rsin B (should all be same)

Percentage Accurate: 76.9% → 99.5%

Time: 18.1s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. associate-*r/ 76.7%

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

   2. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 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 76.7%

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

   1. associate-*r/ 76.7%

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

   2. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 3: 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 76.7%

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

   1. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 4: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -200000.0) (not (<= a 4.9e-5)))
   (/ (* r (sin b)) (cos a))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -200000.0) || !(a <= 4.9e-5)) {
		tmp = (r * sin(b)) / cos(a);
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-200000.0d0)) .or. (.not. (a <= 4.9d-5))) then
        tmp = (r * sin(b)) / cos(a)
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -200000.0) || !(a <= 4.9e-5)) {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -200000.0) or not (a <= 4.9e-5):
		tmp = (r * math.sin(b)) / math.cos(a)
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -200000.0) || !(a <= 4.9e-5))
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -200000.0) || ~((a <= 4.9e-5)))
		tmp = (r * sin(b)) / cos(a);
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -200000.0], N[Not[LessEqual[a, 4.9e-5]], $MachinePrecision]], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $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 < -2e5 or 4.9e-5 < a

   1. Initial program 57.6%

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

      1. associate-*r/ 57.7%

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

      2. +-commutative 57.7%

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

   3. Simplified 57.7%

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

   5. Taylor expanded in b around 0 58.4%

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

   if -2e5 < a < 4.9e-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. Add Preprocessing

   5. 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 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -200000.0) (not (<= a 4.9e-5)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -200000.0) || !(a <= 4.9e-5)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-200000.0d0)) .or. (.not. (a <= 4.9d-5))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -200000.0) || !(a <= 4.9e-5)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -200000.0) or not (a <= 4.9e-5):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -200000.0) || !(a <= 4.9e-5))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -200000.0) || ~((a <= 4.9e-5)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -200000.0], N[Not[LessEqual[a, 4.9e-5]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e5 or 4.9e-5 < a

   1. Initial program 57.6%

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

      1. +-commutative 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in b around 0 58.4%

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

   if -2e5 < a < 4.9e-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. Add Preprocessing

   5. 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 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% 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 76.7%

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

   1. associate-*r/ 76.7%

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

   2. +-commutative 76.7%

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

3. Simplified 76.7%

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

Alternative 7: 76.8% 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 76.7%

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

   1. associate-*r/ 76.7%

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

   2. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. *-commutative 76.7%

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

   2. associate-/l* 76.7%

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

6. Applied egg-rr 76.7%

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

Alternative 8: 76.9% 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 76.7%

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

3. Final simplification 76.7%

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

Alternative 9: 54.3% 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 76.7%

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

   1. +-commutative 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in b around 0 55.2%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a}} \]
6. Add Preprocessing

Alternative 10: 54.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -118000000 \lor \neg \left(b \leq 4.5\right):\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -118000000.0) (not (<= b 4.5)))
   (/ 1.0 (/ (/ 1.0 r) (sin b)))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -118000000.0) || !(b <= 4.5)) {
		tmp = 1.0 / ((1.0 / r) / sin(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 <= (-118000000.0d0)) .or. (.not. (b <= 4.5d0))) then
        tmp = 1.0d0 / ((1.0d0 / r) / sin(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 <= -118000000.0) || !(b <= 4.5)) {
		tmp = 1.0 / ((1.0 / r) / Math.sin(b));
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -118000000.0) or not (b <= 4.5):
		tmp = 1.0 / ((1.0 / r) / math.sin(b))
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -118000000.0) || !(b <= 4.5))
		tmp = Float64(1.0 / Float64(Float64(1.0 / r) / sin(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 <= -118000000.0) || ~((b <= 4.5)))
		tmp = 1.0 / ((1.0 / r) / sin(b));
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -118000000.0], N[Not[LessEqual[b, 4.5]], $MachinePrecision]], N[(1.0 / N[(N[(1.0 / r), $MachinePrecision] / N[Sin[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 < -1.18e8 or 4.5 < b

