.2e+19]], $MachinePrecision]], N[(r / N[(N[(1.0 / b), $MachinePrecision] - N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.5 or 6.2e19 < b

   1. Initial program 56.8%

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

      1. associate-/l* 56.7%

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

      2. +-commutative 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in b around 0 10.5%

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

      1. neg-mul-1 10.5%

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

      2. +-commutative 10.5%

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

      3. unsub-neg 10.5%

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

   6. Simplified 10.5%

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

   7. Taylor expanded in a around 0 10.5%

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

   if -9.5 < b < 6.2e19

   1. Initial program 97.8%

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

      1. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in b around 0 96.5%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \lor \neg \left(b \leq 6.2 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 11: 54.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -10000000 \lor \neg \left(b \leq 4.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -10000000.0) (not (<= b 4.2e+23)))
   (/ (- r) (sin a))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -10000000.0) || !(b <= 4.2e+23)) {
		tmp = -r / sin(a);
	} 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 <= (-10000000.0d0)) .or. (.not. (b <= 4.2d+23))) then
        tmp = -r / sin(a)
    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 <= -10000000.0) || !(b <= 4.2e+23)) {
		tmp = -r / Math.sin(a);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -10000000.0) or not (b <= 4.2e+23):
		tmp = -r / math.sin(a)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -10000000.0) || !(b <= 4.2e+23))
		tmp = Float64(Float64(-r) / sin(a));
	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 <= -10000000.0) || ~((b <= 4.2e+23)))
		tmp = -r / sin(a);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -10000000.0], N[Not[LessEqual[b, 4.2e+23]], $MachinePrecision]], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e7 or 4.2000000000000003e23 < b

   1. Initial program 56.1%

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

      1. associate-/l* 56.0%

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

      2. +-commutative 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in b around 0 10.5%

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

      1. neg-mul-1 10.5%

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

      2. +-commutative 10.5%

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

      3. unsub-neg 10.5%

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

   6. Simplified 10.5%

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

   7. Taylor expanded in b around inf 10.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{r}{\sin a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 10.2%

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

      2. neg-mul-1 10.2%

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

   9. Simplified 10.2%

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

   if -1e7 < b < 4.2000000000000003e23

   1. Initial program 97.8%

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

      1. associate-*r/ 97.7%

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

      2. *-commutative 97.7%

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

      3. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in b around 0 95.0%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -10000000 \lor \neg \left(b \leq 4.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 12: 54.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -100000000 \lor \neg \left(b \leq 4.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -100000000.0) (not (<= b 4.2e+23)))
   (/ (- r) (sin a))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -100000000.0) || !(b <= 4.2e+23)) {
		tmp = -r / sin(a);
	} 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 <= (-100000000.0d0)) .or. (.not. (b <= 4.2d+23))) then
        tmp = -r / sin(a)
    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 <= -100000000.0) || !(b <= 4.2e+23)) {
		tmp = -r / Math.sin(a);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -100000000.0) or not (b <= 4.2e+23):
		tmp = -r / math.sin(a)
	else:
		tmp = (r * b) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -100000000.0) || !(b <= 4.2e+23))
		tmp = Float64(Float64(-r) / sin(a));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -100000000.0) || ~((b <= 4.2e+23)))
		tmp = -r / sin(a);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -100000000.0], N[Not[LessEqual[b, 4.2e+23]], $MachinePrecision]], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e8 or 4.2000000000000003e23 < b

   1. Initial program 56.1%

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

      1. associate-/l* 56.0%

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

      2. +-commutative 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in b around 0 10.5%

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

      1. neg-mul-1 10.5%

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

      2. +-commutative 10.5%

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

      3. unsub-neg 10.5%

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

   6. Simplified 10.5%

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

   7. Taylor expanded in b around inf 10.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{r}{\sin a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 10.2%

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

      2. neg-mul-1 10.2%

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

   9. Simplified 10.2%

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

   if -1e8 < b < 4.2000000000000003e23

   1. Initial program 97.8%

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

      1. associate-*r/ 97.7%

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

      2. *-commutative 97.7%

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

      3. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in b around 0 95.1%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -100000000 \lor \neg \left(b \leq 4.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 13: 38.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \lor \neg \left(b \leq 4.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.35) (not (<= b 4.2e+23))) (/ (- r) (sin a)) (* r b)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.35) || !(b <= 4.2e+23)) {
		tmp = -r / sin(a);
	} else {
		tmp = r * 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 <= (-1.35d0)) .or. (.not. (b <= 4.2d+23))) then
        tmp = -r / sin(a)
    else
        tmp = r * b
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.35) || !(b <= 4.2e+23)) {
		tmp = -r / Math.sin(a);
	} else {
		tmp = r * b;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.35) or not (b <= 4.2e+23):
		tmp = -r / math.sin(a)
	else:
		tmp = r * b
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.35) || !(b <= 4.2e+23))
		tmp = Float64(Float64(-r) / sin(a));
	else
		tmp = Float64(r * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.35) || ~((b <= 4.2e+23)))
		tmp = -r / sin(a);
	else
		tmp = r * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.35], N[Not[LessEqual[b, 4.2e+23]], $MachinePrecision]], N[((-r) / N[Sin[a], $MachinePrecision]), $MachinePrecision], N[(r * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3500000000000001 or 4.2000000000000003e23 < b

   1. Initial program 56.2%

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

      1. associate-/l* 56.1%

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

      2. +-commutative 56.1%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in b around 0 10.6%

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

      1. neg-mul-1 10.6%

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

      2. +-commutative 10.6%

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

      3. unsub-neg 10.6%

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

   6. Simplified 10.6%

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

   7. Taylor expanded in b around inf 10.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{r}{\sin a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 10.3%

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

      2. neg-mul-1 10.3%

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

   9. Simplified 10.3%

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

   if -1.3500000000000001 < b < 4.2000000000000003e23

   1. Initial program 98.4%

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

      1. associate-*r/ 98.3%

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

      2. *-commutative 98.3%

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

      3. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in b around 0 96.3%

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

   5. Taylor expanded in a around 0 64.5%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \lor \neg \left(b \leq 4.2 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot b\\ \end{array} \]

Alternative 14: 35.6% 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 77.1%

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

   1. associate-*r/ 77.0%

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

   2. *-commutative 77.0%

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

   3. +-commutative 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in b around 0 49.5%

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

5. Taylor expanded in a around 0 33.6%

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

6. Final simplification 33.6%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023228 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))