   1. Initial program 54.1%

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

      1. +-commutative 54.1%

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

   3. Simplified 54.1%

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

      1. associate-*r/ 54.1%

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

      2. clear-num 54.0%

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

      3. *-commutative 54.0%

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

   6. Applied egg-rr 54.0%

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

   7. Taylor expanded in a around 0 53.9%

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

      1. associate-/r* 53.9%

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

   9. Simplified 53.9%

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

   10. Taylor expanded in b around 0 11.1%

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

   if -1.18e8 < b < 4.5

   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. Add Preprocessing

   5. Taylor expanded in b around 0 98.1%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -118000000 \lor \neg \left(b \leq 4.5\right):\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -118000000 \lor \neg \left(b \leq 7800\right):\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -118000000.0) (not (<= b 7800.0)))
   (/ 1.0 (/ (/ 1.0 r) (sin b)))
   (* r (/ b (cos (+ b a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -118000000.0) || !(b <= 7800.0)) {
		tmp = 1.0 / ((1.0 / r) / sin(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 <= (-118000000.0d0)) .or. (.not. (b <= 7800.0d0))) then
        tmp = 1.0d0 / ((1.0d0 / r) / sin(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 <= -118000000.0) || !(b <= 7800.0)) {
		tmp = 1.0 / ((1.0 / r) / Math.sin(b));
	} else {
		tmp = r * (b / Math.cos((b + a)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -118000000.0) or not (b <= 7800.0):
		tmp = 1.0 / ((1.0 / r) / math.sin(b))
	else:
		tmp = r * (b / math.cos((b + a)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -118000000.0) || !(b <= 7800.0))
		tmp = Float64(1.0 / Float64(Float64(1.0 / r) / sin(b)));
	else
		tmp = Float64(r * Float64(b / cos(Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -118000000.0) || ~((b <= 7800.0)))
		tmp = 1.0 / ((1.0 / r) / sin(b));
	else
		tmp = r * (b / cos((b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -118000000.0], N[Not[LessEqual[b, 7800.0]], $MachinePrecision]], N[(1.0 / N[(N[(1.0 / r), $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.18e8 or 7800 < b

   1. Initial program 54.1%

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

      1. +-commutative 54.1%

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

   3. Simplified 54.1%

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

      1. associate-*r/ 54.1%

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

      2. clear-num 54.0%

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

      3. *-commutative 54.0%

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

   6. Applied egg-rr 54.0%

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

   7. Taylor expanded in a around 0 53.9%

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

      1. associate-/r* 53.9%

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

   9. Simplified 53.9%

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

   10. Taylor expanded in b around 0 11.1%

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

   if -1.18e8 < b < 7800

   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. Add Preprocessing

   5. Taylor expanded in b around 0 98.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -118000000 \lor \neg \left(b \leq 7800\right):\\ \;\;\;\;\frac{1}{\frac{\frac{1}{r}}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{+34} \lor \neg \left(b \leq 640\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -5.6e+34) (not (<= b 640.0)))
   (/ r (- (sin a)))
   (* r (/ b (cos (+ b a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -5.6e+34) || !(b <= 640.0)) {
		tmp = r / -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 <= (-5.6d+34)) .or. (.not. (b <= 640.0d0))) then
        tmp = r / -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 <= -5.6e+34) || !(b <= 640.0)) {
		tmp = r / -Math.sin(a);
	} else {
		tmp = r * (b / Math.cos((b + a)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -5.6e+34) or not (b <= 640.0):
		tmp = r / -math.sin(a)
	else:
		tmp = r * (b / math.cos((b + a)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -5.6e+34) || !(b <= 640.0))
		tmp = Float64(r / Float64(-sin(a)));
	else
		tmp = Float64(r * Float64(b / cos(Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -5.6e+34) || ~((b <= 640.0)))
		tmp = r / -sin(a);
	else
		tmp = r * (b / cos((b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -5.6e+34], N[Not[LessEqual[b, 640.0]], $MachinePrecision]], N[(r / (-N[Sin[a], $MachinePrecision])), $MachinePrecision], N[(r * N[(b / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.60000000000000016e34 or 640 < b

   1. Initial program 53.3%

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

      1. +-commutative 53.3%

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

   3. Simplified 53.3%

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

      1. associate-*r/ 53.3%

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

      2. clear-num 53.2%

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

      3. *-commutative 53.2%

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

   6. Applied egg-rr 53.2%

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

   7. Taylor expanded in b around 0 8.6%

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

      1. fma-define 8.6%

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

      2. associate-/l* 8.5%

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

   9. Simplified 8.5%

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

   10. Taylor expanded in b around inf 10.5%

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

       1. associate-*r/ 10.5%

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

       2. neg-mul-1 10.5%

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

   12. Simplified 10.5%

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

   if -5.60000000000000016e34 < b < 640

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in b around 0 92.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{+34} \lor \neg \left(b \leq 640\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1450000000000 \lor \neg \left(b \leq 1.1\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1450000000000.0) (not (<= b 1.1)))
   (/ r (- (sin a)))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1450000000000.0) || !(b <= 1.1)) {
		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 <= (-1450000000000.0d0)) .or. (.not. (b <= 1.1d0))) 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 <= -1450000000000.0) || !(b <= 1.1)) {
		tmp = r / -Math.sin(a);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1450000000000.0) or not (b <= 1.1):
		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 <= -1450000000000.0) || !(b <= 1.1))
		tmp = Float64(r / Float64(-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 <= -1450000000000.0) || ~((b <= 1.1)))
		tmp = r / -sin(a);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1450000000000.0], N[Not[LessEqual[b, 1.1]], $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 < -1.45e12 or 1.1000000000000001 < b

   1. Initial program 54.7%

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

      1. +-commutative 54.7%

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

   3. Simplified 54.7%

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

      1. associate-*r/ 54.7%

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

      2. clear-num 54.6%

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

      3. *-commutative 54.6%

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

   6. Applied egg-rr 54.6%

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

   7. Taylor expanded in b around 0 8.3%

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

      1. fma-define 8.3%

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

      2. associate-/l* 8.3%

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

   9. Simplified 8.3%

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

   10. Taylor expanded in b around inf 10.3%

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

       1. associate-*r/ 10.3%

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

       2. neg-mul-1 10.3%

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

   12. Simplified 10.3%

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

   if -1.45e12 < b < 1.1000000000000001

   1. Initial program 98.3%

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

      1. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in b around 0 98.0%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1450000000000 \lor \neg \left(b \leq 1.1\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -600000000000 \lor \neg \left(b \leq 1.3\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -600000000000.0) (not (<= b 1.3)))
   (/ r (- (sin a)))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -600000000000.0) || !(b <= 1.3)) {
		tmp = r / -sin(a);
	} 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 <= (-600000000000.0d0)) .or. (.not. (b <= 1.3d0))) then
        tmp = r / -sin(a)
    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 <= -600000000000.0) || !(b <= 1.3)) {
		tmp = r / -Math.sin(a);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -600000000000.0) or not (b <= 1.3):
		tmp = r / -math.sin(a)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -600000000000.0) || !(b <= 1.3))
		tmp = Float64(r / Float64(-sin(a)));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -600000000000.0) || ~((b <= 1.3)))
		tmp = r / -sin(a);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -600000000000.0], N[Not[LessEqual[b, 1.3]], $MachinePrecision]], N[(r / (-N[Sin[a], $MachinePrecision])), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6e11 or 1.30000000000000004 < b

   1. Initial program 54.7%

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

      1. +-commutative 54.7%

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

   3. Simplified 54.7%

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

      1. associate-*r/ 54.7%

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

      2. clear-num 54.6%

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

      3. *-commutative 54.6%

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

   6. Applied egg-rr 54.6%

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

   7. Taylor expanded in b around 0 8.3%

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

      1. fma-define 8.3%

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

      2. associate-/l* 8.3%

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

   9. Simplified 8.3%

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

   10. Taylor expanded in b around inf 10.3%

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

       1. associate-*r/ 10.3%

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

       2. neg-mul-1 10.3%

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

   12. Simplified 10.3%

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

   if -6e11 < b < 1.30000000000000004

   1. Initial program 98.3%

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

      1. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in b around 0 98.0%

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

      1. clear-num 97.9%

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

      2. un-div-inv 97.8%

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

   7. Applied egg-rr 97.8%

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

      1. associate-/r/ 98.0%

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

   9. Simplified 98.0%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -600000000000 \lor \neg \left(b \leq 1.3\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -600000000000 \lor \neg \left(b \leq 1.5\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -600000000000.0) (not (<= b 1.5)))
   (/ r (- (sin a)))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -600000000000.0) || !(b <= 1.5)) {
		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 <= (-600000000000.0d0)) .or. (.not. (b <= 1.5d0))) 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 <= -600000000000.0) || !(b <= 1.5)) {
		tmp = r / -Math.sin(a);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -600000000000.0) or not (b <= 1.5):
		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 <= -600000000000.0) || !(b <= 1.5))
		tmp = Float64(r / Float64(-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 <= -600000000000.0) || ~((b <= 1.5)))
		tmp = r / -sin(a);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -600000000000.0], N[Not[LessEqual[b, 1.5]], $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 < -6e11 or 1.5 < b

   1. Initial program 54.7%

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

      1. +-commutative 54.7%

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

   3. Simplified 54.7%

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

      1. associate-*r/ 54.7%

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

      2. clear-num 54.6%

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

      3. *-commutative 54.6%

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

   6. Applied egg-rr 54.6%

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

   7. Taylor expanded in b around 0 8.3%

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

      1. fma-define 8.3%

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

      2. associate-/l* 8.3%

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

   9. Simplified 8.3%

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

   10. Taylor expanded in b around inf 10.3%

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

       1. associate-*r/ 10.3%

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

       2. neg-mul-1 10.3%

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

   12. Simplified 10.3%

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

   if -6e11 < b < 1.5

   1. Initial program 98.3%

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

      1. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in b around 0 98.0%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -600000000000 \lor \neg \left(b \leq 1.5\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+14} \lor \neg \left(b \leq 1.4\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -3.8e+14) (not (<= b 1.4))) (/ r (- (sin a))) (* r b)))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -3.8e+14) || !(b <= 1.4)) {
		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 <= (-3.8d+14)) .or. (.not. (b <= 1.4d0))) 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 <= -3.8e+14) || !(b <= 1.4)) {
		tmp = r / -Math.sin(a);
	} else {
		tmp = r * b;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -3.8e+14) or not (b <= 1.4):
		tmp = r / -math.sin(a)
	else:
		tmp = r * b
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -3.8e+14) || !(b <= 1.4))
		tmp = Float64(r / Float64(-sin(a)));
	else
		tmp = Float64(r * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -3.8e+14) || ~((b <= 1.4)))
		tmp = r / -sin(a);
	else
		tmp = r * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -3.8e+14], N[Not[LessEqual[b, 1.4]], $MachinePrecision]], N[(r / (-N[Sin[a], $MachinePrecision])), $MachinePrecision], N[(r * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.8e14 or 1.3999999999999999 < b

   1. Initial program 55.1%

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

      1. +-commutative 55.1%

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

   3. Simplified 55.1%

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

      1. associate-*r/ 55.1%

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

      2. clear-num 55.0%

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

      3. *-commutative 55.0%

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

   6. Applied egg-rr 55.0%

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

   7. Taylor expanded in b around 0 8.3%

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

      1. fma-define 8.3%

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

      2. associate-/l* 8.3%

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

   9. Simplified 8.3%

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

   10. Taylor expanded in b around inf 10.3%

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

       1. associate-*r/ 10.3%

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

       2. neg-mul-1 10.3%

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

   12. Simplified 10.3%

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

   if -3.8e14 < b < 1.3999999999999999

   1. Initial program 97.6%

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

      1. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in b around 0 97.3%

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

   6. Taylor expanded in a around 0 61.7%

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

      1. *-commutative 61.7%

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

   8. Simplified 61.7%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+14} \lor \neg \left(b \leq 1.4\right):\\ \;\;\;\;\frac{r}{-\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 34.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{1}{r} - a \cdot \frac{b}{r}}{b}} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (/ 1.0 (/ (- (/ 1.0 r) (* a (/ b r))) b)))

C

double code(double r, double a, double b) {
	return 1.0 / (((1.0 / r) - (a * (b / 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 = 1.0d0 / (((1.0d0 / r) - (a * (b / r))) / b)
end function

Java

public static double code(double r, double a, double b) {
	return 1.0 / (((1.0 / r) - (a * (b / r))) / b);
}

Python

def code(r, a, b):
	return 1.0 / (((1.0 / r) - (a * (b / r))) / b)

Julia

function code(r, a, b)
	return Float64(1.0 / Float64(Float64(Float64(1.0 / r) - Float64(a * Float64(b / r))) / b))
end

MATLAB

function tmp = code(r, a, b)
	tmp = 1.0 / (((1.0 / r) - (a * (b / r))) / b);
end

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. associate-*r/ 76.7%

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

   2. clear-num 76.2%

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

   3. *-commutative 76.2%

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

6. Applied egg-rr 76.2%

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

7. Taylor expanded in b around 0 53.2%

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

   1. fma-define 53.2%

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

   2. associate-/l* 53.2%

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

9. Simplified 53.2%

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

10. Taylor expanded in a around 0 32.4%

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

    1. +-commutative 32.4%

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

    2. mul-1-neg 32.4%

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

    3. unsub-neg 32.4%

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

    4. associate-/l* 33.0%

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

12. Simplified 33.0%

    \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{r} - a \cdot \frac{b}{r}}}{b}} \]
13. Add Preprocessing

Alternative 18: 34.4% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. +-commutative 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in b around 0 51.0%

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

6. Taylor expanded in a around 0 32.9%

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

   1. *-commutative 32.9%

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

8. Simplified 32.9%

   \[\leadsto \color{blue}{r \cdot b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))